A weighted least-squares technique for the analysis of kinetic data and its application to the study of renal xenon 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-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 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. 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 From the Departments ,of Pediatrics and Medicine, College of Physicians and Surgeons, Columbia University, 630 West 168th Street, New York, New York 10032. This work was supported by U. S. Public Health Service Crauts HD 03993, HE 10182, HL 14148, and HL 14238 from the National Institutes of Health. Dr. Cannon is a recipient of Research Career Development Award 5 K3 HL 15031-07 from the National Heart and Lung Institute. Received June 19, 1972. Accepted for publication October 13, 1972. Grc&iuH, Reutrch, Vol. XXXII, Jmttr, 1973 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 cnfidence, (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 parameters were computed by a multiple comparison technique, and the significance of including additional terms in the equation or model was tested by a residual error analysis. This program is described in the first part of this paper 71 by gest on N ovem er 8, 2017 http://circhajournals.org/ D ow nladed from 72 DELL, SCIACCA, LIEBERMAN, CASE, CANNON along with its application to the estimation of the parameters of the muJtiexponential equation describing renal TMxenon (SXe) washout curves. The second part of the paper presents more detailed results of the application of the computer program to analysis of renal aXe washout curves in dogs and in human subjects. Thorburn et al. (3) first applied the inert gas technique developed by Kety (4) to the study of the intrarenal distribution of blood flow; they estimated the capillary blood flow in different regions of the kidney from analysis of the externally monitored washout curve of •^'krypton. The curve which depicts this washout can be described by a multiexponcntial equation of the general form Each term in the equation is assumed to represent a region or a compartment of the kidney. Using the assumptions of Kety (4), the A parameters can be related to the volume of the compartment (t* compartment as percent of total = 100 X A(/SA() and the a parameters can be related to blood flow through the compartment (flow = X • al p, where A is the partition coefficient for the tracer and p is the density of the tissue). Although most investigators agree that the equation which describes the washout curve in normal dogs contains four terms, there is some uncertainty concerning the number of terms needed to describe the washout curve in dogs during various experimental procedures. There is similar uncertainty in both normal and abnormal man with some investigators reporting three terms (5, 6) and others reporting four (7-9). In view of these discrepancies, the weighted nonlinear regression analysis described in the first part of the paper was applied to the analysis of renal Xe washout curves obtained in studies of normal dogs and of dogs subjected to graded hemorrhage or partid occlusion of the thoracic inferior vena cava. Washout curves obtained from human kidney donors and from patients with hypertensioii, congestive heart failure, or oliguric renal failure were also analyzed. Data from these studies were included principally to illustrate the detailed operation of the technique with respect to the characteristics of both dog and human Xe washout curves.