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9.4 Summary of theoretical comparison of methods

A general conclusion from the foregoing analyses is that the two-point method for calculating CO₂ production is the most robust in the presence of systematic changes in CO₂ and water production, provided that N_O and N_D are correctly measured, and that the non-linear regression method is most sensitive to systematic changes. Both multi-point methods perform better when there is a high degree of covariance in the ²H and ¹⁸O data. Evidence from most data sets (see Chapter 5) and physiological commonsense indicates that a degree of covariance will normally exist. However, in all cases where large systematic changes occur, initial isotope distribution spaces will be incorrectly estimated from the zero-time intercepts of fitted curves. Thus bias can occur with this procedure. On the other hand, at all levels of analytical error, combining slopes and intercepts substantially improves precision.

Upper panel - Linear regression
Lower panel - Non-linear regression

Table 9.3. Effect of moderate and large random errors in isotopic enrichment on the precision of the doubly-labelled water method for measuring energy expenditure during a 2 half-life metabolic period

¹ sd_O = 0.18 + (d_O/260) x 0.79, and sd_D = 1.2 + (d_D/600) x 2.

² Large random error = 3 x moderate random error.

Table 9.4. Combined effects of random analytical error in isotopic enrichment and baseline isotopic abundance on the precision of the doubly-labelled water method for measuring energy expenditure during a 2 half-life metabolic period

Superscripts as in Figure 9.3.

The non-linear regression method further improves precision in the face of analytical error but this improvement is small and does not compensate for the potential inaccuracies in the presence of the particular systematic changes in the isotope turnover rates specified in this simulation. However, as stated in Chapters 4 and 5 the fitting procedure should be chosen after inspecting the residual plot. Using this approach only some of the simulations would be identified as appropriate material for this type of fit: e.g. the variable r_CO2 (middle) and variable r_H2O (end) simulations in which the non-linear regression method performed marginally better than linear regression (see Tables 9.1 and 9.2).

The apparent advantages of the two-point method portrayed by the above analyses may however be somewhat misleading. As has been mentioned, it was assumed that one space or the other (N_O or N_D) was correctly measured from plateau determinations and the relationship between them was such that N_D/N_O = 1.03. There are therefore two questions that must be asked. Firstly, what level of precision and accuracy can be achieved by making measurements at plateaus of enrichment: near to the start of an experiment? Secondly, are we justified in fixing a relationship at N_D/N_O = 1.03 for all conceivable subjects. As already indicated (Section 4.3.1) our knowledge in this area is relatively scanty. For American adults a combination of analytical accuracy and noise about plateau values gives a precision of about 1.5% for volume determinations using saliva samples at 3-6 hr after a dose (recalculated from ref ¹⁰) but we do not know if this time interval is appropriate in all physiological circumstances. This error need not worry us too much however since if N_D/N_O really does equal 1.03 estimates of CO₂ production will only be incorrect by the % inaccuracy in the plateau determination of body-water.

Table 9.5. Energy expenditure values calculated on the assumption that N_D/N_O = 1.03 compared to true values obtained when N_D/N_O ¹ 1.03

What is of greater concern is the appropriateness of N_D/No = 1.03. The evidence is that this is a reasonable average figure (see Section 4.3 and Table 9.6) but we really do not know its physiological range for all conceivable subjects. Table 9.5 indicates the problem for a case where k_D and k_O are 0.1050 and 0.1300 respectively. It can be seen that even small physiological variations away from N_D/N_O = 1.03 could cause a substantial inaccuracy in estimates of CO₂ production.

9.5 Practical comparisons of the two-point and slope-intercept methods

We have shown in Chapter 4 and earlier in this chapter that the two procedures are quite different from the point of view of what is intended to be measured. The two-point method will produce the correct value for total flux between two time points even in circumstances where systematic variations occur to such an extent that the calculation of an average flux rate per day could almost be said to be inappropriate. In contrast a multi-point method only produces a correct result when variations in rate constants are random. The multi-point method does, however, provide an estimate of the variation about the daily average production rate which combines effects of instrumental and physiological variation (see Chapter 5).

There is little data that allows comparisons to be made between two-point and multi-point methodologies. In the comparison provided by Coward ¹¹ there was no significant bias between the two methods (see Table 9.6). The SDs of the differences between the two procedures are slightly higher than would be predicted from the theoretical uncertainties of about 3% for the slope/intercept approach and about 4.5% for the two-point method. In this comparison the small average difference between the two procedures almost certainly originated from the fact that the volume ratio N_D/N_O averaged 1.037 in the slope/intercept method but was normalised to 1.03 in the two-point method.

In contrast Schoeller & Taylor ¹² compared two-point methods in which either a plateau method was applied to measure N_D and N_O and the values normalised to N_D/N_O = 1.03, or individual N_D and N_O were calculated from the intercepts of disappearance curves generated between two points. In these circumstances the differences between the results obtained using each method can only be attributed to different estimates of N_D and N_O because slopes used were the same. For a period 0-7 days the intercept method produced an average value that was 15% lower than that obtained using the plateau procedure (see Table 9.7) and if values of k_D and k_O are taken to be 0.105 and 0.130 respectively this difference is equivalent to an average N_D/N_O ratio of about 1.06. In the worst case (Subject D) where the difference was 26%, the same assumptions produce an N_D/N_O ratio of 1.08. These differences from the value of 1.03 on which spaces were normalised are clearly important and contrast markedly with the data from Table 9.6 where mean N_D/N_O ratios were 1.037 ± 0.012 SD and with the more extensive data in Table 4.2. It is impossible to say for certain whether marked deviations from the value of 1.03 are genuine physiological differences, in which case the use of 1.03 as a normalising factor is unlikely to be inappropriate, or whether such large ratios have a non-physiological origin, in which case normalisation is an adequate procedure.

Table 9.6. Comparisons between rate-constants for ¹⁸O and ²H disappearance (k_O and k_D, d^-1) isotope distribution volumes (N_O and N_D, g) and carbon dioxide production rate F_CO2, estimated using the slope-intercept and two-point methods (n = 50)

Mean difference in F_CO2 (as % of A) = -1.9 (SD 7.4), paired-t = 1.82.

Table 9.7. Energy expenditure (MJ/day) for 7 subjects calculated using the two-point method with N_D and N_O estimated from plateau or intercept

Data from Schoeller & Taylor ¹².

9.6 Conclusions

It will now be evident that the main difficulty with the methodologies we have been discussing lies not with the measurement of slopes of isotope disappearance curves but with the estimates of volume.

The use of multi-point data with fitting procedures appropriate to the error structure will provide a good estimate of the average difference between rate constants when there is a high degree of covariance between ²H and ¹⁸O data and even in cases where only CO₂ production is increased and there is no such covariance the errors are fairly small even where 30% changes in CO₂ production occur that persist for one third of a total measurement period. If such a change happened at regular intervals, such as every third day as might occur with a recreational runner, errors are less than 1% and can be ignored. There is also the additional factor to consider that, in the case of any level of analytical error, multi-point methods will reduce errors to about half of those obtained with a two-point method assuming that great care is taken to minimise analytical error in the measurement of baseline abundance. Furthermore if no systematic physiological deviations from linearity occur negative covariance will further improve precision. However, in the presence of large systematic variations in either CO₂ production or water output, bias will occur in estimates of CO₂ production if isotope distribution spaces are measured from the zero-time intercepts of isotope disappearance curves. This error will not always be observed in an estimate of a regression coefficient as these are almost invariably better than 0.99 but will be observable in plots of residuals near time zero (see Figure 9.2). This illustrates the importance of drawing and inspecting a residual plot for all multi-point studies. When systematic deviations are detected, then it will be theoretically preferable to obtain intercept data from the analysis of a subset of data near to zero-time.

In many ways the conclusions that should be drawn from the inadequacies of the two-point methodology are very similar to those outlined for the multi-point method. Provided there is the assurance of analytical accuracy in the determination of a slope the problem is likely to lie with the measurement of volume. The adequacy of this value will only be improved by making several measurements at different times on a plateau near to zero-time; in other words on a similar subset of data to that suggested for the multi-point method when systematic variation occurs.

There remains the problem of the relationship between N_O and N_D. The balance of current evidence suggests a value for N_D/N_O of about 1.035 and if this relationship cannot be established during an experiment it is not unreasonable to use this average value, on the understanding that if the true value for any subject is 1% different from this, the error produced in the measurement of CO₂ production will be about 5%. On the other hand if it is possible to measure both N_D and N_O with an accuracy of 0.5 - 1.5% in experiments it is theoretically preferable to use these values. However, experience from the literature and from the data exchange exercise prior to this meeting suggests that values of 1.03 (plus or minus some small SD) are not always found. Until consistent findings for particular populations indicate otherwise it is commonsense to treat N_D/N_O ratios differing markedly from these values with some suspicion. The IDECG Workshop recommended that 1.015 - 1.060 should be adopted as the acceptable range.

At this meeting Speakman suggested that a general rule might be applied when chosing methods for calculating energy expenditure measurements and his view fairly encapsulates all the arguments propounded in this chapter. That is, that with low isotopic enrichments and relatively small temporal variation in water turnover or carbon dioxide production it will be preferable to use a multi-point slope-intercept method. This is because relatively low enrichments put a premium on the analyses and, in the absence of much temporal variation, fitting single exponentials to data is a satisfactory procedure. When temporal variation is large, as it may often be in wild animals, a two-point method will certainly be preferable from both the theoretical and practical point of view ¹³. The theoretical reasons are evident, the practical reason is that wild animals are difficult to catch more than a few times in any measurement period. There is the further advantage that relatively high enrichments are often used in experiments such as these and this is an advantage when analytical precision is considered, especially in the two-point method.

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