## 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

 Model Samples k (%B) N (%B) r' (%B) Deuterium A Two-pool All 156.38 63.72 99.61 B One-pool 6, 7, 8 hr 1-12d 100.00 100.00 100.00 C One-pool 1-12d 99.10 101.16 100.24 Oxygen-18 A Two-pool All 148.52 67. 21 99.82 B One-pool 6, 7, 8 hr 1-12d 100.00 100.00 100.00 C One-pool 1-12d 99.13 100.96 100.07

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 fixed ND/NO 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 ND and NO are measured the observed values can be appropriately weighted. If ND/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 ND and NO values if we are dealing with a genuine physiological variation in ND/NO.

It is important to determine the likely physiological range of ND/NO. The small SDs for ND/NO, 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 (ND/NO) in a variety of studies performed by the Dunn Nutrition Unit

 Subjects Ratio (SD) n Pregnant women (Cambridge) 1.036 (0.012) 8 subjects 42 measurements Lactating women (Cambridge) 1.034 (0.012) 12 subjects 36 measurements NPNL women (Cambridge) 1.037 (0.011) 17 subjects Obese women (Cambridge) 1.040 (0.013) 8 subjects Infants < 6 months (Cambridge) 1.035 (0.020) 136 subjects Men (Northern Ireland) 1.034 (0.011) 16 subjects Women (Northern Ireland) 1.036 (0.010) 16 subjects Elderly women patients 1.028 (0.009) 14 subjects Men (Gambia) 1.030 (0.024) 16 subjects 29 measurements

NPNL = non-pregnant, non-lactating.

Data from Coward.

Table 4.3. Exchangeable H in a reference man

 kg % exchangeable H Total exchangeable H %Total exchangeable H Water 41 11.11 4.555 94.758 Fat 14 0.35 0.049 1.019 Protein 13 1.49 0.194 4.036 CHO 0.5 1.85 0.009 0.187 Minerals 1.5 0 0 0 Total 70 4.807

Predicted ND/NO = 1.055

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

 kg Total exchangeable H %Total exchangeable H Example 1 Water 41 4.555 95.734 Fat 0 0 0 Protein 13 0.194 4.077 CHO 0.5 0.009 0.189 Minerals 1.5 0 Total 56 4.758 Predicted ND/No = 1.045 Example 2 Water 41 4.555 93.801 Fat 28 0.098 2.018 Protein 13 0.194 3.995 CHO 0.5 0.009 0.185 Minerals 1.5 0 Total 84 4.856 Predicted ND/No = 1.067 Example 3 Water 41 4.555 96.709 Fat 14 0.049 1.040 Protein 6.5 0.097 2.059 CHO 0.5 0.009 0.191 Minerals 1.5 0 Total 63.5 4.710 Predicted ND/NO = 1.034 Example 4 Water 41 4.555 92.883 Fat 14 0.049 0.999 Protein 19.5 0.291 5.934 CHO 0.5 0.009 0.184 Minerals 1.5 0 Total 4.904 Predicted ND/NO = 1.077

4.3.1 Biological basis for differences between ND and NO

Culebras and Moore 5 provided a theoretical basis for the differences between ND and NO. Assuming that NO correctly predicts body water, then ND/NO 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 ND/NO value should be used as an initial screen in the detection of analytical problems. If extreme values for ND/NO 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 ND/NO.)

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.