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
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 ^{2}H and ^{18}O 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 = 8d^{18}O +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}.
Let C_{r}(t) = C_{o}(t)/C_{d}(t) = [C_{o}'(t) - d^{18}O]/[ 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 ^{18}O and ^{2}H vary along the line given by the equation:
d²H = S x d^{18}O + K
where for the meteoric water line S = 8 and K = 10. The differential of C_{r}(t) with respect to d^{18}O 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 d^{18}O 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 ^{2}H and p is the ratio k_{o}/k_{d}, then the optimal ratio of ^{2}H to ^{18}O as measured by the increment in enrichment immediately post dose, is the reciprocal of C_{r} (0) and is given by:
_{}
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 (d^{18}O) 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 ^{18}O and ^{2}H are important and, if it is necessary to increase ^{2}H enrichment in order to improve analytical precision, maintaining an appropriate ratio could mean that ^{18}O 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/d^{18}O net of background) for different numbers of ^{2}H half-lives and k_{o}/k_{d} ratios
k_{o}/k_{d} |
||||
1.1 |
1.2 |
1.3 |
||
Number of half-lives |
2 |
6.68 |
5.61 |
4.74 |
3 |
6.32 |
5.03 |
4.02 |
Initial enrichment for ^{18}O relative to SMOW (net of background) was 142.59‰ and d²H/d^{18}O 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 ^{2}H.
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.
1. Schoeller DA (1983) Energy
expenditure from doubly labelled water: some fundamental considerations in humans. Am J
Clin Nutr; 38: 999-1005.