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11.3 Procedure for calculating multi-point data

Due to the extra complexities inherent in multi-point analysis this section presents worked examples using 3 of the datasets exchanged prior to the workshop. These have been selected to represent good (Subject 1), moderate (Subject 2) and indifferent (Subject 3) data.

It must again be emphasised that the choice between the 3 different fitting procedures represents a refinement of the method which need only be invoked under unusual circumstances. Readers who are new to the method should initially concentrate on the simple log fitting procedure (or the Poisson fit if they have the computing capacity, since this represents an excellent compromise approach), and be aware that an alternative method is available to deal with data showing large residuals at the end of the measurement period.

11.3.1 Initial treatment of mass spectrometric data

The unprocessed mass spectrometric results for these subjects are presented in Table 11.1. The most convenient way of handling enrichment data is to represent all d's as a fraction of the initial dose given. This is achieved using the formula:

where d is the enrichment of the sample (d_s), pre-dose baseline (d_p), dose (d_a) and tap water (d_t); a is the amount of dose diluted for analysis (g); W is the amount of water used to dilute the dose (g); A is the amount of dose administered (g); and 18.02 converts g water into moles.

There are two advantages to this procedure. Firstly, the pool sizes (N_D and N_O) are derived simply as the reciprocal of the intercept (or plateau value). Similarly the intercept of the d_O/d_D plot is the ratio of the spaces N_D/N_O. Secondly, the procedure 'normalises' the results so that both d_O and d_D values can be plotted on the same scale. Table 11.1 contains the converted data for Subjects 1 - 3, and Figure 11.1 shows the untransformed plots of d_O and d_D.

At this stage the data can be screened for any obviously abberant points which may be due to sample contamination, mislabelling with respect to time or faulty analysis. If the data-points in question remain outliers after re-analysis it may be acceptable to exclude them from the analysis always remembering that the fewer the number of data points the less justifiable this procedure becomes. In practice untransformed plots need not be drawn since the same use can be made of log transformed plots (see below).

Table 11.1a. Mass spectrometric data for worked examples - Subject 1

d values are relative to SMOW (¹⁸O/¹⁶O = 0.0020052, ²H/¹H = 0.00015576)

Normalised values are calculated as described in Section 11.3.1.

Other values are: a = 0.6785 g, W = 249.3215 g, A = 116.39 g.

For ¹⁸O: d_p = -4.72, d_a = 181.55, d_t = -7.64.

For ²H: d_p = -22.34, d_a = 895.89, d_t = -45.84.

Table 11.1b. Mass spectrometric data for worked examples - Subject 2

d values are relative to SMOW (¹⁸O/¹⁶O = 0.0020052, ²H/¹H =0.00015576)

Normalised values are calculated as described in Section 11.3.1.

Other values are: a = 0.5605 g, W = 249.4395 g, A = 114.33 g.

For ¹⁸O: d_p = -4.4, d_a = 163.23, d_t = -7.64.

For ²H: d_p = -27.41, d_a = 544.85, d_t = -45.84.

Table 11.1c. Mass spectrometric data for worked examples - Subject 3

d values are relative to SMOW (¹⁸O/¹⁶O = 0.0020052, ²H/¹H = 0.00015576)

Normalised values are calculated as described in Section 11.3.1.

Other values are: a = 0.5000 g, W = 249.5000 g, A = 70.00 g.

For ¹⁸O: d_p = -1.496, d_a = 77.11, d_t = -3.00.

For ²H: d_p = -39.87, d_a = 487.95, d_t = -14.00.

Table 11.2a. Estimates of pool sizes, rate constants, products and ratios for the worked examples - Subject 1

Units: N = moles; k = day^-1; Nk = moles x day^-1.

Difference columns list % offset compared to the Log fit.

Units: N = moles; k = day^-1; Nk = moles x day^-1.

Difference columns list % offset compared to the Log fit.

Table 11.2b. Estimates of pool sizes, rate constants, products and ratios for the worked examples - Subject 2

Units: N = moles; k = day^-1; Nk = moles x day^-1.

Difference columns list % offset compared to the Log fit.

Table 11.2c. Estimates of pool sizes, rate constants, products and ratios for the worked examples - Subject 3

Units: N = moles; k = day^-1; Nk = moles x day^-1.

Difference columns list % offset compared to the Log fit.

11.3.2 Data transformation and curve fitting

Figure 11.2 illustrates plots of the log transformed enrichments and fitted regression lines. Similar plots could be drawn using the exponential or Poisson fits (Chapters 4 and 5) if preferred. Whichever method is used, the rate constants (k_D and k_O) are represented by the slopes of the regression, and the pool sizes (N_D and N_O) are represented by the reciprocal of the intercepts. These are listed in Table 11.2.

From Subject 1 to Subject 3 the data are progressively less tidy, and although covariance between the d_O and d_D plots is evident for Subject 1 and Subject 2, this is not the case for Subject 3.

11.3.3 Checking pool space ratios

As discussed in Sections 4.3 and 9.6, N_D/N_O ratios lying outside the range 1.015 - 1.060 should be treated with scepticism and most probably indicate analytical or dosing error (the latter will only hold true if the doses are administered separately). Table 11.2 shows that Subjects 1 and 2 have acceptable pool space ratios irrespective of which fitting procedure is used. Subject 3, on the other hand, has unacceptably high ratios by all methods of calculation (ranging from 1.087 - 1.114), and according to the IDECG guidelines should be rejected.

11.3.4 Residual plots

The next step is to calculate the residuals (i.e. the difference between the model and each of the experimental points). These are plotted for each of the 3 models in Figures 11.3 - 11.5. The residual plots are extremely useful in three respects.

Firstly, they highlight deviations from the Lifson model. For instance, positive residuals at each end of the measurement with negative residuals in the middle, or vice versa, would indicate curvature due to changing flux or pool size.

Secondly, they provide information about the error stucture exhibited by the data and hence about which fitting procedure is preferable. For instance, if the residuals from a log plot increase through the experiment it is an indication that errors are not proportional to d_O or d_D values and that an exponential fit may be more appropriate.

Thirdly, they provide the optimum way of assessing covariance between the ²H and ¹⁸O data, on the log scale used, a value of +0.1 indicates that the observed value is 10% higher than the fitted value. For Subject 1 the residuals are small and highly covariant; for Subject 2 the residuals are larger but usually covariant; but for Subject 3 the residuals are both large and frequently not covariant. This provides clear evidence that the data from Subject 3 is far from ideal. The poor level of covariance suggests possible analytical errors which should be checked by re-analysing all of the samples. If it persists then other explanations, such as gross changes in water flux, must be sought. Whatever the explanation, the inspection of residuals has identified the data from this subject as problematical and needing cautious interpretation or outright rejection especially when considered together with the pool space anomaly (Section 11.3.3).

An examination of residuals for early time points is also informative. If these are not close to zero, it indicates that isotope distribution spaces calculated from the intercepts would differ from those calculated from the early time points. This is certainly true for the theoretical situations described in Chapter 9. In the present: examples Subjects 2 and 3 show differences but these are relatively small, and in the same direction for both isotopes. (Note that there was no early data for Subject 3.)

It should be stressed that comparison of the results of the three fitting procedures gives little information on which is the appropriate choice or on the 'quality' of the data. As discussed in Chapter 9 only the residuals can be used to decide on the appropriate fit. Even with no measurement error the 3 fitting methods will give rise to different estimates for the parameters. In general, the Poisson fit will give parameter values intermediate between the log and exponential estimates. This will always hold for the rate constants, almost always for the intercept, pool sizes and r_H2O, and for r_CO2 in most cases. Thus any of the 3 fitting procedures can be used to obtain a residual plot before deciding which is the best for the final analysis.

11.3.5 Product and ratio plots

Figure 11.6 shows plots of d_O/d_D ratios, and Figure 11.8 shows their residuals. (Note that Ratios and Products can also be fitted using Poisson or exponential procedures, but to save space these are not illustrated.) A good fit with small residuals indicates relatively constant rates of CO₂ production (irrespective of possible variations in water turnover) combined with adequate analytical technique. Poor fit indicates the opposite.

A further point about d_O/d_D plots is that their intercept on the Y axis directly indicates the N_D/N_O ratio. Thus an intercept of +0.03 on the log scale is equivalent to N_D/N_O = 1.03. The lines in Figure 11.6 indicate ratios of 1.031, 1.026 and 1.083 respectively. Once again Subjects 1 and 2 are within the range considered acceptable by IDECG, but Subject 3 is well outside this range. This generates a danger of bias in the CO₂ production estimate as indicated in Section 5.7.

Similarly, deviation of the residuals for early time points away from zero indicates that ratios of volume calculated from intercepts are different from those calculated from early points. If this is the case, then curvature leading to bias may be the cause.

Finally, Product plots of d_O.d_D and their residuals (Figs 11.7 and 11.8) provide information about the constancy of water turnover. Again, good fits and small residuals indicate the absence of such fluctuations. These need not be important if CO₂ production is relatively constant, as indicated by the d_O/d_D plot. However, there is the possibility that curvature on the d_O and d_D plots, even if it is covariant, may produce biased estimates of N_O and N_D from the intercepts of the fitted lines. This is the possibility considered in Table 9.2 and discussed in Chapter 9.

In Figure 11.8 residuals for time points early in the disappearance curve for Subjects 1 and 2 indicate that N_O x N_D calculated from the early time points is 4% smaller than N_O x N_D calculated from the intercept (residual = +0.04). Provided N_D/N_O is near the expected value of 1.03 this represents a difference of 2% between values for CO₂ production calculated from intercept isotope distribution spaces and calculated using early values for N_O and N_D. This degree of bias is small enough to be of no concern, but users of the methodology should always check for bias of this type in particular groups of subjects or particular experimental protocols associated with them, in case it leads to erroneous conclusions. In other words some bias on individual subjects is to be expected, but the bias should be randomly distributed about zero for groups of subjects.

11.3.6 Further calculation of results

Table 11.2 contains further intermediate results for the 3 worked examples. Pool spaces (moles) are calculated as the reciprocal of the zero-time intercept derived from the chosen fitting procedure. Rate constants (d^-1) are the slope of the disappearance curves. The percentage differences between results from the different fitting procedures are listed. Considering each subject separately, the following observations can be made:

Subject 1

The data are well-behaved in all respects. Using simple fitting procedures (ie not using the product-ratio method), the standard errors for estimates of pool spaces average about 0.5%, and for rate constants about 0.8 - 1.0%. None of the pool sizes or rate constants differ by more than 1% when calculated using the 3 different fitting procedures, and N_Ok_O - N_Dk_D (ie 2r'_CO2) differs by only 0.71% between the 3 procedures. Note that the estimate of k_rA_r + k_pA_p obtained from the product-ratio method (equivalent to N_Ok_O - N_Dk_D) is always the same as that obtained from the log plot with the exception of minor rounding-error differences. The variance calculated from the product-ratio method is ± 1.3%.

Taken together with the tight residual plots, almost perfect covariance between the two isotopes and acceptable N_D/N_O value of close to 1.03, the results represent a model case and can be considered very secure.

Subject 2

The data are moderately well-behaved. Using simple fitting procedures, the standard errors for estimates of pool spaces are higher than for Subject 1 and are 0.6 - 1.0%. The same is true for the errors on the rate constants which are between 0.8 and 1.5%. The answers obtained from the different fitting procedures are quite similar, the largest difference being 2.45%. However, when calculated through to N_Ok_O - N_Dk_D the log and Poisson methods differ by 2.40%, but the exponential and log methods differ by 8.24%. The variance calculated from the product-ratio method is ± 3.98%.

The wider standard errors on the estimates of pool size and rate constants propagate through to the higher final estimate of error of about ± 4. This is still quite acceptable and is backed up by the existence of an acceptable N_D/N_O ratio of about 1.035, and respectable residual plots. The only cause for concern is that the exponential fit gives a higher answer (+ 6-8%) than the other two fitting procedures. However, the residuals give no a priori evidence that an exponential fit is required, and the good agreement between the other two fitting procedures suggest that they are preferable the Poisson fit would represent a sensible compromise solution.

Subject 3

Inspection of any of the data plots (Figs 11.1 - 11.8) immediately shows that the data from this subject are very variable. Using the simple fitting procedures the standard errors for estimates of pool spaces vary between 1.8 and 3.3%, and for rate constants between 2.9 and 3.6%. The pool sizes and rate constants differ by up to 4.2% when calculated using the 3 different fitting procedures, but N_Ok_O - N_Dk_D differs by only 1.91% between the 3 procedures. This contrasts with the 8.24% difference for this parameter in Subject 2 and serves to illustrate that a difference in derived parameters between the 3 fitting methods is not a reliable indicator or data quality. The variance calculated from the product-ratio method is ± 7.8%.

When interpreting the results of this analysis it is important to consider them in the context of the experiment. In Subject 3 the starting enrichment of ¹⁸O was lower than that for Subjects 1 and 2. This in itself would give rise to greater variance on the derived parameters. Also, Subject 3 was an athlete in training who exhibited large daily changes in both r_CO2 and r_H2O when these were assessed by independent measurements. Therefore the standard errors on the derived parameters and a variance of ± 7.8% on r_CO2 may well be reasonable. Indeed this estimate is within the recommended cutoff of 8% for the two-point method (Section 11.2.2). However, the magnitude of the residuals from the ratio plot and, more importantly the large pool size ratio, indicate that there may be an analytical problem with this data, and that the samples and the dose should be reanalysed before proceeding.

11.3.7 Incorporation of fractionation corrections and conversion of r_CO2 into energy expenditure

Full guidance for converting r'_CO2 into r_CO2 is provided in Chapters 4, 5 and 6. Further conversion into energy expenditure is described in Chapter 9.

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