## 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 (ds), pre-dose baseline (dp), dose (da) and tap water (dt); 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 (ND and NO) are derived simply as the reciprocal of the intercept (or plateau value). Similarly the intercept of the dO/dD plot is the ratio of the spaces ND/NO. Secondly, the procedure 'normalises' the results so that both dO and dD 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 dO and dD.

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

 Time d Normalised enrichment (d) (‰) (fraction of dose x 107) 18O 2H 18O 2H 0.232 177.4956 859.6359 4058 3946 0.525 169.9968 829.4619 3891 3811 1.400 153.6073 768.6668 3526 3539 2.000 140.2262 719.4943 3228 3319 3.000 128.0576 663.6164 2957 3069 4.000 111.5333 598.5746 2589 2778 5.000 101.6996 557.4485 2370 2594 6.000 90.2494 512.5227 2115 2393 7.300 80.1463 470.5026 1890 2205 8.000 73.0517 437.8699 1732 2059 9.100 65.4631 397.8614 1563 1880 10.100 55.2702 352.9356 1336 1679 11.000 51.4983 331.0315 1252 1581 12.600 42.1136 290.5759 1043 1400 12.900 40.6767 281.8590 1011 1361 13.900 36.9947 258.8373 929 1258

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: dp = -4.72, da = 181.55, dt = -7.64.

For ²H: dp = -22.34, da = 895.89, dt = -45.84.

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

 Time d Normalised enrichment (d) (‰) (fraction of dose x 107) 18O 2H 18O 2H 0.172 179.4361 582.7563 3811 3659 1.000 156.5229 513.5519 3336 3244 2.000 136.8417 462.8576 2928 2940 3.000 113.7356 400.3234 2446 2565 4.000 99.4571 360.9686 2153 2329 6.100 76.6886 287.0950 1681 1886 7.000 64.6290 245.9058 1431 1639 8.100 56.8144 215.0556 1269 1454 10.200 42.0535 164.3613 963 1150 11.100 36.8438 143.5166 855 1025 12.200 30.9104 123.3389 732 904 13.300 26.1831 105.8293 634 799

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: dp = -4.4, da = 163.23, dt = -7.64.

For ²H: dp = -27.41, da = 544.85, dt = -45.84.

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

 Time d Normalised enrichment (d) (‰) (fraction of dose x 107) 18O 2H 18O 2H 1.000 83.2588 450.4150 5458 5039 2.000 71.5347 375.0090 4703 4264 3.000 57.5124 379.1928 3800 4307 4.000 47.2791 331.1275 3141 3813 5.000 41.1919 312.0571 2749 3617 6.000 39.0801 266.5216 2613 3149 7.000 34.1265 234.9970 2294 2825 8.000 29.0486 202.1103 1967 2487 9.000 25.9119 184.0128 1765 2301 10.000 24.3124 170.5857 1662 2163 11.000 21.1135 164.7478 1456 2103 12.000 16.8898 143.2449 1184 1882 13.000 14.2499 129.4286 1014 1740 14.000 11.4393 109.2879 833 1533

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: dp = -1.496, da = 77.11, dt = -3.00.

For ²H: dp = -39.87, da = 487.95, dt = -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.

 Log Poisson Diff (%) Exp'l Diff (%) NO 2450.60 2444.19 -0.26 2437.58 -0.53 (CV%) 0.71 0.59 0.52 ND 2527.81 2523.72 -0.16 2519.38 -0.33 (CV%) 0.56 0.50 0.45 ND/NO 1.0264 1.0325 +0.59 1.0336 +0.70 kO 0.10775 0.10818 +0.40 0.10883 +1.00 (CV%) 0.82 0.88 1.03 kD 0.08282 0.08307 +0.31 0.08343 +0.74 (CV%) 0.84 0.90 1.01 NOkO 264.042 264.403 +0.14 265.282 +0.47 NDkD 209.341 209.643 +0.14 210.189 + 0.41 NOkO - NDkD 54.702 54.760 +0.11 55.093 +0.71 kp 0.19056 (CV%) 0.81 Ip (x 107) 1.61429 (CV%) 1.24 kr 0.02493 (CV%) 1.72 Ir 1.03152 (CV%) 0.34 krAr + kpAp 54. 700

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

 Log Poisson Diff (%) Exp'l Diff (%) NO 2639.91 2620.24 -0.75 2602.66 -1.41 (CV%) 1.02 0.92 0.80 ND 2711.05 2712.46 +0.05 2711.62 +0.02 (CV%) 0.68 0.62 0.57 ND/NO 1.0269 1.0352 +0.81 1.0419 +1.46 kO 0.13494 0.13630 +1.01 0.13824 +2.45 (CV%) 0.96 1.19 1.48 kD 0.11489 0.11477 -0.04 0.11482 -0.06 (CV%) 0.76 0.90 1.12 NOkO 356.230 357.139 +0.26 359.792 +1.00 NDkD 311.471 311.309 -0.05 311.348 -0.04 NOkO - NDkD 44.758 45.830 +2.40 48.444 +8.24 kp 0. 24983 (CV%) 0.76 Ip (x 107) 1.39724 (CV%) 1.50 kr 0.02005 (CV%) 5.59 Ir 1.02694 (CV%) 0.88 krAr + kpAp 44.761 (See Section 5.4)

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

 Log Poisson Diff (%) Exp'l Diff (%) NO 1708.04 1685.64 -1.31 1650.94 -3.34 (CV%) 3.33 2.80 2.42 ND 1856.41 1845.50 -0.59 1838.86 -0.95 (CV%) 2.16 1.94 1.80 ND/NO 1.0869 1.0948 +0.73 1.1138 +2.47 kO 0.13415 0.13580 +1.23 0.13978 +4.20 (CV%) 2.91 3.14 3.55 kD 0.08920 0.08986 +0.74 0.09028 +1.21 (CV%) 2.84 2.99 3.33 NOkO 229.134 228.910 -0.10 230.768 +0.71 NDkD 165.592 165.837 +0.15 166.012 +0.25 NOkO - NDkD 63.542 63.073 -0.74 64.756 +1.91 kp 0.22335 (CV%) 1.91 Ip (x 107) 3.15376 (CV%) 1.24 kr 0.04495 (CV%) 11.17 Ir 1.08676 (CV%) 4.27 krAr + kpAp 63.561 (See Section 5.4)

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

Difference columns list % offset compared to the Log fit.

Figure 11.1. Untransformed data plots Figure 11.2. Log transformed data plots Figure 11.3. Residuals from log fit Figure 11.4. Residuals from Poisson fit Figure 11.5. Residuals from exponential fit Figure 11.6. Log transformed ratio plots Figure 11.7. Log transformed product plots Figure 11.8. Residuals from log transformed ratio and product plots 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 (kD and kO) are represented by the slopes of the regression, and the pool sizes (ND and NO) 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 dO and dD 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, ND/NO 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 dO or dD 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 rH2O, and for rCO2 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 dO/dD 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 dO/dD plots is that their intercept on the Y axis directly indicates the ND/NO ratio. Thus an intercept of +0.03 on the log scale is equivalent to ND/NO = 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 dO.dD 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 dO/dD plot. However, there is the possibility that curvature on the dO and dD plots, even if it is covariant, may produce biased estimates of NO and ND 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 NO x ND calculated from the early time points is 4% smaller than NO x ND calculated from the intercept (residual = +0.04). Provided ND/NO 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 NO and ND. 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 NOkO - NDkD (ie 2r'CO2) differs by only 0.71% between the 3 procedures. Note that the estimate of krAr + kpAp obtained from the product-ratio method (equivalent to NOkO - NDkD) 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 ND/NO 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 NOkO - NDkD 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 ND/NO 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 NOkO - NDkD 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 rCO2 and rH2O when these were assessed by independent measurements. Therefore the standard errors on the derived parameters and a variance of ± 7.8% on rCO2 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 rCO2 into energy expenditure

Full guidance for converting r'CO2 into rCO2 is provided in Chapters 4, 5 and 6. Further conversion into energy expenditure is described in Chapter 9.