Thermoacoustic imaging methods for breast cancer detection

Breast cancer is the most common cancer type for women. It is common and dangerous also because of the high death rates. To decrease the number of deaths caused by breast cancer, early diagnosis has critical importance. In recent years, there has been an intensive attention on detection of breast cancer. As a result of this attention, some detection methods appeared on literature such as mammography, ultrasound, thermoacoustic, etc. Thermoacoustic imaging is the main topic of this work. Thermoacoustic tomography (TAT) is a noninvasive hybrid imaging technique and has a big potential for breast imaging and cancer detection. TAT combines the properties of electromagnetism and acoustics. It gets high contrast feature from electromagnetism and high spatial resolution from the acoustics. The thermoacoustic effect occurs in a biological tissue after the tissue is illuminated by microwave pulse. At first, tissue expands in volume. After the microwave pulse stops, the tissue tends to go back to its original state. During this expansion and shrinkage process, an acoustic wave is generated. The principle of thermoacoustic imaging is creating the map of tissue using these acoustic fields. Transducers are used to measure the acoustic fields. This procedure works well to detect the cancerous tissues in breast because of the different properties of cancer and fat tissues. Microwave absorption is related to the microwave pulse and the conductivity of the tissue. Cancerous tissues has higher conductivity values than fat tissues. That's why they absorb more energy and has high potential to generate acoustic field afterward. There are a lot of solution methods for thermoacoustic imaging problem such as backprojection, time reversal, MUSIC, conjugate gradient (CG), tomographic solution, etc. The tomographic approach is well known and it uses Fourier transform to find a solution. The main statement of the tomographic solution is that one dimensional Fourier transform of the measured acoustic field is related with the two dimensional Fourier transform of the source (cancerous tissue) that generated the acoustic field. The theory behind the tomographic approach is explained in this work and numerical results are given. Numerical results show that conjugate gradient which is an optimization based method can give better results than the tomographic solution. For this reason, CG method is selected to be analyzed in detail and develop better methods from. At the beginning, the relation between TAT and inverse source problem explained. In the inverse source problem, an unknown source is reconstructed from the measured fields which come from that source. This procedure occurs in thermoacoustic in a similar way. Unknown cancerous tissue acts like a source and generates an acoustic field. From that point, TAT is referred as an inverse source problem and examined from that perspective. Sparsity approach has become popular in recent years and used in many imaging methods. Sparsity means that number of significant (nonzero) components are relatively small compared to signal length. If the search domain is sparse, sparsity methods can work well for an imaging application. Early detection of breast cancer application of TAT can be directly analyzed from sparsity approach. In this work, two reconstruction methods based on sparsity approach for thermoacoustic imaging is presented. The proposed methods combine thresholding with conjugate gradient method to solve the inverse source problem of thermoacoustic tomography. In Conjugate Gradient Hard Thresholding (CG-HT) method cost function is minimized with conjugate gradient algorithm while hard thresholding holds the sparsity. Similarly, in Conjugate Gradient Soft Thresholding (CG-ST) method soft thresholding is used instead of hard thresholding. CG-HT uses hard thresholding and it corresponds to $l_0$ norm solution. CG-ST uses soft thresholding and it solves the $l_1$ norm included convex problem. Numerical results for different scenarios verify that the proposed methods improve the image reconstruction quality of classical CG method. It is also observed from the numerical results that CG-ST is more robust to the noise than CG-HT. If the source (cancerous tissue) is bigger than the sparsity bounds, these two methods can not give good results. At this point, the wavelet transform can be used to provide sparsity criteria. Haar wavelet transform is used in this work to make the search domain sparse enough. In the wavelet domain, Haar coefficients will be sparse so CG-HT and CG-ST can be applied to these coefficients. Once the coefficients are updated, inverse wavelet transform can be used to find the source. Numerical results for big sources showed that sparse reconstruction in wavelet domain is possible for TAT. At the second part of this work, inverse scattering problem is studied. This study of inverse scattering problem is planned to be similar to the inverse source problem. Inverse scattering can be described as determining properties of inaccessible objects from their response to a known electromagnetic excitation. Geometric (location and shape) and physical (dielectric permittivity, conductivity, etc.) properties of immersed objects can be reconstructed with electromagnetic inverse scattering problems. At the beginning of this part of the thesis, diffraction tomography is explained theoretically. Diffraction tomography is based on Born approximation and uses Fourier transform to solve the inverse scattering problem. After Born approximation, one dimensional Fourier transform of scattered field is related with the two dimensional Fourier transform of object function. Next, Modified Gradient Method (MGM) which is an optimization method is explained for inverse scattering application. Numerical results showed that MGM can provide better results than diffraction tomography. From that point, MGM is selected to develop a sparsity based solution method. Two methods Modified Gradient Hard Thresholding (MG-HT) and Modified Gradient Soft Thresholding (MG-ST) are developed. These methods solve cost function of inverse scattering problem with MGM and hold sparsity with a thresholding function. Numerical results showed that both MG-HT and MG-ST have considerable improving on the results of classical MGM. In the case of the big object, these methods can not be applied to inverse scattering problem directly because of the nonsparsity of search domain. For this situation wavelet transform can be used to make the problem to become sparse. After transferring to the wavelet domain, MG-HT and MG-ST can be used to find optimum wavelet coefficients. After that, object function is reconstructed with inverse wavelet transform. Numerical results are given to verify the effect of using wavelet transform. To sum up, sparsity approach is used to improve classical methods of both inverse source and inverse scattering problems. Thermoacoustic imaging for breast cancer is examined as an application of inverse source problem. Two methods are developed for each problem. For nonsparse situations, the problem is transformed to a sparse domain with the aid of wavelet transform.