Title:

The analysis of isotope clearance data in biological systems

Clearance curves resulting from biological studies using radioactive isotopes are frequently described mathematically in terms of the summation of a number of exponential terms. This allows the curves to be interpreted by reference to the physical characteristics of a model of the biological system. Numerous exponential curve fitting methods are now available which make use of digital computers. Despite the very widespread application of exponential curve analysis, a systematic study of the relative importance of the factors which affect the parameter errors has not yet been described. A quantitative statistical study of the problem is described in this thesis with particular reference to the special limitations encountered in biological investigations. These limitations are firstly, the limited number of samples and, secondly, the relatively poor accuracy normally associated with such studies. The accuracy would not normally be better than +/ 2% nor would the number of samples exceed 60. Of the ten principal factors which affect the errors in the estimated parameters, two of these, the exponent and amplitude ratios, are intrinsic factors dependent on the system under study. The principal factors under the control of the investigator are the number of samples, the data accuracy, the sampling frequency, and the duration of sampling, which determines the extent to which the data define the function under study. Other factors of lesser importance were not investigated in the same detail as those mentioned above. Artificial data, on which a controlled random error was superimposed, were generated by a computer programme and recorded on magnetic tape in a format suitable for exponential analysis by the Berman SAAM22 computer programme. The parameter errors were estimated by a statistical analysis of twenty curve fitting operations carried out on twenty different sets of data with a constant controlled random error. Twice the coefficient of variation, expressed as a percentage, was taken to be the parameter error. A range of exponential and amplitude ratios was investigated for two exponential and three exponential functions with data errors from 2  10%. The study has indicated, in quantitative terms, the effects of the various factors on the errors associated with the estimated parameters, and also the relative importance of these factors. The results indicate the conditions which must be fulfilled if sellable results are to be obtained by exponential analysis. The information is also of value in designing investigations which will subsequently involve exponential analysis of the data. In view of the parameter errors encountered in the study of two and three exponential functions, it appears unlikely that analysis of biological data in terms of a greater number of exponentials will be helpful unless further independent information is available concerning the biological system under study. Two clinical applications of exponential curve fitting procedures are described. In a study of uric acid metabolism, two different computer programmes were used to examine the same data. A mathematical significance test was used to indicate which sets of data were better fitted by a double rather than a single exponential function. It was found that with one of these programmes only, when using a particular weighting factor on the data, a strong correlation exists between the indication of a double exponential function in the data and the clinical diagnosis of gout. This is interpreted as evidence of the existence of uric acid in two different physiological forms in gouty patients. In the second study, the detailed investigation of a depth focusing radioisotope collimator, and its use in the measurement of local cortical blood flow in the brain, is described. By using this collimator, clearance curves of radioactivity from a very small volume of Min tissue in the cortex were obtained. The curves were analysed empirically by means of a double exponential curve fitting procedure, in order to determine the initial slope. No biological significance is assigned to the individual exponential terms. Since the collimator is designed to accept radiation originating specifically in the cortex, the detector is particularly sensitive to changes of flow in this tissue. The results obtained for cortical tissue in normals agree with the values of grey matter flow determined by other workers on much larger regions of the brain containing both grey and white tissues.
