Implications of the variation in biological 18O natural abundance in body water to inform use of Bayesian methods for modelling total energy expenditure when using doubly labelled water

Rationale Variation in 18O natural abundance can lead to errors in the calculation of total energy expenditure (TEE) when using the doubly labelled water (DLW) method. The use of Bayesian statistics allows a distribution to be assigned to 18O natural abundance, thus allowing a best‐fit value to be used in the calculation. The aim of this study was to calculate within‐subject variation in 18O natural abundance and apply this to our original working model for TEE calculation. Methods Urine samples from a cohort of 99 women, dosed with 50 g of 20% 2H2O, undertaking a 14‐day breast milk intake protocol, were analysed for 18O. The within‐subject variance was calculated and applied to a Bayesian model for the calculation of TEE in a separate cohort of 36 women. This cohort of 36 women had taken part in a DLW study and had been dosed with 80 mg/kg body weight 2H2O and 150 mg/kg body weight H2 18O. Results The average change in the δ18O value from the 99 women was 1.14‰ (0.77) [0.99, 1.29], with the average within‐subject 18O natural abundance variance being 0.13‰2 (0.25) [0.08, 0.18]. There were no significant differences in TEE (9745 (1414), 9804 (1460) and 9789 (1455) kJ/day, non‐Bayesian, Bluck Bayesian and modified Bayesian models, respectively) between methods. Conclusions Our findings demonstrate that using a reduced natural variation in 18O as calculated from a population does not impact significantly on the calculation of TEE in our model. It may therefore be more conservative to allow a larger variance to account for individual extremes.


| INTRODUCTION
The doubly labelled water (DLW) method is considered to be the "gold standard" for measuring free-living total energy expenditure (TEE) in humans. 1,2 A bolus dose of 2 H 2 18 O is given and the dilution spaces and rate constants for 2 H and 18 O are calculated. Carbon dioxide (CO 2 ) production is calculated from the difference in the elimination rates, with the 2 H being lost as water, and the 18 O as both water and CO 2 via the bicarbonate-water exchange in the blood. TEE is then estimated from CO 2 production ( R CO2 ) and the To calculate the elimination rates of 2 H or 18 O, it is first necessary to account for the natural abundance of isotope already present in the system. For the DLW method, it is typical to obtain a single pre-dose sample, which can be plasma, saliva or urine, prior to the experiment, and take this as representative of the natural abundance throughout the measurement period. For ease of collection, this is commonly urine. The underlying assumption of the DLW method is that the natural abundances of both isotopes remain unchanged over the period of measurement. This is a consequence of two of the assumptions of the method; first that water leaves the body unfractionated and secondly that the intake is at the same isotopic enrichment as the body water. [5][6][7] While these assumptions are known to be untrue, [7][8][9] they are generally accepted, as it is not possible to directly measure natural abundance for either isotope during the measurement period.
Therefore, either the natural abundance must be assumed to be unchanged or indirect methods must be used to overcome the likely variation.
To date there have been four such indirect methods: (1) dosing an individual to result in an optimal ratio between the two isotopes at the end-period of the measurement. This has been shown to reduce the error due to natural variation by matching the slope of covariance between the isotopes. However, it is dependent on the size of the analytical error. 9 Hence ideal ratios have varied between 6:1 and 12:1 delta values per mil of 2 H to 18 O. Whilst this takes into account the variation over the DLW period for the post-dose samples, it still assumes that the measured pre-dose value is a representative value in the calculation of TEE. (2) As an alternative to method (1), the use of a highly enriched DLW dose would mask the variation in natural abundance. 10 However, this is an expensive method, which may be further complicated by concerns of accuracy in measuring such high enrichments and, as a result, it has not been utilised frequently. 11 (3) Another proposal has been to use the natural variation in undosed participants to give a proxy of the natural variation within the dosed participant. 12,13 However, it has recently been shown that there is no inter-individual correlation in time that would allow for this. 14 (4) Interestingly, Berman et al 14  In the absence of any practical method to determine natural variation during a DLW experiment, here we investigate the use of modelling software to allow the natural abundance to vary from the measured value in the calculations to a best-fit value over the period of measurement. This paper looks at calculating TEE using a Bayesian model in the free software WinBUGS. 15 The WinBUGS software has been applied successfully to a wide range of physiological models, from gastric emptying 16 to insulin sensitivity and the glucose minimal model. 17 Bayesian methods allow for the incorporation of a priori knowledge (often referred to as priors) into the model and for uncertainty to be quantified; this is then modelled with the existing data to produce posterior probability distributions for the parameters of interest. For TEE there are a number of parameters for which prior knowledge is available, e.g. R CO2 must be greater than zero and the fraction of body fat must lie between zero and one. The priors given may be informative or vague (otherwise known as non-informative) depending on what is known about the probable distributions or how reliant the model is upon observed information. Within our model, tauO defines the variance for the distribution of 18

| Participants
Participant data used for this study came from two previous cohorts.
The first cohort was 99 UK women (Table 1) originally recruited as part of a breast milk intake study for the Diet and Nutrition Survey of Infants and Young Children (DNSIYC). The second cohort was 36 UK women (Table 1)

| General design
Within-subject variance in 18

| Calculations
All data considered in this paper are expressed in ‰ with respect to Vienna Standard Mean Ocean Water (vSMOW) on the delta scale: where R samp is the 18

| Total energy expenditure
Rate constants and dilution spaces are calculated from the slopes and intercepts of the log-transformed data, with the rate of CO 2 production, R CO2 given by: 7 WinBUGS. 19 Parameter priors were assigned to the following: CO 2 production rate, R CO2 ; space ratio, S; water turnover, R W ; and fraction of body fat, F . The priors were vague with the following distributions given: for R CO2 , a uniform distribution between 0 and 100 mol/day; for R W , a uniform distribution between 0 and 1000 mol/day; and, for F , a uniform distribution between 0 and 1. However, the prior S was given to be informative and assigned a normal distribution with a mean of 1.035 and standard deviation of 0.01.
The pool sizes and rate constants for H and O were described in terms of R CO2 , S, R W , F and body weight, with these described in the kinetic calculations for first-order disappearance.
The within-subject variance as calculated from the DNSIYC cohort was used to modify the basal 18

| Statistical analysis
The primary outcome measurement was the total energy expenditure (TEE) determined using the non-Bayesian, Bluck Bayesian and modified Bayesian models. Secondary outcome measurements were S and R CO2 (mol/day). Data analysis was performed using IBM SPSS Statistics for Windows, version 22.0 (IBM Corp., Armonk, NY, USA).
The data are presented as means and standard deviation with 95% confidence intervals, and were checked for normality using the Kolmogorov-Smirnov test. To compare potential differences in the TEE calculated using the three methods, a one-way repeated measures ANOVA was conducted. Agreement between the two Bayesian models was assessed using Bland-Altman plots with significance assessed using Student's t-test. The level of significance was set at P <0.05.

| Participants
The women from both the DNSIYC and the NDNS cohorts matched for all variables except for age. It can be seen that the women from the NDNS cohort were significantly older than those from the DNSIYC cohort (Table 1). The calculated variable of N H was not significantly different between cohorts; however, k H was 10% higher in the DNSIYC than in the NDNS cohort. Bayesian model was set to 4 and the modified value was set to 7.6. There were no significant differences in S, R CO2 or TEE when calculated using each of the three methods ( Table 2). There is a significant correlation between the two Bayesian models (Figure 2A

| DISCUSSION
The aim of the present study was to quantify natural abundance variation in 18 O within a cohort of UK women, and incorporate this into our working Bayesian model to allow for a more robust determination of total energy expenditure.
The observed data shows that there are no differences between TEE for our NDNS cohort when calculated using either the Bluck or the modified Bayesian model. This would suggest that the Bluck Bayesian model has sufficient ability to allow for 18 O variation in the model, so restricting tauO is unnecessary.
Typically, in non-Bayesian methods of calculating TEE, the largest proportion of the error of the TEE estimate comes from natural abundance variation. 26 It can be seen from our reported NDNS data that, when TEE is calculated using non-Bayesian methods, the total error is 4.77 ± 1.29% as calculated according to Ritz et al;26 of this, the error arising from natural abundance variation is 4.36 ± 1.22%. This total error is comparable with that found in other studies. 26,27 It is calculated using regression statistics on the isotope enrichments and their products and ratios to calculate internal precision and, in addition, makes assumptions regarding the associated error of the single pre-

| Limitations and future work
Both cohorts were subsets of nationally representative surveys; however, neither subset has been chosen to be nationally representative and as such may be biased geographically. Darling et al 28 reported that the isotopic composition of the UK groundwater varies depending on location within the UK and, as water source does have an effect on the isotopic natural abundance of total body water, further work in this area would be of interest and could include analysis regarding natural abundance variation across the UK and even further afield.
Although we are aware that the two studies were spaced approximately 10 years apart, this seems unlikely to matter as the same instrument and methods were used for the 18 O analysis.
The DNSIYC cohort are younger than the NDNS cohort and it has been reported that water turnover is affected by age in children. 29 However, our own data in adults (unreported NDNS Y1 and 3 30 ) show that water turnover increases with age from about 20 years to 50 years of age, and so we would expect that the NDNS cohort would have greater water turnover if it was not for the fact that the DNSIYC cohort were breast-feeding. Levels of breast-feeding varied considerably from almost none to exclusively breast-feeding, as the average age of the infants was 11 months. We have previously observed an increased rate of water turnover in our laboratory for breast-feeding women (unpublished data), which in turn may impact natural abundance 18 O and 2 H variation. It is possible, therefore, that the increased water turnover observed in the DNSIYC cohort would result in swifter and more visible changes in natural abundance variation; however, this remains to be investigated.

| CONCLUSIONS
The application of Bayesian methods is a superior methodology to calculate total energy expenditure (TEE) when a single pre-dose has been taken, due to the ability to assign probability distributions to the known parameters. We sought to calculate 18