## John N. Aarsvold, Ph.D.

Nuclear Medicine Service

Atlanta VAMC #115

1670 Clairmont Road

Decatur, GA 30033

Phone: 404-321-6111 ext. 6156

Fax: 404-728-4846

E-mail: jaarsvo@emory.edu

John Aarsvold is an Assistant Professor of Radiology (Nuclear Medicine) at Emory University and the Atlanta Veterans Affairs Medical Center. He has performed medical imaging research in the Mathematics and Biomedical Engineering Departments at The University of Michigan, in The Franklin McLean Memorial Research Institute at The University of Chicago, and in the Radiology Research Laboratory at The University of Arizona. He has studied and/or taught mathematics at St. Olaf College, Colorado State University, Purdue University, St. John's University (MN), The University of Arizona (Ph.D., Applied Mathematics), and The University of Michigan. The primary objective of his research is the development of novel application-specific medical imaging devices, particularly planar gamma-ray imagers and single-photon emission computed tomographs. His work includes analytic modeling and numerical simulation of imaging devices, solution of inverse problems, development of reconstruction algorithms, and the design, construction, and testing of prototype imaging systems. He is presently involved in research to develop novel imagers for brain imaging and breast imaging. Dr. Aarsvold co-edited the recently released book, Emission Tomography: The Fundamentals of PET and SPECT, on the physics and engineering of emission tomography.

**Mathematics and Medical Imaging: An Introduction**

Most medical images are not photographs. That is, most diagnostic images are not maps of reflected light. Most are maps of tissue densities (TCT); some are maps of radiopharmaceutical uptakes in various tissues (ECT); some are maps of acoustic reflectivities of tissues (UL); some are maps of hydrogen concentrations in tissues (MRI). All are created via computer implementations of mathematical algorithms that convert data to "images". All are created from mathematical models of the physics, chemistry, and biology of an imaging process and mathematical models of the tissues and physiological processes being imaged.

This presentation is an "images" excursion into mathematics and medical imaging. It will be a presentation of medical images, images of medical imaging research, and images of mathematical concepts used in medical imaging. Images shown are from efforts to design novel imagers for dynamic imaging of the heart and brain and from efforts to develop novel approaches for the clinical management of electrical trauma victims.

TCT -- Transmission Computed Tomography

ECT -- Emission Computed Tomography

PET -- Positron Emission Tomography

SPECT -- Single-Photon Emission Computed Tomgraphy

UL -- Ultrasound

MRI -- Magnetic Resonance Imaging

**Group Representations, Symmetries, and Tomography**

(

*An introduction to dihedral groups and their representations through a discussion of their relationship to a tomographic imager.*)

The symmetries of single-slice tomographic imaging systems are identical to those of regular polygons. Thus, the symmetry groups related to such systems are dihedral groups and cyclic groups. This presentation is an introduction to the dihedral group of order 8 in the context of the symmetries of a tomograph having square symmetries. The presentation is primarily visual, as images rather than formulas are the primary means for the conveyance of the relevant mathematical concepts.

The presentation includes a discussion of a prototype of a novel emission tomograph, a discussion of properties of a matrix operator that models the system, an introduction to eigenvectors, a visual display of the system's symmetries manifested in the eigenvectors of the matrix operator, and a discussion of relationships between the symmetries of the system and the dihedral group of order 8.

Because much of the presentation is visual, the presentation is accessible to most undergraduates and can be adjusted to fit the audience.

**Singular-Value Decomposition and Single-Photon Emission Computed Tomography**

(

*An introduction to singular-value decomposition and tomography.*)

Matrix operators are often used to model tomographic imaging; thus, analysis of appropriate matrix operators can provide information useful in the design of tomographic systems. This presentation is an introduction to eigenanalysis and singular-value analysis in the context of matrix operators that model novel emission tomographic systems. The systems are multiple-pinhole, modular-camera, single-photon emission computed tomography (MPMC SPECT) systems.

The presentation includes a discussion of MPMC SPECT systems, a discussion of matrices that model such systems, an introduction to eigenanalysis and singular-value analysis, and a visual presentation of several of the properties of the singular vectors of the matrix operators.

Because much of the presentation is visual, the presentation is accessible to most undergraduates and can be adjusted to fit the audience.