# Novel Stopping Criteria for Optimization-Based Microwave Breast Imaging Algorithms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. DGM-CSI Algorithm

#### 2.2. Frequency-Cycling Tissue-Dependent Mapping Technique

#### 2.3. Stopping Criteria for Single-Frequency Reconstructions

#### 2.4. Global Termination of Multi-Frequency Reconstructions

#### 2.5. Full Description of Multi-Frequency Imaging Procedure

- The real and imaginary parts of the complex permittivity are reconstructed using the lowest frequency data available (e.g., 1.0 GHz). The termination point of this reconstruction is dictated by the results of successive two-sample K-S tests performed on both the real and imaginary parts of the data error separately, comparing the current iteration to those of a sliding window of past iterations, governed by a choice of parameters for p-value, window size, and percentage of windowed iterations reaching this p-value threshold. For robustness, a back-up termination condition may be implemented, either related to the relative change in domain error for CSI-based algorithms as described in Section 2.4, or a maximum number of iterations.
- A point-by-point search through the reconstructed real part of each nodal basis coefficient in the DGM-CSI mesh (or more generally, each mesh element or pixel of the reconstructed image) classifies the type of breast tissue. This classification is based solely on the range of expected values of dielectric constant at that frequency, as outlined in [25].
- An initial guess for the next imaging frequency (e.g., 2.0 GHz) is generated using the tissue-dependent mapping process [24,25]. It consists of the unmodified real parts of the reconstructed ${\epsilon}_{r}$ at the mesh nodal points, and a new imaginary part created from a simple linear interpolation of the expected range of dielectric loss values, based on the appropriate Cole-Cole models of tissues classified in Step 2. This technique preserves the geometry of the real and imaginary parts of the solution.
- This new initial guess for the complex permittivity is used to run the inversion algorithm at the next frequency (e.g., 2.0 GHz). As per the procedure outlined in [25], the user may choose to keep the imaginary part constant during this inversion and update only the real part to converge to a new solution. This “anchoring” process has been shown to improve overall imaging results due to the tendency of CSI-based inversion algorithms to cause significant deterioration of the imaginary part at high-frequency reconstructions. Again, the aforementioned parameterized stopping criteria would be primarily employed to determine the appropriate point to halt this reconstruction.
- If more than two frequencies are used in the frequency hop, steps 2–4 are repeated as necessary until the reconstruction of the final frequency of the succession is complete (e.g., 3.0 GHz). This succession may include “frequency cycling”; that is, returning the inversion algorithm to the lowest frequency data and incrementally stepping through each frequency again [25].

- When each available dataset in the frequency cycle has been used at least once to contribute to the overall image reconstruction, a global termination criterion will become active, which will monitor the relative change in the domain error between successive iterations (Section 2.4). If this relative change falls below 0.1% at any point, the current reconstruction is halted and the frequency cycle is broken.
- Regardless of the frequency at which the algorithm was halted by this relative domain error threshold, if the imaginary part of the solution has been continuously held constant during the frequency cycle after Step 1, one last initial guess is generated as in Step 3 and a final reconstruction is run at the lowest frequency available (e.g., 1.0 GHz) with both the real and imaginary parts allowed to converge to a solution (i.e., the imaginary part is no longer “anchored”). This final inversion is terminated by the parameterized stopping criteria or a relative change of domain error between successive iterations falling below 0.1%, whichever occurs first. The purpose of this final run is to demonstrate the stability of the final solution and ensure that its imaginary part, despite being originally based on the geometry and tissue properties of the real part, does indeed satisfy full CSI optimization.

#### 2.6. Synthetic Breast Models

#### 2.7. Error Calculation

## 3. Results and Discussion

#### 3.1. Imaging with Open Boundaries

#### 3.2. Imaging with PEC Boundaries

#### 3.3. Effect of Noise Levels

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Trentham-Dietz, A.; Kerlikowske, K.; Stout, N.K.; Miglioretti, D.L.; Schechter, C.B.; Ergun, M.A.; van den Broek, J.J.; Alagoz, O.; Sprague, B.L.; van Ravesteyn, N.T.; et al. Tailoring breast cancer screening intervals by breast density and risk for women aged 50 years or older: Collaborative modeling of screening outcomes. Ann. Intern. Med.
**2016**, 165, 700–712. [Google Scholar] [CrossRef] [PubMed] - D’Orsi, C.J.; Mendelson, E.B.; Morris, E.A. (Eds.) ACR BI-RADS Atlas: Breast Imaging Reporting and Data System, 5th ed.; American College of Radiology: Reston, VA, USA, 2012. [Google Scholar]
- Boerner, W.M.; Brand, H.; Cram, L.A.; Gjessing, D.T.; Jordan, A.K.; Keydel, W.; Schwierz, G.; Vogel, M. (Eds.) Inverse Methods in Electromagnetic Imaging; D. Reidel Publishing: Dordrecht, The Netherlands, 1985. [Google Scholar]
- Larsen, L.E.; Jacobi, J.H. (Eds.) Medical Applications of Microwave Imaging; IEEE Press: New York, NY, USA, 1986. [Google Scholar]
- Kosmas, P.; Crocco, L. Introduction to Special Issue on “Electromagnetic Technologies for Medical Diagnostics: Fundamental Issues, Clinical Applications and Perspectives”. Diagnostics
**2019**, 9, 19. [Google Scholar] [CrossRef] - Sill, J.M.; Fear, E.C. Tissue sensing adaptive radar for breast cancer detection—Experimental investigation of simple tumor models. IEEE Trans. Microw. Theory Tech.
**2005**, 53, 3312–3319. [Google Scholar] [CrossRef] - Delbary, F.; Brignone, M.; Bozza, G.; Aramini, R.; Piana, M. A visualization method for breast cancer detection using microwaves. SIAM J. Appl. Math.
**2010**, 70, 2509–2533. [Google Scholar] [CrossRef] - Cakoni, F.; Colton, D.; Monk, P. Qualitative Methods in Inverse Electromagnetic Scattering Theory: Inverse Scattering for Anisotropic Media. IEEE Antennas Propag. Mag.
**2017**, 59, 24–33. [Google Scholar] [CrossRef] - Colton, D.; Kress, R. Inverse Acoustic and Electromagnetic Scattering Theory; Springer Science & Business Media: Berlin, Germany, 2012; Volume 93. [Google Scholar]
- Engl, H.W.; Hanke, M.; Neubauer, A. Regularization of Inverse Problems; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996. [Google Scholar]
- Van den Berg, P.M.; Kleinman, R.E. A contrast source inversion method. Inverse Probl.
**1997**, 13, 1607–1620. [Google Scholar] [CrossRef] - Abubakar, A.; Van den Berg, P.M.; Mallorqui, J.J. Imaging of biomedical data using a multiplicative regularized contrast source inversion method. IEEE Trans. Microw. Theory Tech.
**2002**, 50, 1761–1771. [Google Scholar] [CrossRef][Green Version] - Zakaria, A.; Gilmore, C.; LoVetri, J. Finite-element contrast source inversion method for microwave imaging. Inverse Probl.
**2010**, 26, 115010. [Google Scholar] [CrossRef] - Jeffrey, I.; Geddert, N.; Brown, K.; LoVetri, J. The Time-Harmonic Discontinuous Galerkin Method as a Robust Forward Solver for Microwave Imaging Applications. Prog. Electromagn. Res.
**2015**, 154, 1–21. [Google Scholar] [CrossRef] - Brown, K.G.; Geddert, N.; Asefi, M.; LoVetri, J.; Jeffrey, I. Hybridizable Discontinuous Galerkin Method Contrast Source Inversion of 2-D and 3-D Dielectric and Magnetic Targets. IEEE Trans. Microw. Theory Tech.
**2019**, 67, 1766–1777. [Google Scholar] [CrossRef] - Baran, A.; Kurrant, D.; Zakaria, A.; Fear, E.; LoVetri, J. Breast Imaging Using Microwave Tomography with Radar-Based Tissue-Regions Estimation. Prog. Electromagn. Res.
**2014**, 149, 161–171. [Google Scholar] [CrossRef] - Kurrant, D.J.; Baran, A.; Fear, E.C.; LoVetri, J. Integrating prior information into microwave tomography Part 1: Impact of detail on image quality. Med. Phys.
**2017**, 44, 6461–6481. [Google Scholar] [CrossRef] - Kurrant, D.; Baran, A.; LoVetri, J.; Fear, E. Integrating prior information into microwave tomography part 2: Impact of errors in prior information on microwave tomography image quality. Med. Phys.
**2017**, 44, 6482–6503. [Google Scholar] [CrossRef] - Abdollahi, N.; Kurrant, D.; Mojabi, P.; Omer, M.; Fear, E.; LoVetri, J. Incorporation of Ultrasonic Prior Information for Improving Quantitative Microwave Imaging of Breast. IEEE J. Multiscale Multiphys. Comput. Tech.
**2019**, 4, 98–110. [Google Scholar] [CrossRef] - Nemez, K.; Baran, A.; Asefi, M.; LoVetri, J. Modeling Error and Calibration Techniques for a Faceted Metallic Chamber for Magnetic Field Microwave Imaging. IEEE Trans. Microw. Theory Tech.
**2007**, 65, 4347–4356. [Google Scholar] [CrossRef] - Asefi, M.; Baran, A.; LoVetri, J. An Experimental Phantom Study for Air-Based Quasi-Resonant Microwave Breast Imaging. IEEE Trans. Microw. Theory Tech.
**2019**. [Google Scholar] [CrossRef] - Shea, J.D.; Kosmas, P.; Van Veen, B.D.; Hagness, S.C. Contrast-enhanced microwave imaging of breast tumors: A computational study using 3D realistic numerical phantoms. Inverse Probl.
**2010**, 26, 074009. [Google Scholar] [CrossRef] - Chew, W.; Lin, J. A frequency-hopping approach for microwave imaging of large inhomogeneous bodies. IEEE Microw. Guided Wave Lett.
**1995**, 5, 439–441. [Google Scholar] [CrossRef] - Kaye, C.; Jeffrey, I.; LoVetri, J. Enhancement of multi-frequency microwave breast images using a tissue-dependent mapping technique with discontinuous Galerkin contrast source inversion. In Proceedings of the AES 2016—4th Advanced Electromagnetics Symposium, Malaga, Spain, 26–28 July 2016; pp. 37–39. [Google Scholar]
- Kaye, C.; Jeffrey, I.; LoVetri, J. Improvement of Multi-Frequency Microwave Breast Imaging through Frequency Cycling and Tissue-Dependent Mapping. IEEE Trans. Antennas Propag.
**2019**. submitted. [Google Scholar] - Abubakar, A.; van den Berg, P.M.; Habashy, T.M. Application of the multiplicative regularized contrast source inversion method on TM- and TE-polarized experimental Fresnel data. Inverse Probl.
**2005**, 21, S5–S13. [Google Scholar] [CrossRef][Green Version] - Van den Berg, P.M.; van Broekhoven, A.L.; Abubakar, A. Extended contrast source inversion. Inverse Probl.
**1999**, 15, 1325–1344. [Google Scholar] [CrossRef] - Bloemenkamp, R.F.; Abubakar, A.; van den Berg, P.M. Inversion of experimental multi-frequency data using the contrast source inversion method. Inverse Probl.
**2001**, 17, 1611–1622. [Google Scholar] [CrossRef] - Jeffrey, I.; Zakaria, A.; LoVetri, J. Microwave Imaging by Mixed-Order Discontinuous Galerkin Contrast Source Inversion. In Proceedings of the 2014 XXXIst URSI General Assembly and Scientific Symposium (URSI GASS 2014), Beijing, China, 16–23 August 2014; pp. 255–258. [Google Scholar]
- Lazebnik, M.; Popovic, D.; McCartney, L.; Watkins, C.B.; Lindstrom, M.J.; Harter, J.; Sewall, S.; Ogilvie, T.; Magliocc, M.A.; Breslin, T.M.; et al. A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries. Phys. Med. Biol.
**2007**, 52, 6093–6115. [Google Scholar] [CrossRef] [PubMed] - Kaye, C. Development and Calibration of Microwave Tomography Imaging Systems for Biomedical Applications Using Computational Electromagnetics. Master’s Thesis, University of Manitoba, Winnipeg, MB, Canada, 2009. [Google Scholar]
- Massey, F.J. The Kolmogorov-Smirnov Test for Goodness of Fit. J. Am. Stat. Assoc.
**1951**, 46, 68–78. [Google Scholar] [CrossRef] - Miller, L.H. Table of Percentage Points of Kolmogorov Statistics. J. Am. Stat. Assoc.
**1956**, 51, 111–121. [Google Scholar] [CrossRef] - Marsaglia, G.; Tsang, W.; Wang, J. Evaluating Kolmogorov’s Distribution. J. Stat. Softw.
**2003**, 8, 1–4. [Google Scholar] [CrossRef] - MathWorks. Two-Sample Kolmogorov-Smirnov Test (R2018a). Available online: www.mathworks.com/help/stats/kstest2.html (accessed on 16 March 2018).
- Kurrant, D.J.; Fear, E.C. Regional estimation of the dielectric properties of inhomogeneous objects using near-field reflection data. Inverse Probl.
**2012**, 28, 075001. [Google Scholar] [CrossRef] - Nemez, K.; Asefi, M.; Baran, A.; LoVetri, J. A faceted magnetic field probe resonant chamber for 3D breast MWI: A synthetic study. In Proceedings of the 2016 17th International Symposium on Antenna Technology and Applied Electromagnetics (ANTEM), Montreal, QC, Canada, 10–13 July 2016; pp. 322–324. [Google Scholar]

**Figure 1.**Two-dimensional representation of the imaging problem. The object of interest (OI) has an unknown contrast $\chi $, where $\mathcal{D}$ is the imaging domain and $\mathcal{S}$ is the surface containing the transmitters (Tx) and receivers (Rx).

**Figure 2.**Examples of histograms demonstrating estimates of the probability distribution of the real and imaginary parts of the data error (${\rho}_{\mathrm{E}}$) during a DGM-CSI inversion of an arbitrary 2D synthetic breast model early in the reconstruction process (iteration 20).

**Figure 3.**Examples of histograms demonstrating estimates of the probability distribution of the real and imaginary parts of the data error (${\rho}_{\mathrm{E}}$) during the same DGM-CSI inversion as Figure 2, later in the reconstruction process (iteration 190).

**Figure 4.**Complex dielectric properties of University of Calgary 2D synthetic breast model at 1.0 GHz (

**top**), 2.0 GHz (

**middle**) and 3.0 GHz (

**bottom**).

**Figure 5.**Complex dielectric properties of University of Wisconsin 2D synthetic breast model at 1.0 GHz (

**top**), 2.0 GHz (

**middle**) and 3.0 GHz (

**bottom**).

**Figure 6.**Relative error norms of open-boundary 2D DGM-CSI reconstructions across several choices of stopping criteria parameter values. Each curve in the figure corresponds to a frequency-cycled inversion for a particular choice of the three parameters (Section 3.1), with the window sizes coded by color for convenience. Data points on each curve correspond to REN values for the reconstructed model at each frequency change. Arrows point to the final REN values of simulations that terminated with the lowest relative error norms, indicating the best combination of parameters among those tested for this imaging scenario.

**Figure 7.**Relative error norms of frequency-hopping and frequency-cycled reconstructions: Scenario A (solid line)—without tissue-dependent mapping at 400 iterations per frequency terminating after first inversion of 3.0 GHz, Scenario B (dashed line)—with tissue-dependent mapping at 200 iterations per frequency, cycling through reconstruction frequencies once (with imaginary component “anchored” following initial 1.0 GHz inversion), Scenario C (dotted line)—with tissue-dependent mapping and stopping criteria in place, terminating after two consecutive inversions of 1.0 GHz data (one with the imaginary component “anchored” and the final run with the imaginary component freely optimized according to the guidelines of Section 2.5). See Table 1 for further details.

**Figure 8.**Final results of DGM-CSI frequency-hopping and frequency-cycled complex dielectric property reconstruction of synthetic breast model using 1.0–3.0 GHz data, without modification of intermediate initial guesses (Scenario A—

**top**), using tissue-dependent mapping at fixed 200 iterations per frequency (Scenario B—

**middle**), and employing stopping criteria and tissue-dependent mapping (Scenario C—

**bottom**).

**Figure 9.**Final results of DGM-CSI frequency-hopping and frequency-cycled complex dielectric property reconstruction of PEC-bounded synthetic breast model using 1.0–1.5 GHz data, without modification of intermediate initial guesses (Scenario D—

**top**), using tissue-dependent mapping at fixed 200 iterations per frequency (Scenario E—

**middle**), and employing stopping criteria and tissue-dependent mapping (Scenario F—

**bottom**).

**Figure 10.**Relative error norms of reconstructions with PEC boundaries: Scenario D (solid line)—without tissue-dependent mapping at 400 iterations per frequency terminating after first inversion of 1.5 GHz data, Scenario E (dashed line)—with tissue-dependent mapping at 200 iterations per frequency, cycling through reconstruction frequencies once (with imaginary component “anchored” following initial 1.0 GHz inversion), Scenario F (dotted line)—with tissue-dependent mapping and stopping criteria in place, terminating after two consecutive inversions of 1.0 GHz data (one with the imaginary component “anchored” and the final run with the imaginary component freely optimized according to the guidelines of Section 2.5). See Table 2 for further details.

**Figure 11.**Final results of DGM-CSI frequency-hopping and frequency-cycled complex dielectric property reconstruction of open-boundary Category D synthetic breast model using 1.0–3.0 GHz data: (

**a**) Scenario G—without tissue-dependent mapping at 400 iterations per frequency terminating after first inversion of 3.0 GHz (5% noise), (

**b**) Scenario H—with tissue-dependent mapping and stopping criteria in place at 3% noise, (

**c**) Scenario I—with tissue-dependent mapping and stopping criteria in place at 5% noise, (

**d**) Scenario J—with tissue-dependent mapping and stopping criteria in place at 7.5% noise, (

**e**) Scenario K—with tissue-dependent mapping and stopping criteria in place at 10% noise. See Table 3 for further details.

**Figure 12.**Relative error norms of frequency-hopping and frequency-cycled reconstructions of Category D model: Scenario G (blue solid line)—without tissue-dependent mapping at 400 iterations per frequency terminating after first inversion of 3.0 GHz (5% noise), Scenario H (red dashed line)—with tissue-dependent mapping and stopping criteria in place at 3% noise, Scenario I (black dotted line)—with tissue-dependent mapping and stopping criteria in place at 5% noise, Scenario J (magenta solid line)—with tissue-dependent mapping and stopping criteria in place at 7.5% noise, Scenario K (green dashed line)—with tissue-dependent mapping and stopping criteria in place at 10% noise. See Table 3 for further details.

**Table 1.**Reconstruction progression of open boundary scenarios (Figure 7).

Frequency: | |||||||||
---|---|---|---|---|---|---|---|---|---|

Scenario | TM | SC | No. of Iterations (Stopping Condition) | Total | |||||

Components Reconstructed | |||||||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | |||||||

A | No | No | 400 (F) | 400 (F) | 400 (F) | 1200 | |||

Re, Im | Re, Im | Re, Im | |||||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | 1.0 GHz: | 2.0 GHz: | 3.0 GHz: | ||||

B | Yes | No | 200 (F) | 200 (F) | 200 (F) | 200 (F) | 200 (F) | 200 (F) | 1200 |

Re, Im | Re | Re | Re | Re | Re | ||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | 1.0 GHz: | 2.0 GHz: | 1.0 GHz: | ||||

C | Yes | Yes * | 140 (KS) | 190 (KS) | 255 (KS) | 53 (KS) | 89 (DE) | 41 (KS) | 768 |

Re, Im | Re | Re | Re | Re | Re, Im |

**Table 2.**Reconstruction progression of PEC-bounded scenarios (Figure 10).

Frequency: | |||||||||
---|---|---|---|---|---|---|---|---|---|

Scenario | TM | SC | No. of Iterations (Stopping Condition) | Total | |||||

Components Reconstructed | |||||||||

1.0 GHz: | 1.25 GHz: | 1.5 GHz: | |||||||

D | No | No | 400 (F) | 400 (F) | 400 (F) | 1200 | |||

Re, Im | Re, Im | Re, Im | |||||||

1.0 GHz: | 1.25 GHz: | 1.5 GHz: | 1.0 GHz: | 1.25 GHz: | 1.5 GHz: | ||||

E | Yes | No | 200 (F) | 200 (F) | 200 (F) | 200 (F) | 200 (F) | 200 (F) | 1200 |

Re, Im | Re | Re | Re | Re | Re | ||||

1.0 GHz: | 1.25 GHz: | 1.5 GHz: | 1.0 GHz: | 1.0 GHz: | |||||

F | Yes | Yes * | 123 (KS) | 124 (KS) | 97 (KS) | 14 (DE) | 48 (DE) | 406 | |

Re, Im | Re | Re | Re | Re, Im |

**Table 3.**Reconstruction progression of Category D model noise-variant scenarios (Figure 12).

Frequency: | |||||||||
---|---|---|---|---|---|---|---|---|---|

Scenario (Noise %) | TM | SC | No. of Iterations (Stopping Condition) | Total | |||||

Components Reconstructed | |||||||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | |||||||

G (5%) | No | No | 400 (F) | 400 (F) | 400 (F) | 1200 | |||

Re, Im | Re, Im | Re, Im | |||||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | 1.0 GHz: | 2.0 GHz: | 1.0 GHz: | ||||

H (3%) | Yes | Yes * | 155 (KS) | 140 (KS) | 265 (KS) | 50 (KS) | 52 (DE) | 43 (DE) | 759 |

Re, Im | Re | Re | Re | Re | Re, Im | ||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | 1.0 GHz: | 1.0 GHz: | |||||

I (5%) | Yes | Yes * | 139 (KS) | 140 (KS) | 265 (KS) | 34 (DE) | 8 (DE) | 586 | |

Re, Im | Re | Re | Re | Re, Im | |||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | 1.0 GHz: | 1.0 GHz: | |||||

J (7.5%) | Yes | Yes * | 91 (KS) | 112 (KS) | 234 (KS) | 37 (DE) | 13 (DE) | 487 | |

Re, Im | Re | Re | Re | Re, Im | |||||

1.0 GHz: | 2.0 GHz: | 3.0 GHz: | 1.0 GHz: | 1.0 GHz: | |||||

K (10%) | Yes | Yes * | 87 (KS) | 90 (KS) | 228 (KS) | 11 (DE) | 32 (DE) | 448 | |

Re, Im | Re | Re | Re | Re, Im |

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**MDPI and ACS Style**

Kaye, C.; Jeffrey, I.; LoVetri, J. Novel Stopping Criteria for Optimization-Based Microwave Breast Imaging Algorithms. *J. Imaging* **2019**, *5*, 55.
https://doi.org/10.3390/jimaging5050055

**AMA Style**

Kaye C, Jeffrey I, LoVetri J. Novel Stopping Criteria for Optimization-Based Microwave Breast Imaging Algorithms. *Journal of Imaging*. 2019; 5(5):55.
https://doi.org/10.3390/jimaging5050055

**Chicago/Turabian Style**

Kaye, Cameron, Ian Jeffrey, and Joe LoVetri. 2019. "Novel Stopping Criteria for Optimization-Based Microwave Breast Imaging Algorithms" *Journal of Imaging* 5, no. 5: 55.
https://doi.org/10.3390/jimaging5050055