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

Abstract

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-Baphson 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 l33 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) t w o 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). Obvio u sly, this form of analysis can be easily applied to other sets of data which are described by nonlinear equations.KEY WOBDS inert gas washout curves essential hypertension model fitting nonlinear regression analysis intr»renal distribution of blood flow • Mathematical expressions describing quantitative physiological data have a demonstrated value in modern physiological research. Such expressions are usually based on either an empirically derived equation (s) which summarizes the phenomenon under study or a theoretically derived mathematical model. In either case, the mathematical expression contains parameters with unknown values, and the problem is to estimate values for these parameteis from experimental data. Frequently, well-known techniques of linear regression or multiple regression can be used for parameter estimation providing the equation is linear or can be linearized. In many cases, however, nonlinear equations cannot be linearized, e.g., an equation containing the sum of exponentials. In these instances nonlinear regression techniques for parameter estimation are necessary (1). Graphic techniques have been extensively used in the past for nonlinear parameter estimation (2), but this technique has several limitations: (a) it is subjective, (b) it is unable to separate parameters of similar magnitude with a high degree of c o nfidence, (c) it cannot provide confidence limits for the parameter estimates, and (d) it assumes that all of the data points have an equal variance.Because of these limitations we developed a digital computer technique for nonlinear regression analysis by which parameters were estimated by a east-squares procedure with each data point being weighted if the variance was given. Confidence limits for the paramete...

Parameters of the three-pool model of the turnover of plasma cholesterol in normal and hyperlipidemic humans.

“…using the method N-24a for cholesterol (13), and a modification (14) of the Kessler and Lederer technique (15) for triglycerides. The data for each subject were analyzed by digital computer, using a weighted, least-squares, nonlinear regression technique as described elsewhere (6,16), to determine the parameters of a three-pool mammillary model which would provide the best fit. The model used is illustrated in Fig.…”

Lack of effect of lovastatin therapy on the parameters of whole-body cholesterol metabolism.

“…In each study, the specific radioactivity data were analyzed by a weighted, least squares technique (15) to determine the parameters of a three-pool mammillary model that would provide the best fit. The model and these analyses have been described in detail previously (8,9).…”

Radioimmunoassay studies of human apolipoprotein E.

“…The radioimmunoassay results were calculated by a modification of the procedure of Rodbard and Hutt (24) using a previously described nonlinear estimation procedure for determination of the parameters of the standard curve (25). The residual standard deviation ofthe points ofthe standard curves about the fitted expression ranged from 1.8 to 3.5%.…”