Hybrid Methods of Imaging

Since its invention in the 1960s, computed tomography (CT) has become an indispensable technique of biomedical imaging. Numerous modalities have been introduced since then, including the traditional Xray CT scan, SPECT, MRI, Optical-, Ultrasound-, and Electrical Impedance Tomography, with many others being currently developed. All these techniques are used to obtain images of the internal structure of the living tissue in humans or animals. They differ by the underlying physics, by costs (sometimes very significant) and health hazards to the patient (also significant in some modalities). But most importantly, they differ by the biomedical features they can (or cannot) detect. In spite of the wide variety of existing modalities, such tasks as detection of cancerous tumors in soft tissues still present a significant challenge. Thus, the search continues for new, more capable, more sensitive, and more affordable imaging techniques. Recently, several “hybrid” of “coupled physics” modalities have been introduced. They remain a subject of intensive research activity since then, due to the great promises they hold for medical imaging. By combining two or three different types of waves (or physical fields) these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information — at a lesser cost and with less harm to the patient. Among these methods are the Thermoacoustic Tomography, Photo-Acoustic Tomography, Ultrasound Modulated Optical and Impedance Tomographies, Magneto-Acousto-Electric Tomography (MAET) and several other modalities combining magnetic fields with ultrasound scanning of the tissue. Closely related to these methods are so-called combined physics modalities such as Current Density Imaging and Elastography. As a rule, the images in these modalities are obtained by complex mathematical procedures, rather than through direct acquisition. In most cases, the necessary mathematics involves sophisticated techniques of integral geometry, contemporary theory of partial differential equations, spectral theory, microlocal analysis, numerical analysis, etc. Each time a new modality is introduced, researchers face new mathematical challenges, ranging from the theoretical questions about the existence, uniqueness and stability of the solutions to the equations (representing the images that need to be reconstructed), to the more practical tasks of developing computer algorithms and programs capable of computing the images fast and in high resolution, as required by modern medical practice. These tasks are not trivial; obtaining these results requires collaboration of theoretical and applied mathematicians, as well as an interdisciplinary dialog between the mathematicians and physicists, engineers, and medical practitioners who build and use this state-of-the-art equipment.