A Weighted Least-Squares Technique for the Analysis of Kinetic Data and Its Application to the Study of Renal 133Xenon Washout in Dogs and Man

A computer program was written in PL/1 to successively fit the sum of two, three, and four exponential terms to data by an iterative least-squares technique, using a combination of the steepest-descent and the Newton-Raphson methods for convergence. Each data point was weighted by the reciprocal of its variance, assuming that the errors followed a Poisson distribution. A compartment, i.e., an exponential term, was declared nonsignificant if it did not significantly reduce the least-squares error about the fitted line as judged by an F test. Validity of the data was assessed by a “runs” test and by the frequency with which data points fell outside the 95% confidence range. Results of the analysis showed that (1) 9 of 12 normal human kidney Xe washout curves were best described by a four-compartment model, (2) 18 of 38 studies in patients with essential hypertension yielded a four-compartment curve with significant reduction in compartment-1 flow, (3) nine patients with congestive heart failure all had three-compartment washout curves, (4) two patients with oliguric renal failure had washout curves described best by a two-exponential equation (one of these patients responded to an injection of furosemide with the appearance of a third, more rapid compartment). Obviously, this form of analysis can be easily applied to other sets of data which are described by nonlinear equations.