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Two procedures are commonly used for determining flux rates. These have become known as the two-point and multi-point methods. It may often be supposed that these procedures are basically the same, the two-point procedure being an abbreviated multi-point method. This is not, however, the case for although the underlying assumptions outlined in Section 4.2 apply in each case, the two-point procedure is more tolerant of deviations from the model than the multi-point method, essentially because the two-point method measures total flux between two points several days apart whereas the multi-point method attempts to provide a mean value for daily flux rates over the same time interval.
4.4.1 Two-point method
If N remains constant during an experimental period total outflow r' is correctly calculated for a period between t1 and tt as:
..........5
where the subscripts refer to times 1 and t and pre-dose (p).
This is true even if the rate of outflow per unit time is not constant and varies in a random fashion or systematically. Imagine, for example, that r' measured over successive similar time intervals varies. In this case total flow for the period 1 to t is given by:
or
..........6
Cross-cancellation gives: on gives:
.........7
The two-point methodology is also resilient when N changes with time as a consequence of rin not equalling rout. Coward et al 7 have shown that if rin does not equal rout, but both are constant so that N changes linearly, or if rin(t)/N(t) and rout(t)/N(t) are unequal but constant and N thus changes exponentially, total outflow is given by:
.........8
(A full derivation of this Equation can be found in Coward et al 7 and an alternative version is presented in Appendix 3.)
Because daily variation in the slope of the line between the two points does not matter in the two-point method obtaining the correct answer for the overall slope is entirely a matter of analytical accuracy and precision.
4.4.2 Multi-point (slope-intercept) method
The multi-point method is fundamentally different from the two-point procedure because the basic intention is to obtain a mean value for flux rate with a standard deviation indicating the extent of random physiological variation.
Pool spaces (N) are measured from the zero-time intercept of a plot of isotopic enrichment versus time. The most important advantage of this manoeuvre is that it reduces the error on calculation of the products kONO and kDND because of the covariance of slopes and intercepts. For example, an error tending to make a slope steeper will increase the value of the intercept and consequently reduce N. Allowance can also be made for covariance between errors in the 2H and 18O data in the determination of the SE of the estimate. These aspects are covered more fully in Chapter 5. Additionally random analytical error becomes less important than it is in the two-point method. This aspect is also covered fully in Chapter 5.
There are many possible ways of handling multi-point data. Three of these were explored in detail during the workshop and in the data analysis exercise leading up to it. They are summarised below. A fourth variant (the Product-Ratio Method) was developed by Cole and Coward and introduced at the workshop. It gives the same answer as the simple log fit described below, but provides a better estimate of error and the most informative way of inspecting the raw data. It is described in full in Chapter 5.
4.4.2.i Curve fitting to multi-point data
In order to obtain values for rate constants and intercepts (or volumes) using multi-point: data it is necessary to fit an appropriate curve to the data which describes the mean decay over time. Different fitting procedures attach more or less importance to individual points and the choice of the most appropriate treatment should ideally be based on knowledge of the system's error structure. At extreme ends of a range of possibilities are: a) that errors are proportional to the level of enrichment; or b) that the errors have a constant absolute value and are not dependent on enrichment. The first case might be represented by physiological variations such as random changes in water turnover. The second case could arise from constant analytical errors. In practice however it could be suggested that an ideal fitting procedure would be one that was appropriate to an error structure lying somewhere between the two extremes. A detailed discussion of this problem is given in Appendix 4, and further coverage appears in Chapter 5. However a few general statements can be made at this juncture and the following sections give a simple summary of 3 different data-fitting techniques and guidance as to which one is appropriate under different error conditions.
4.4.2.ii Log transformation
This procedure is the simplest that can be used since any computer package or calculator programme will fit the best straight line through log transformed data. The method assumes that errors are proportional; that is to say that residuals (differences between observed points and the fitted enrichment curve) become smaller as enrichment decreases or, in other words, are constant in size relative to enrichment. This fitting procedure is thus appropriate for errors described in case (a) above and has been used by most groups 8, 12-15. The curve fitting procedure described by Feldman 16 is used by the Cambridge group.
4.4.2.iii Exponential fit
Haggarty et al 17 at the Rowett have estimated volumes and rate constants by fitting exponentials directly to untransformed data. This approach was developed after inspecting residual plots which indicated a constant error structure throughout the labelling period in the particular group of subjects studied 18. It was this process of scrutinising residual plots which alerted these authors to the need for alternative fitting methods under certain circumstances. In contrast to log transformation the exponential fit allows early points (higher enrichment) to have a greater effect on the outcome of the fitting than later points (lower enrichment). The curve fitting is not as readily available, but it is found in a number of more sophisticated statistical packages for computers. These workers use "Maximum Likelihood Program" (Numerical Algorithms Group, Oxford, UK) to fit models to data.
4.4.2.iv Poisson fit
Franklin, also from the Rowett, suggested a third "Poisson" model at the workshop which assumes that the error structure lies somewhere between the constant: CV and constant SD situations covered above. The simplest ways to achieve this are to use either a weighted linear regression method or a 'generalised linear model' 19 with a logarithmic link function and a Poisson error distribution coupled with a heterogeneity factor. Essentially this allows the fitting of the log or exponential models with the standard deviation at each point being proportional to the square root of the enrichment. "Maximum Likelihood Program" also provides the facilities to carry out this type of analysis.
4.4.2.v Choosing the appropriate model
Firstly, it should be re-emphasised
that with well-behaved data-sets it is virtually irrelevant as to which method of data
reduction is employed since they will yield very similar answers. The workshop therefore
recommended that users should adapt their software to calculate all data-sets in each of
the 3 possible ways. They can then predefine an acceptable level of agreement (say 3%). If
the 3 results agree to within this tolerance then the particular version favoured by that
laboratory can safely be used. If the results fail to satisfy this criteria then the data
must be carefully scrutinised.
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