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Nondestructive evaluation (NDE) is to materials and structures as CAT-scanning is to the human body--an attempt to look inside without opening it. As in CAT-scanning, modern NDE requires sophisticated mathematical software to perform the mathematical "inversion" operations that allow one to infer the internal state of a structure from external measurments. In this note we will show how VIC-3D® fulfills this need. In reading this example, keep in mind that the data could be taken with a conventional impedance analyzer, such as the HP4194A. For higher frequencies, up to 180MHz,or so, the HP4915A series can be used. There is nothing exotic about the instrumentation; the exotic stuff is in VIC-3D®.

Heating cancer tumors facilitates their treatment by conventional means. The treatment of tumors by raising their temperatures is called hyperthermia, and can be accomplished by using microwaves if the antennas in the figure are excited with such phases and amplitudes that the resulting field is focused within the body. This allows selected regions of the body to be heated (at least more than the surrounding tissue). This phasing is also the basis of phased-array radars, in which a radar beam is steered electronically rather than mechanically. This is nice when the antenna is a huge billboard mounted on a naval ship.

As part of the planning stage for hyperthermia, the patient is scanned by a CT system, which produces a very precise map of his innards, with a resolution of fractions of a millimeter. Thus, the location of the tumor is precisely known, which then allows a precise computation of the excitation of each of the antennas, so that the tumor is irradiated. Of course, the nearby tissue will also be heated, but it is the duty of the designer of the antenna control system to minimize this heating. This is itself an interesting optimization problem, because it requires the joint solution of an electromagnetic and bio-thermal problems.

It is desired to run the hyperthermia system as a closed loop system, in which the temperature is sensed, and then fed-back to the applicator, sort of like a temperature regulator system that utilizes a thermostat and furnace to keep our houses warm. At the present time, the only way to measure the temperature of the tissue is to insert six to ten thermocouples, that are mounted on the ends of optical fibers, but this is invasive. Magnetic resonance imaging (MRI) is being studied for noninvasive thermometry, but this technique is likely to be slow and expensive. What is needed in the industry is a noninvasive method that is reliable, robust, and speedy, and that's what we're going to provide.

Noninvasive Thermometry by Electromagnetic Inversion.

We are going to apply the nonlinear parameter estimation inversion algorithm, that was described in Thermal Barrier Coating Inversion and Forward and Inverse Problems, to the idealized model, or `phantom,' of the human body, shown in this figure.

This three dimensional phantom was designed by the Center for Devices and Radiological Health (CDRH) Division of the United States Food and Drug Administration (FDA). It comprises a truncated elliptical cylinder, whose length is 57 cm, major axis 32 cm, and minor axis 22 cm, with a fatty layer of 1 cm thickness. The tumor is modeled as a sphere whose radius is 2 cm. The electrical conductivity of the fat is 0.07 S/m, and of the muscle 0.74 S/m.

The parameter estimation algorithm is limited to estimating a few parameters, perhaps ten, which is roughly the number of thermocouples currently used in hyperthermia. In order to apply the algorithm, we assume that we know the electromagnetic parameters of the tissue when the temperature is known, and the job is to solve the inverse problem, i.e., to estimate the temperature from noninvasive measurements of these electromagnetic parameters.

As in Thermal Barrier Coating Inversion and Forward and Inverse Problems we will assume that the system (the phantom) is excited by a single probe coil that induces eddy-currents into the tissue. (We will leave the more realistic assumption of probing by dipole antennas to a later study.) As the probe is scanned past the phantom, an `impedance signature' is obtained. If we perform a scan under normal temperature conditions, we can subtract that impedance signature from the impedance signature obtained under treatment conditions, where the temperature of the tumor is elevated. The result of this `digital subtraction' is a signature of the heated tumor alone, and this is the information that will be presented to the parameter estimator algorithm for inversion. The data returned will be the conductivity of the tumor, which can then be translated into an estimate of the temperature of the tumor.

We should note that digital subtraction of two large signals to get a small signal requires precise instrumentation, that allows six or seven significant digits to be recorded. This is reasonable for modern impedance and network analyzers, and is well within the capabilities of VIC-3D®.

In the numerical experiments performed with these models, we assume that the coil contains 10,000 turns, has an inner radius of 25 mm, an outer radius of 35 mm, and a height of 10 mm. It is excited at 100 kHz, and is scanned from -250 mm to +250 mm across the midpoint of the phantom in 51 steps.

Example 1. Reconstruction of a Two-Layer Tumor.

Imagine that the centered-tumor is nonuniformly heated, with the top-half cooler than the bottom-half, resulting in the upper-hemisphere having a conductivity of 0.67 S/m, and the lower-hemisphere having a conductivity of 0.65 S/m, as in the figure.

When the impedance data are presented to the nonlinear parameter estimation inversion algorithm, we get the results shown in the table. The conductivity (S/m) is shown in the right column.

%Noise in Data (top, bottom)
0 (0.6681, 0.6553)
1 (0.6685, 0.6536)
10 (0.6698, 0.6447)

If the conductivity of the tumor varies from 0.7 S/m to 0.63 S/m over a six degree Centigrade rise in temperature, which is typical, then a change in tumor temperature of one-half degree Centrigrade corresponds to a change of 0.00583 S/m. From this table, therefore, we conclude that even with additive noise of ten percent (which is huge), we are still able to resolve temperatures within one-half degree Centigrade, which is a requirement for hyperthermia.

Reconstruction Based on the Idealized CDRH Phantom.

In this example, and the next, we expedite the calculations by replacing the CDRH phantom with the idealization shown in this figure, comprising plane-parallel layers of fat and muscle.

Example 2. Reconstruction of a Four-Layer Tumor.

  Let the tumor now comprise the four layers shown in the figure. The tumor is non-uniformly heated, with the top warmer than the bottom. Each layer is one centimeter high, so we are going to determine if we can reconstruct tumors with a spatial resolution of one centimeter, while maintaining an accuracy of one-half degree Centigrade. These are the requirements of hyperthermia.

Upon taking data, as before, and inverting it, using the nonlinear parameter estimation algorithm, we get the results shown in the table. The conductivity (S/m) of each layer is shown in the right column.

%Noise in Data (layer1, layer2, layer3, layer4)
0 (0.6511, 0.6595, 0.6707, 0.6799)
1 (0.6545, 0.6602, 0.6711, 0.6697)
10 (0.6546, 0.6596, 0.6706, 0.6695)

Once, again, we note that we are able to accurately reconstruct the temperatures, while meeting the one-centimeter spatial resolution requirement, even when the noise reaches ten percent, which is quite large. Breakdown occurs in the fourth layer, which is the one that is most shielded from the source by the other layers, as shown in the figure. If we had scanned the body from below, as well as above, we would have gotten a better, more stable, reconstruction of the bottom layer of the two-layer and four-layer tumors.

Example 3. Reconstruction of a Complex Tumor.

Consider the complex tumor shown in the figure, in which a spherical portion, at one temperature, lies within a cube at another temperature. The results that are shown in the next table indicate that we can meet the temperature accuracy requirement, even with ten percent noise. The conductivity (S/m) of the sphere, sigma1, and cube, sigma2, are shown in the right column.

%Noise in Data (sigma1, sigma2)
0 (0.6315, 0.6986)
1 (0.6295, 0.7007)
10 (0.6306, 0.6971)