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Appendix 5: Determination of optimal dosing ratios

App 5.1: Background
App 5.2: Derivation of a formula to predict optimal dose ratios
App 5.3: Practical consequences
App 5.4: Reference

Contributor:

Andy Coward
Tim Cole

App 5.1: Background

The consequences of changing isotopic backgrounds have been discussed in Chapter 8. Strictly speaking the background values to be subtracted from post-dose enrichment values to determine isotope disappearance rates should be those to which the exponentials are decaying at infinite time. These values are assumed to be the same as those existing at the start of the experiment and this assumption may not be correct. If it is not correct error will be introduced. However, the background changes for ²H and ¹⁸O are likely to be covariant (see Chapter 8) and the slope of background variation may have a slope similar to that of the meteoric water line (d²H = 8d¹⁸O +10). Intuitively, one would expect the magnitude of the error in k_o-k_d caused by the assumption of an incorrect background to depend on the size of the doses given and the length of time over which measurements are made. Large doses will minimise the effect of background changes, but if experiments continue for a long time background effects will be more significant. There is however another important consideration. Since background changes are likely to be covariant it ought to be possible to minimise the effects of background variation by giving appropriate amounts of dose as first suggested by Schoeller ¹.

App 5.2: Derivation of a formula to predict optimal dose ratios

Let C_r(t) = C_o(t)/C_d(t) = [C_o'(t) - d¹⁸O]/[ C_d'(t) - d²H]

where C' is the absolute enrichment, C is the enrichment net of background and (t) indicates time t. Assume that the background enrichments of ¹⁸O and ²H vary along the line given by the equation:

d²H = S x d¹⁸O + K

where for the meteoric water line S = 8 and K = 10. The differential of C_r(t) with respect to d¹⁸O is given by:

[S x C_r(t)-1]/C_d(t)

indicating dependence on S but not K.

Now C_o(t) = C_o(0)e^-ko.t, C_d(t) = C_d(0)e^-kd.t and C_r(t) = C_d(0)e^-kr.t where

k_r = k_o - k_d and (0) indicates time zero.

So e^kr.t = C_r(0)/C_r(t) and the differential of e^-kr.t with respect to is given by:

C_r(t)/C_d(0)[(e^ko.t-1) - (e^kd.t-1) x S x C_r(0)]

For e^kr.t and hence k_r to be least affected by d¹⁸O its differential should be zero. Setting it to zero and solving for C_r (0) gives the optimal ratio of post-dose concentrations:

C_r(0) = (e^ko.t -1)/S x (e^kd.t -1)

If t is chosen to be n half-lives for ²H and p is the ratio k_o/k_d, then the optimal ratio of ²H to ¹⁸O as measured by the increment in enrichment immediately post dose, is the reciprocal of C_r (0) and is given by:

App 5.3: Practical consequences

In practice n will usually be in the range 2 - 3 and p is unlikely to be smaller than 1.1 or greater than 1.3. For a value of S = 8 (the same as the meteoric water line) these ranges produce the optimal ratios shown in Table 1. Using dose regimes that produce these initial enrichments will provide protection against both random and unidirectional changes in background during the experiments.

The advantages of using appropriate dose regimes is illustrated in Fig App 5.1. Here, the correct background values are -4 (d¹⁸O) and -22 d²H and the correct k_o - k_d difference is 0.02 (k_o = 0.12 and k_d =.10). Other values are incorrect but covariantly so. Thus, for example, if pre-dose values were -6 and -38 but in reality the subject was equilibrating to backgrounds of -4 and 22 errors of -3.3 and +7.5% would be produced for initial ratios of 6.68 and 4.02 respectively (Curves D and E). Clearly inappropriately tailored doses such as those producing ratios of 11.22 and 2.81 (Curves B and C) are dangerous to use compared to the ideal ratio of 5.61 (Curve A). Fig App 5.2 shows the expected effect of generally increasing dose levels but maintaining a variety of ratios. If that is done errors are reduced but the general shape of the curves remains the same.

Unfortunately, although adopted here for simplicity, these are not the only considerations. The relative measurement precisions for ¹⁸O and ²H are important and, if it is necessary to increase ²H enrichment in order to improve analytical precision, maintaining an appropriate ratio could mean that ¹⁸O costs limit the amount of work that can be done. Clearly balances need to be found but gross deviations away from ideal dose regimes are not advisable.

Table App 5.1. Optimal initial isotopic ratios (d²H/d¹⁸O net of background) for different numbers of ²H half-lives and k_o/k_d ratios

Initial enrichment for ¹⁸O relative to SMOW (net of background) was 142.59‰ and d²H/d¹⁸O ratios are 5.61 (Curve A), 11.22 (Curve B), 2. 81 (Curve C), 6.68 (Curve D) and 4.02 (Curve E). The figure assumes covariant background changes along the meteroic water line and that the duration of the experiment was 2 half-lives for ²H.

Curves A, B and C and other assumptions are the same as in Fig App 5.1. A', B' and C' ratios correspond to A, B and C but initial enrichments net of background have been doubled.

App 5.4: Reference

1. Schoeller DA (1983) Energy expenditure from doubly labelled water: some fundamental considerations in humans. Am J Clin Nutr; 38: 999-1005.

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