Iterative Determination of Kinetic Constants from PET Data Utilizing Total Least Squares Estimates

Rosemary Renaut

Department of Mathematics, Arizona State University, PO Box 87 1804, Tempe, AZ 85287-1804


Abstract

Time series data acquired at the pixel level from positron emission tomography images can be utilized to provide visual images of parameters defining the physiological and biochemical function of differing structures/tissues within the brain. In studies of Alzheimer's disease it is of interest to quantify the data such that statistical studies identifying significant change can be performed.It is thus important that quantification can be made both reliable and computationally efficient. The standard approach for the solution of this problem assumes a single input/single output model, in which parameter estimates are made independently for each pixel. Even at this level, there are still several questions concerning accuracy of the estimation that need to be addressed, and in particular how to address the contamination of the results due to the various measurement errors. The first approach reported in the medical literature uses nonlinear least squares data fitting for the parameters defining the model. This method does not account at all for the error and has been found to be inefficient computationally. More recent works discuss the linear least squares problem which results from the Laplace transform of the underlying differential equation. Although more efficient, this approach also does not account for the measurement error. To overcome this problem a generalized least squares method can be used. The method is further refined by the incorporation of a penalty term which enforces the parameters to physiologically realistic values. This latter approach, while requiring the determination of an appropriate penalty parameter, can be incorporated in both the linear and nonlinear models. The need to estimate the penalty can be avoided by the utilization of a total least squares formualation. In this paper we evaluate the use of the total least squares method in this context as compared to generalized least squares with penalty, and investigate the formulation of the problem as single-input multiple output. The latter model is designed to take advantage of the locality of the data in the plane, hence treating the image as a whole, rather than on the pixel by pixel level. A multilevel iterative algorithm for the solution of this problem is proposed. Its effectiveness for the appropriate account of measurement error and its computational efficiency will also be discussed.