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4.3 Calculation of pool sizes

Two different techniques are currently used for the determination of volume. In the slope/intercept multi-point procedure time zero distribution space is calculated from the same data as that used to measure the slope. In the two-point methodology volume is calculated from observed isotopic dilution at a plateau shortly after dose administration. In order to appreciate the differences between these two procedures it is instructive to consider the only circumstance in which they will produce identical results. That is when the rate constants for output are zero. In this case the intercept: of values for all samples collected will be the same as the average of all values because a permanent plateau of isotope concentration exists. In all other conceivable circumstances the methods will not produce the same answers, and each individual method will be correct, assuming no analytical errors, in certain achievable circumstances.

The slope/intercept methods will be correct, ignoring analytical errors, if mixing is instantaneous and if the rate constants for output do not change over time. Unfortunately output is never absolutely constant, but varies both within and between days, and an absolutely correct value cannot therefore be obtained. If the variations are random, however, the error can usually be reduced to 1% or less.

The plateau method will be correct, within the limits of analytical error, irrespective of the rate of isotope mixing if all isotope losses occurring during the equilibration period can be accounted for and subtracted from the dose administered. Unfortunately, only a fraction of all losses can be accounted for by urine collection for example, thus this procedure cannot ever produce an absolutely correct value. With collection of urine and estimation of other losses this error can also be reduced to 1% or less.

Table 4.1. Effect of changing the model or sampling times or both on estimates of rate constants (k), initial distribution volumes (N), or outflow rates (r' = kN) in an adult male subject orally dosed with ²H₂¹⁸O

The question is: Do these differences from correct values and potential differences between the two methods matter? From first principles it would seem reasonable to suppose that the plateau method may slightly over-estimate volume because the dose remaining in the body may be over-estimated. Conversely, because of the time taken for mixing, an intercept procedure that ignores mixing will over-estimate initial isotope concentration and under-estimate volume.

For the plateau method Schoeller et al ³ recommend that a fixe_d N_D/N_O ratio of 1.03 is assumed to exist and that if only one space measurement is made the other should be calculated from it. Alternatively, when both N_D and N_O are measured the observed values can be appropriately weighted. If N_D/No is 1.03 plus or minus some small SD for all conceivable subjects this procedure can be justified, but it ought to be preferable to use the observed N_D and N_O values if we are dealing with a genuine physiological variation in N_D/N_O.

It is important to determine the likely physiological range of N_D/N_O. The small SDs for N_D/N_O, observed by both Coward 6(see Table 4.2) and Schoeller et al ³ suggest a combination of small analytical errors and a relatively minor degree of between-subject variation. However, values obtained in the data sets exchanged prior to this meeting ranged from 0.939 to 1.329 (mean 1.045 ± 0.081 SD). In view of the theoretical considerations outlined in the next section it appears likely that a number of the outlying values were incorrect as a result of analytical errors.

Table 4.2. Ratios of isotope distribution spaces (N_D/N_O) in a variety of studies performed by the Dunn Nutrition Unit

NPNL = non-pregnant, non-lactating.

Data from Coward.

Table 4.3. Exchangeable H in a reference man

Predicted N_D/N_O = 1.055

Table 4.4. Exchangeable H in a reference man with modified body composition

4.3.1 Biological basis for differences between N_D and NO

Culebras and Moore ⁵ provided a theoretical basis for the differences between N_D and N_O. Assuming that N_O correctly predicts body water, then N_D/N_O ratios of 1.055 are predicted by exchange of ²H with ¹H in fat, protein and carbohydrate in a reference man (see Table 4.3). Clearly variations in body composition will alter the extent of ²H exchange. However, these effects are relatively small and changing assumptions about body composition within very wide limits (see Table 4.4) does not produce the range of values that were found in the present data sets. One is driven to the view that the observed range was physiologically unreasonable, and this is the basis for the assumption that many of the outliers must have been wrong.

In later chapters it is proposed that the N_D/N_O value should be used as an initial screen in the detection of analytical problems. If extreme values for N_D/N_O ratios are obtained, the origin of these differences should be investigated and only if there is a good physiological explanation should they be used. (See Chapter 9 for further discussion of this point, and Chapter 11 for recommendations concerning the screening of data sets according to the between-measurement variance in N_D/N_O.)

4.3.2 Equation for calculating pool size from isotope dilution

The general equation recommended for use in human studies is:

where N is the pool space; W is the amount of water used to dilute the dose; A is the amount of dose administered; a is the dose diluted for analysis; and d is enrichment of dose (a), tap water (t), post-dose sample (s) and pre-dose baseline (p). Since it possible to use this equation without fully understanding its origin the full derivation has been put in Appendix 2.

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