Due to the extra complexities inherent in multipoint 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}), predose 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 datapoints in question remain outliers after reanalysis 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 10^{7}) 

^{18}O 
^{2}H 
^{18}O 
^{2}H 

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 (^{18}O/^{16}O = 0.0020052, ^{2}H/^{1}H = 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 ^{18}O: 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
Time 
d 
Normalised enrichment 

(d) 
(‰) 
(fraction of dose x 10^{7}) 

^{18}O 
^{2}H 
^{18}O 
^{2}H 

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 (^{18}O/^{16}O = 0.0020052, ^{2}H/^{1}H =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 ^{18}O: 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
Time 
d 
Normalised enrichment 

(d) 
(‰) 
(fraction of dose x 10^{7}) 

^{18}O 
^{2}H 
^{18}O 
^{2}H 

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 (^{18}O/^{16}O = 0.0020052, ^{2}H/^{1}H = 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 ^{18}O: 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.
Log 
Poisson 
Diff (%) 
Exp'l 
Diff (%) 

N_{O} 
2450.60 
2444.19 
0.26 
2437.58 
0.53 

(CV%) 
0.71 
0.59 
0.52 

N_{D} 
2527.81 
2523.72 
0.16 
2519.38 
0.33 

(CV%) 
0.56 
0.50 
0.45 

N_{D}/N_{O} 
1.0264 
1.0325 
+0.59 
1.0336 
+0.70 

k_{O} 
0.10775 
0.10818 
+0.40 
0.10883 
+1.00 

(CV%) 
0.82 
0.88 
1.03 

k_{D} 
0.08282 
0.08307 
+0.31 
0.08343 
+0.74 

(CV%) 
0.84 
0.90 
1.01 

N_{O}k_{O} 
264.042 
264.403 
+0.14 
265.282 
+0.47 

N_{D}k_{D} 
209.341 
209.643 
+0.14 
210.189 
+ 0.41 

N_{O}k_{O}  N_{D}k_{D} 
54.702 
54.760 
+0.11 
55.093 
+0.71 

k_{p} 
0.19056 

(CV%) 
0.81 

I_{p} 
(x 10^{7}) 
1.61429 

(CV%) 
1.24 

k_{r} 
0.02493 

(CV%) 
1.72 

I_{r} 
1.03152 

(CV%) 
0.34 

k_{r}A_{r} + k_{p}A_{p} 
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 (%) 

N_{O} 
2639.91 
2620.24 
0.75 
2602.66 
1.41 

(CV%) 
1.02 
0.92 
0.80 

N_{D} 
2711.05 
2712.46 
+0.05 
2711.62 
+0.02 

(CV%) 
0.68 
0.62 
0.57 

N_{D}/N_{O} 
1.0269 
1.0352 
+0.81 
1.0419 
+1.46 

k_{O} 
0.13494 
0.13630 
+1.01 
0.13824 
+2.45 

(CV%) 
0.96 
1.19 
1.48 

k_{D} 
0.11489 
0.11477 
0.04 
0.11482 
0.06 

(CV%) 
0.76 
0.90 
1.12 

N_{O}k_{O} 
356.230 
357.139 
+0.26 
359.792 
+1.00 

N_{D}k_{D} 
311.471 
311.309 
0.05 
311.348 
0.04 

N_{O}k_{O}  N_{D}k_{D} 
44.758 
45.830 
+2.40 
48.444 
+8.24 

k_{p} 
0. 24983 

(CV%) 
0.76 

I_{p} 
(x 10^{7}) 
1.39724 

(CV%) 
1.50 

k_{r} 
0.02005 

(CV%) 
5.59 

I_{r} 
1.02694 

(CV%) 
0.88 

k_{r}A_{r} + k_{p}A_{p} 
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 (%) 

N_{O} 
1708.04 
1685.64 
1.31 
1650.94 
3.34 

(CV%) 
3.33 
2.80 
2.42 

N_{D} 
1856.41 
1845.50 
0.59 
1838.86 
0.95 

(CV%) 
2.16 
1.94 
1.80 

N_{D}/N_{O} 
1.0869 
1.0948 
+0.73 
1.1138 
+2.47 

k_{O} 
0.13415 
0.13580 
+1.23 
0.13978 
+4.20 

(CV%) 
2.91 
3.14 
3.55 

k_{D} 
0.08920 
0.08986 
+0.74 
0.09028 
+1.21 

(CV%) 
2.84 
2.99 
3.33 

N_{O}k_{O} 
229.134 
228.910 
0.10 
230.768 
+0.71 

N_{D}k_{D} 
165.592 
165.837 
+0.15 
166.012 
+0.25 

N_{O}k_{O}  N_{D}k_{D} 
63.542 
63.073 
0.74 
64.756 
+1.91 

k_{p} 
0.22335 

(CV%) 
1.91 

I_{p} 
(x 10^{7}) 
3.15376 

(CV%) 
1.24 

k_{r} 
0.04495 

(CV%) 
11.17 

I_{r} 
1.08676 

(CV%) 
4.27 

k_{r}A_{r} + k_{p}A_{p} 
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.
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 ^{2}H and ^{18}O 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 reanalysing 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 CO_{2} 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 CO_{2} 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 CO_{2} 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 CO_{2} 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 zerotime 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 wellbehaved in all respects. Using simple fitting procedures (ie not using the productratio 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_{O}k_{O}  N_{D}k_{D} (ie 2r'_{CO2}) differs by only 0.71% between the 3 procedures. Note that the estimate of k_{r}A_{r} + k_{p}A_{p} obtained from the productratio method (equivalent to N_{O}k_{O}  N_{D}k_{D}) is always the same as that obtained from the log plot with the exception of minor roundingerror differences. The variance calculated from the productratio 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 wellbehaved. 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_{O}k_{O}  N_{D}k_{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 productratio 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 (+ 68%) 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_{O}k_{O}  N_{D}k_{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 productratio 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 ^{18}O 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 twopoint 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.