**3. Results**

#### **3.1 Algorithm convergence**

While the final desired result of the iterative image reconstruction algorithm is the associated property images, the best way to assess the quality of the process is by analyzing the computed electric fields as compared with the measurement data [47]. As can be seen from Eq. (2), the minimization statement should converge to a point where measured and computed magnitudes and phases achieve a reasonable match. In essence, the final image can really be considered a bi-product of this process. A key reason why this distinction is important is that there are ample situations where researchers employ a range of a priori information to achieve reasonable results. In a laboratory or experimental setting this might appear reasonable; however, these ad hoc measures often introduce biases which can artificially skew the final results. The best unbiased way to examine the process is to focus on the actual terms being compared.

**Figure 7** shows the normalized (a) magnitude and (b) phase values for the receiver positions associated with a single transmitter for both the measured values (heavy black lines) and computed values (narrower colored lines) at selected iterations [50]. The data was collected at 1300 GHz and both the magnitude and phase values have been normalized to the cases where there is no target in the bath—i.e. that the reconstruction process starts with an assumption of an empty bath which is reasonable since it is possible to exactly know the bath properties in all situations. Note that the measured phases exceed the +/ 180 degree bounds in some cases, implying that our algorithm is robust even for large, high contrast targets which often generate some measurements outside of the primary Riemann sheet. As a useful analogy, we refer to *Theoretical Premises and Contemporary Optimizations of Microwave Tomography DOI: http://dx.doi.org/10.5772/intechopen.103011*

#### **Figure 7.**

*1300 MHz (a) Magnitude and (b) phase projections for the measured data and multiple computed values at representative iterations. Note that the dashed black lines represent the projections at the end of the first step (Permission granted by Wiley Publishers to re-print graphs [50]).*

the curves for each measurement set as a projection, much like the nomenclature used in X-ray CT. It should be noted that in this example, a two-step algorithm was used whereby the first 50 iterations used a standard Levenberg–Marquardt technique to produce a good, albeit smoothed image. After which, a Tikhonov algorithm was used for 20 iterations to more carefully refine the images. The intermediate projection is a black, dashed line for both the magnitude and phase.

For both the magnitude and phase, the projections monotonically approach those for the measurement data. The projections for the intermediate image have not yet matched the measurement data, but their curves retain many of the characteristics of the measurements but with a reduced overall strength and are also somewhat smoothed. Once the second step is employed, the computed magnitudes and phases rapidly approach the measured values. The convergence of the phase values is almost exact, while those for the amplitude are good but not as precise. The discussion in Section 4 provides a possible rationale for this mismatch. Regardless, the match is quite good and gives confidence that the process works well. It should be noted that this is for an actual breast exam where the measurement data inherently has 3D characteristics while the algorithm is a 2D one. In addition, these plots are for only one of the 16 antennas but is reasonably representative of all sets.

#### **3.2 Therapy monitoring**

While microwave imaging is poised to play a substantial role in the detection and diagnosis of breast cancer, it may be that its most important role will be in the area of neoadjuvant chemotherapy monitoring [17]. In this setting, it is important to be able to tell if the tumor is responding adequately to treatment at a sufficiently early time point so that treatment can be altered appropriately. Measures such as a physical exam present morphological information regarding the tumor response—i.e. size—however, these changes are generally quite slow to manifest compared to physiological changes. Early results suggest that the microwave images may be more physiological in nature [17].

**Figure 8a** shows representative sagittal MR scans of the ipsilateral breast for a woman with heterogeneously dense breasts and a 6.5 x 3.7 x 7.1 cm3 tumor in the right breast just before treatment started: (i) T2, (ii) T1 with gadolinium, and (iii) the T1

#### **Figure 8.**

*Sagittal MR images of the right breast of a patient with a complete pathologic response: (a) images prior to therapy and (b) at day 85. For each set, (i) is a T2-weighted image, (ii) is a contrast-enhanced image from a spoiled gradient recalled (SPGR) sequence, and (iii) is a subtraction image between (ii) and a pre-contrast baseline, respectively (Reproduced from [17], with the permission of Springer Nature—License link: https://creativec ommons.org/licenses/by/4.0/).*

subtraction image for the cases with and without gadolinium. (i) Shows a large thickening of the skin surrounding most of the breast, while (ii) and (iii) show the extent of the tumor. **Figure 8b** shows the corresponding images after 85 days of treatment. The skin thickening had diminished substantially and the subtraction image indicates only small, minor enhancements.

**Figure 9** shows the 1300 MHz permittivity and conductivity images for planes 5, 6, and 7, respectively, where plane 1 is closest to the chestwall and plane 7 is closest to the nipple. Set (a) is for the contralateral breast (normal) and sets (b), (c), and (d) are for the ipsilateral breast at different time points during the treatment—prior to treatment, after 44 days of treatment, and just prior to surgery, respectively. In all cases, the outline of the breast is readily visible separating the tissue from the surrounding bath. For the contralateral breast, the majority of the tissue exhibits quite low

#### **Figure 9.**

*1300 MHz microwave tomographic images. Imaging planes five to seven are shown corresponding to the three closest planes to the nipple with the permittivity on the top row and conductivity on the bottom row. (a) Left (contralateral) breast prior to treatment, (b) right (ipsilateral) breast prior to treatment, (c) right breast 44 days into treatment, and (d) right breast immediately prior to surgery (Reproduced from [17], with the permission of Springer Nature—License link: https://creativecommons.org/licenses/by/4.0/).*

permittivity and conductivity which is consistent with adipose tissue, while there are also centralized zones of slightly elevated tissue corresponding to the fibroglandular tissue. For the diseased breast at the first two time points, there is a pronounced ring of highly elevated tissue properties—primarily to the lower left quadrant. This feature is consistent with the edema that would be present from the skin thickening observed in the MR images. For these same time points, the localized, elevated properties of the tumor are also visible, albeit slightly diminished at the 44 day mark. By the final set, both the microwave observable skin thickening and tumor have essentially disappeared in agreement with the MR images shown in **Figure 8b**. It is noteworthy that the tumor has shrunk considerably with respect to size and property intensity by the midway point suggesting that the microwave images are capable of recognizing early tumor response.
