Mathematical Methods for Novel Modalities of Medical Imaging

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KuchmentDMS-0604778 The investigator and his colleague develop mathematicaltechniques, both analytic and numerical, for obtaining images ofbiological tissues. They concentrate on mathematical issues thatarise in recently introduced imaging methods for which there islittle mathematical theory: Thermoacoustic Tomography (TAT),Ultrasound-Modulated Optical Tomography (UOT), andAcousto-Electric Tomography (AET). Computerized tomography has become a major method of medicaldiagnostic imaging and industrial non-destructive testing. Overtime, many modalities have been developed, such as X-raytomography (usual CAT scan), MRI, Emission Tomography, OpticalTomography (OT) and Electrical Impedance Tomography (EIT), toname only a few. All of them have their advantages anddisadvantages in terms of contrast between different biologicaltissues, resolution, cost of devices and procedures, and safetyof the patient. For instance, cheap and safe procedures like EITand OT are plagued by low resolution. Recent years have broughtnovel methods of imaging that intend to overcome these problemsand have a high potential for becoming cheap, safe and effectivediagnostic tools. These are Thermoacoustic Tomography (TAT),Ultrasound-Modulated Optical Tomography (UOT), andAcousto-Electric Tomography (AET). They all combine differenttypes of radiation to overcome the known difficulties of methodsinvolving only one of these modalities (e.g., optical tomography,ultrasound imaging, or electrical impedance tomography), such as,for example, low contrast of ultrasound imaging and highinstability and low resolution of optical and electricalimpedance imaging. Mathematical issues are among the mostcrucial in development of these methods, because images have tobe obtained by sophisticated mathematical procedures rather thanby direct acquisition as in the standard (non-tomographic) X-rayimaging. However, in all these modalities necessary analytic andnumerical tools are essentially absent or at very early stages ofdevelopment. Thus, the investigators develop mathematicalanalytic and numerical techniques for obtaining images ofbiological tissues, with the primary targets being TAT, UOT, andAET. The project aids development and implementation of severalnew, cheap, and safe methods of medical diagnostic imaging to beused in clinics. Some applications for non-destructiveindustrial testing are also possible. Graduate students play asignificant role in the project, and some of the project'stechniques and results are incorporated into graduate-levelclasses. This prepares students for work in excitingcontemporary areas at the junction of exact sciences and medicineand biology.