PET is a procedure in nuclear medicine used to determine the intensity of internal biochemical and metabolic processes in living tissue. This procedure is quite different from traditional X-ray methods, which determine images of the internal anatomical structure by passing high energy beams through the patient. In PET, these high energy rays result from radioactive decay inside the patient. The radioactivity is usually a result of changing the atomic structure of simple substances such as glucose which is readily absorbed by living tissue. Since regions with high metabolic activity absorb a proportionately larger amount of glucose than regions of lower metabolic activity, these highly active regions also result in a larger number of high energy emissions. Data is collected by placing the patient in a scanner which keeps track of the number of these high energy emissions occurring in various regions. However, the scanner cannot tell exactly where these emissions occurred inside the body. This information is obtained by a mathematical algorithm that is an integral part of the PET machine. This algorithm is able to take the "raw data," which looks nothing like the internal biochemical structure, and convert it into a meaningful image that physicians and medical researchers then use to make diagnoses, plan treatments, determine the health of individual organs, and do psychological studies of the brain.
Among other things, the PET procedure has been used to determine
(a) the effects of various drugs on the human brain,
(b) the health of the human heart, by estimating the volume of blood flow
(c) the location and treatment of tumors
(d) the responses of the brain to lesions or surgical resectioning.
PET is being used extensively to determine the response of the brain to various stimuli, and to map which areas of the brain are responsible for functions such as speech, motor skills, cognitive skills, and memory recall. PET provides information that is complementary to other medical imaging modalities such as CAT and MRI, which provide mainly anatomical information. Therefore, a major emphasis in the medical research community is to develop automatic computer guided methods of combining information from the different imaging modalities into a single consistent, aligned image containing all the "pieces of the puzzle" so that the physician can obtain a realistic picture of the brain, or the heart, or any other organ of interest, in a truly active, functional form. And all this, without lifting a scalpel or spilling any blood.
PET is a very effective and versatile imaging method, so there is significant interest in obtaining more accurate numerical algorithms for use in the commercial machines. One of the major obstacles in achieving this is the significant deviation of the raw data from the approximate mathematical model used to derive present reconstruction algorithms. This sometimes results in degraded images that are not able to accurately resolve small but significant details, especially when they are close to regions of high uptake, such as the abdomen.
A major goal of my research is to obtain a precise mathematical formulation and understanding of the underlying model, which has so far been only approximated numerically. I hope that an appropriate formulation will result in a new paradigm from which more accurate, stable, and faster numerical algorithms can be obtained. I also develop algorithms based on the currently adopted numerical model. To perform research in this area, I have had to learn some biology and medicine, and a significant amount of the physics and engineering involved in PET scanners.
I work very closely with two other faculty members, Murali Rao in the Mathematics Department, and John Anderson in Electrical and Computer Engineering and with doctoral students Raymond Carroll (Mathematics) and Chen Hsien Wu (Electrical and Computer Engineering). We also collaborate with Dr. John Votaw at Emory University Hospital on this project. In fact we are preparing to test some of our algorithms on PET data provided to us by Dr. Votaw.
I truly enjoy the interaction between the various disciplines that is a vital component of this project. I try to carry over my enthusiasm for interdisciplinary research to my classes. I believe it is extremely important for our students to understand the motivation and possible applications of the mathematics they are learning. Hopefully, this motivates students to learn the material better and be more enthusiastic about the subject.
In addition to teaching and research, I have been responsible for the graduate program in the Department of Mathematics, since 1996. Over the past five years, the job market for mathematicians with graduate training has been evolving rapidly. No longer can a mathematics doctorate expect to automatically obtain a tenure-track job in a Ph.D. granting research university institution, such as University of Florida. More graduates are seeking (and obtaining) employment in areas such as telecommunications, finance, insurance, research labs, various engineering and medical type industries, and - not surprisingly - the computer industry. It may come as a surprise to many that the National Security Agency is the largest employer of mathematicians in this country (the actual number is classified, but it is a four digit number!). I believe this change is a credit to the discipline as it demonstrates the fundamental reliance of technology and business on mathematics and the acceptance of this view by those in leadership. In order to prepare our students for this changing market, we have modified our graduate program, especially at the Master's level to encourage our students to take courses in other departments such as statistics, physics, finance, electrical and computer engineering, and industrial and systems engineering. We have also introduced new courses in the mathematics departments. These include courses in wavelets, nonlinear optics, mathematics of finance, and medical imaging. As a result, we currently have six graduate students in some form of internship program. Two are with investment companies, three are with high tech industries, and one with the medical school. We intend to continue modifying our program to increase the number of training options available, and improve the quality of our program.