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Chapter 6: Isotope fractionation corrections

6.1 Isotope fractionation
6.2 Experimental determinations of fractionation factors in vivo
6.3 Incorporation into calculations
6.4 Potential errors in r_CO2
6.5 Estimation of fractionated water loss
6.6 References

Contributors:

Dale Schoeller
Andy Coward

6.1 Isotope fractionation

6.1.1 General Terminology

Isotope fractionation is the physical phenomenon which causes changes in the relative abundance of isotopes due to their differences in mass. There are two categories of isotope effects: equilibrium and kinetic.

An equilibrium isotope effect will cause one isotope to concentrate in one component of a reversible system that is in equilibrium. If it is theta heavier isotope that concentrates in the component of interest, then that component is commonly referred to as enriched or heavy. If it is the light isotope that concentrates then the component is referred to as depleted or light. In most circumstances the heavy isotope concentrates in the component in which the element is bound more strongly and thus equilibrium isotope effects usually reflect relative differences in the bond strengths of the isotopes in the various components of the system.

A kinetic isotope effect occurs when one isotope reacts more rapidly than the other in an irreversible system or a system in which the products are swept away from the reactants before they have an opportunity to come to equilibrium. Normally, the lighter isotope will react more rapidly than the heavy isotope and thus the product will be lighter than the reactant.

It should be noted that isotope fractionation will only occur in systems in which there is both an isotope effect and a reaction that does not proceed to completion. Thus, even in the presence of an isotope effect, there will be no isotope fractionation if all the reactant goes to a single product because all the atoms have reacted and thus the ratio of the heavy to light isotope must be the same in the product as it was in the reactant.

The magnitude of an isotope effect is expressed as a fractionation factor. This is defined as the ratio of the heavy to light isotope in the product divided by the ratio of the heavy to light isotope in the reactant. Among users of doubly-labelled water the fractionation factor has generally been represented by the symbol "f". Other common symbols that have been used by other communities of users include a and f.

Stated mathematically:

When f is greater than 1, the product is heavy or enriched. When it is less than 1, the product is light or depleted. Most fractionation factors lie between 0.9 and 1.1, but deuterium isotope effects can result in much smaller or larger fractionation factors. A fractionation factor of 1.050 is often referred to as a 5% isotope effect.

For further basic information on isotope fractionation, the reader is referred elsewhere ^{1, 2}.

6.1.2 Influence of isotope fractionation in vivo

When stable isotopes are used as tracers for in vivo metabolic studies, an investigator is typically measuring the tracer flux and making inferences about the flux of the tracee (i.e. the material being traced). When there is no isotope fractionation, the flux of the heavy tracer is equal to the flux of the lighter major isotope and hence the tracee flux. The rates derived from the isotope kinetic analysis are thus exactly equal to the material flux. When there is isotope fractionation, however, the flux of the heavy tracer and the tracee are not equal and thus flux derived from the kinetic analysis of the tracer is not equal to that of the tracee. For example, ¹⁸O is fractionated between water and carbon dioxide. The fractionation factor is 1.037 at 37°C. Thus, the rate of removal of ¹⁸O by CO₂ is 3.7% greater than the rate of ¹⁶O removal. Because ¹⁶O comprises 99.8% of the oxygen pool, the rate of carbon dioxide production is essentially equal to the rate of ¹⁶O removal by CO₂. Therefore, the true rate of CO₂ production will not be equal to that measured from ¹⁸O, but will be 3.7% less. In other words, isotope fractionation leads to an error in the calculated tracee flux, unless the tracer rate is corrected for the fractionation.

6.1.3 Isotope fractionations influencing doubly-labelled water

One of the basic assumptions of the doubly-labelled water method is that isotope only exits the body water pool as water and carbon dioxide. The processes of interest are therefore the exchange of ¹⁸O between water and carbon dioxide and the elimination of water as vapour or liquid.

The isotopic abundances of plasma water, carbon dioxide, water vapour, urine and other physiologic samples for a subject living in the Chicago metropolitan area are shown in Figure 6.1. As can be seen, there is little or no isotope fractionation with respect to deuterium or ¹⁸O for urine or sweat, which are excreted as liquids. Water vapour, however, is isotopically fractionated with respect to both isotopes and carbon dioxide is fractionated with respect to ¹⁸O. As a matter of convention factors for these three isotope fractionations have been termed f₁, f₂, and f₃, where:

6.2 Experimental determinations of fractionation factors in vivo

6.2.1 Fractionation factors measured in vivo

In the original work by Lifson and coworkers ³, the values of the fractionation factors were taken from in vitro experiments performed at 25°C. In what was probably the first attempt to measure the values in vivo, Nagy ⁴ and Nagy and Costa ⁵ measured the change in isotopic abundance of tritium and ¹⁸O in a toad during progressive dehydration. The results did not support Lifson's contention of in vivo isotope fractionation. However, neither they did they constitute a disproof because the isotope measurements were of only moderate precision. More recently, investigators have taken advantage of the greater precision of gas isotope ratio mass spectrometry to measure f₁, f₂, and f₃ in humans under steady-state conditions ^6-9. In each of these four studies, samples of excreted water vapour, carbon dioxide and/or sweat were quantitatively collected and isotopically compared with either plasma water or urine. In each study samples were expressed in per mil relative to a standard or plasma water. It can be shown from the definition of per mil and the definition of fractionation factor, that the latter can be calculated as:

f = (product + 1000)/(reactant + 10000)

The observed values of f₁, f₂ and f₃ are summarised in Table 6.1. Where there are values from more than one study, they are quite similar. In the case of f₃ a small effect was noted with increasing exercise intensity ⁷. Because this was consistent with the predicted effect for a small increase in temperature, the authors suggested that it simply reflected the increase in core temperature with increasing exercise intensity.

The observed values for the fractionation factors are quite similar to those that have been measured in vitro and adjusted to body temperature (Table 6.1). This is particularly true for breath carbon dioxide and breath water, which are nearly identical to the in vitro values for an equilibrium effect. Water collected from the arm, however, is suggestive of a kinetic isotope effect rather than an equilibrium isotope effect. The value, however, from the total water collected from the arm indicates that even at rest, water loss is a mixture of unfractionated sweat losses and fractionated transcutaneous water loss. When the fractionation factor (as listed in Table 6.1) is recalculated for only the estimated transcutaneous water loss, the value is quite similar to the value determined in vitro. It should be noted that these in vitro values for kinetic isotope effects are not known as precisely as the equilibrium fractionation factors and thus a range is included in Table 6.1.

Table 6.1. Isotope fractionation factors

^a Range 0.917 - 0.93

Recent studies at the Rowett laboratory ¹⁰ have indicated that water lost from the skin during exercise may not behave as a combination of unfractionated sweat loss plus fractionated transcutaneous water loss (Table 6.2). They observed that during a 20 minute period of exercise at 75% VO_2max the entire water vapour lost from the arm (sweat plus transcutaneous) was kinetically fractionated.

Because f₁ and f₂ differ for equilibrium and kinetic fractionation effects, there has been some concern about which value is most appropriate for application in doubly-labelled water studies. Schoeller et al ⁸ have suggested the use of a weighted average, but in truth the use of one or other has little influence on final doubly-labelled water results. This is because the fractionation correction for water vapour loss depends on the difference between f₁ and f₂ rather than on the absolute values. This difference is almost identical for an equilibrium isotope effect (0.992 - 0.941 = 0.051) as for a kinetic isotope effect (0.976 - 0.923 = 0.053).

Because values for the fractionation factors measured in vivo do not differ from literature values, it is recommended that the literature be used because they are based on more extensive and precise measurements.

6.3 Incorporation into calculations

As stated above, isotopic fractionation introduces inequality into the relationship between the kinetics of the isotopic tracer and the tracee. As such, the calculation of CO₂ production rate from the isotopic data must include a correction term for fractionation.

Table 6.2. Isotope fractionation factors at rest and while working at 75% VO_2max

** p<0.01, SE in parentheses.

Data supplied by the Rowett group.

Three more or less equivalent equations have been suggested. Originally it was assumed that a constant proportion of true water output (x) could be regarded as being fractionated ³. In which case:

.........1

and

.........2

From Equation 1:

and substitution in Equation 2 gives:

and

.........3

which can be approximated by:

.........3a

(As f₂>f₁ the effect of fractionation is equivalent to increasing k_D or N_D)

Schoeller ¹⁵ recommends another approach which involves different terminology, and removes the need to supply a value for x in Equation 3. The routes of water loss are separated into non fractionated (r_H2O1) and fractionated routes (r_H2Of):

.........4

and

.........5

Substitution for r_H2O1 in the second equation and rearrangement to solve for r_CO2 yields:

.........6

Using the literature values for the fractionation factors for 37°C for human studies in which the subject has a normal body temperature yields:

.........7

A third elaboration proposed by Coward ¹⁶ further divided fractionated water loss into that occurring as respiratory loss (r_r) and those skin losses (r_s) that are fractionated. Arithmetic analogous to Equations 4 and 5 generates:

.........8

It is then assumed that there is a constant relationship between respiratory water loss and CO₂ loss (r_r = qr_CO2) so rearranging Equation 8 gives:

.........9

Values suggested for q and r_s are 1.1 mole water/mole CO₂ and about 27 mole/day for normal adults in a temperate climate. Thus:

.........10

Substitution of typical values for a human adult (K_ON_O - k_DN_D = 36 mole/day, f₃ = 1.04, f₂ - f₁ = 0.052) yields values of 16.4 from Equation 7 and 16.2 mole/day from Equation 10. The close agreement between the equations in terms of the result is somewhat coincidental because rather different relationships between respiratory and skin water losses are assumed to exist in the establishment of the constants.

6.4 Potential errors in r_CO2

Inspection of the above equation indicates a practical problem, which is that some estimate of the rate of fractionated water loss is required for the calculation of CO₂ production. In the vast majority of applications of doubly-labelled water, it is impossible to know the exact rate of fractionated water loss. The error made in estimating the rate of fractionated water loss will therefore introduce some error or uncertainty into the calculated rate of carbon dioxide production.

The potential error is presented graphically in Figure 6.2 for virtually any potential case in human studies. For healthy adults, the potential range of uncertainty is rather small. Indeed, for a typical adult living in a moderate climate (N_Ok_O/N_Dk_D = 0.095/0.07 = 1.35), the relative rate of fractionated water loss only needs to be estimated to the nearest 15% of total water output to reduce the uncertainty in the calculated carbon dioxide production rate to less than 2%. Potential for error in neonates is larger, but the largest potential error probably occurs in burn patients who may have a very large water turnover and hence a large deuterium elimination relative to the rate of oxygen elimination (N_Ok_O/N_Dk_D = 0.33/0.30 = 1.10). In burn patients, the relative rate of fractionated water loss must be known to the nearest 5% to reduce the uncertainty in the calculated carbon dioxide output rate to less than 2% (see Chapter 13 for further consideration of this point).

Curves represent different levels of error (5, 10 and 20%) in the proportion of water turnover assumed to be fractionated.

Hatching shows the range within which most measurements will fall.

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