The results presented in this dissertation concern two different types of tomographic imaging. The first part of the dissertation is devoted to the time reversal method for approximate reconstruction of images in thermoacoustic tomography. A thorough numerical study of the method is presented. Error estimates of the time reversal approximation are provided. In the second part of the dissertation a type of emission tomography, called Compton camera imaging is considered. The mathematical problem arising in Compton camera imaging is the inversion of the cone transform. We present three methods for inversion of this transform in two dimensions. Numerical examples of reconstructions by these methods are also provided. Lastly, we turn to a problem of significance in homeland security, namely the detection of geometrically small, low emission sources in the presence of a large background radiation. We consider the use of Compton type detectors for this purpose and describe an efficient method for detection of such sources. Numerical examples demonstrating this method are also provided.

An exogenous contrast agent has been developed to enhance the contrast achievable in Thermoacoustic Tomography (TAT). TAT utilizes the penetration depth of microwave energy while producing high resolution images through acoustic waves. A sample irradiated by a microwave source expands due to thermoelastic expansion. The acoustic wave created by this expansion is recorded by an ultrasonic transducer. The water content in biological samples poses an obstacle, as it is the primary absorber of microwave radiation. The addition of an exogenous contrast agent improves image quality by more effectively converting microwave energy to heat. The use of iron oxide nanoparticles in MRI applications has been explored but super paramagnetic iron oxide nanoparticles (SPION) have benefits in microwave applications, as well. Through ferromagnetic resonance, SPION samples more effectively convert microwave energy into heat. This transduction to heat creates significantly larger thermoacoustic waves than water, alone. Characterization of the SPION samples is executed through TAT, TEM, XPS, EDS, and a vector network analyzer with a dielectric probe kit. Onedimensional and phantom model imaging with an iron oxide nanoparticle contrast agent provide a two-fold improvement in contrast at current system configurations. Further enhancement is possible through adjustments to the nanoparticles and TAT system.

We consider the inverse source problem arising in thermo-and photoacoustic tomography. It consists in reconstructing the initial pressure from the boundary measurements of the acoustic wave. Our goal is to extend versatile time reversal techniques to the case when the boundary of the domain is perfectly reflecting, effectively turning the domain into a reverberant cavity. Standard time reversal works only if the solution of the direct problem decays in time, which does not happen in the setup we consider. We thus propose a novel time reversal technique with a nonstandard boundary condition. The error induced by this time reversal technique satisfies the wave equation with a dissipative boundary condition and, therefore, decays in time. For larger measurement times, this method yields a close approximation; for smaller times, the first approximation can be iteratively refined, resulting in a convergent Neumann series for the approximation.

In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have emerged. By combining two or three different types of waves these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Thermoacoustic (and photoacoustic) tomography is the most developed multi-physics imaging modality. Thermo- and photo-acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods cannot be used. In chapter 2 we present a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity with a constant speed of sound. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. In chapter 3 we consider the more general problem of an arbitrarily shaped resonant cavity with a non constant speed of sound and present the gradual time reversal method for computing solutions to the inverse source problem. It consists in solving back in time on the interval [0, T] the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.

Thermoacoustic tomography (TAT) is a newly emerging modality in biomedical imaging. It combines the good contrast of electromagnetic and good resolution of ultrasound imaging. The mathematical model of TAT is the observability problem for the wave equation: one observes the data on a hyper-surface and reconstructs the initial perturbation. In this dissertation, we consider several mathematical problems of TAT. The first problem is the inversion formulas. We provide a family of closed form inversion formulas to reconstruct the initial perturbation from the observed data. The second problem is the range description. We present the range description of the spherical mean Radon transform, which is an important transform in TAT. The next problem is the stability analysis for TAT. We prove that the reconstruction of the initial perturbation from observed data is not H¨older stable if some observability condition is violated. The last problem is the speed determination. The question is whether the observed data uniquely determines the ultrasound speed and initial perturbation. We provide some initial results on this issue. They include the unique determination of the unknown constant speed, a weak local uniqueness, a characterization of the non-uniqueness, and a characterization of the kernel of the linearized operator.

This thesis is a collection of the three projects I have worked on at Purdue. The first is a paper on thermoacoustic tomography involving circular integrating detectors that was published in Inverse Problems and Imaging. Results from this paper include demonstrating that the measurement operators involved are Fourier integral operators, as well as proving microlocal uniqueness in certain cases, and also stability. The second paper, submitted to the Journal of Inverse and Ill-Posed Problems, is much more of an application of sampling theory in to the specific case of thermoacoustic tomography. Results from this paper include demonstrating resolution limits imposed by sampling rates, and showing that aliasing artifacts appear in predictable locations in an image when the measurement operator is under sampled in either the time variable or space variables. We also show an application of a basic anti aliasing scheme based on averaging of data. The last project moves slightly away from microlocal analysis and considers the uniqueness in medical imaging of the restricted Radon transform in even dimensions. This is the classical interior problem, and we show a characterization of the range of the Radon transform, and from this are able to obtain a characterization of the kernel of the restricted Radon transform. We include figures throughout to illustrate results.

Photoacoustic tomography (PAT), as well as thermoacoustic tomography (TAT), utilize electromagnetic radiation in its visible, near infrared, microwave, and radiofrequency forms, respectively, to induce acoustic waves in biological tissues for imaging purposes. Combining the advantages of both the high image contrast that results from electromagnetic absorption and the high resolution of ultrasound imaging, these new imaging modalities could be the next successful imaging techniques in biomedical applications. Basic research on PAT and TAT, and the relevant physics, is presented in Chapter I. In Chapter II, we investigate the imaging mechanisms of TAT in terms of signal generation, propagation and detection. We present a theoretical analysis as well as simulations of such imaging characteristics as contrast and resolution, accompanied by experimental results from phantom and tissue samples. In Chapter III, we discuss the further application of TAT to the imaging of biological tissues. The microwave absorption difference in normal and cancerous breast tissues, as well as its influence on thermoacoustic wave generation and the resulting transducer response, is investigated over a wide range of electromagnetic frequencies and depths of tumor locations. In Chapter IV, we describe the mechanism of PAT and the algorithm used for image reconstruction. Because of the broad bandwidth of the laser-induced ultrasonic waves and the limited bandwidth of the single transducer, multiple ultrasonic transducers, each with a different central frequency, are employed for simultaneous detection. Chapter V further demonstrates PAT’s ability to image vascular structures in biological tissue based on blood’s strong light absorption capability. The photoacoustic images of rat brain tumors in this study clearly reveal the angiogenesis that is associated with tumors. In Chapter VI, we report on further developing PAT to image deeply embedded optical heterogeneity in biological tissues. The improved imaging ability is attributed to better penetration by NIR light, the use of the optical contrast agent ICG (indocyanine green) and a new detection scheme of a circular scanning configuration. Deep penetrating PAT, which is based on a tissue’s intrinsic contrast using laser light of 532 nm green light and 1.06 µm near infrared light, is also presented.