Two procedures are commonly used for
determining flux rates. These have become known as the two-point and multi-point methods.
It may often be supposed that these procedures are basically the same, the two-point
procedure being an abbreviated multi-point method. This is not, however, the case for
although the underlying assumptions outlined in Section 4.2 apply in each case, the
two-point procedure is more tolerant of deviations from the model than the multi-point
method, essentially because the two-point method measures __total flux__ between two
points several days apart whereas the multi-point method attempts to provide a __mean
value__ for daily flux rates over the same time interval.

__4.4.1 Two-point method__

If N remains constant during an
experimental period total outflow r' is correctly calculated for a period between t_{1}
and t_{t} as:

_{}

..........5

where the subscripts refer to times 1 and t and pre-dose (p).

This is true even if the rate of outflow per unit time is not constant and varies in a random fashion or systematically. Imagine, for example, that r' measured over successive similar time intervals varies. In this case total flow for the period 1 to t is given by:

_{}

or

_{}

..........6

Cross-cancellation gives: on gives:

_{}

.........7

The two-point methodology is also
resilient when N changes with time as a consequence of r_{in} not equalling r_{out}.
Coward *et al* ^{7} have shown that if r_{in} does not equal r_{out},
but both are constant so that N changes linearly, or if r_{in}(t)/N(t) and r_{out}(t)/N(t)
are unequal but constant and N thus changes exponentially, total outflow is given by:

_{}

.........8

(A full derivation of this Equation
can be found in Coward *et al* ^{7} and an alternative version is presented
in Appendix 3.)

Because daily variation in the slope of the line between the two points does not matter in the two-point method obtaining the correct answer for the overall slope is entirely a matter of analytical accuracy and precision.

__4.4.2 Multi-point
(slope-intercept) method__

The multi-point method is fundamentally different from the two-point procedure because the basic intention is to obtain a mean value for flux rate with a standard deviation indicating the extent of random physiological variation.

Pool spaces (N) are measured from
the zero-time intercept of a plot of isotopic enrichment versus time. The most important
advantage of this manoeuvre is that it reduces the error on calculation of the products k_{O}N_{O}
and k_{D}N_{D} because of the covariance of slopes and intercepts. For
example, an error tending to make a slope steeper will increase the value of the intercept
and consequently reduce N. Allowance can also be made for covariance between errors in the
^{2}H and ^{18}O data in the determination of the SE of the estimate.
These aspects are covered more fully in Chapter 5. Additionally random analytical error
becomes less important than it is in the two-point method. This aspect is also covered
fully in Chapter 5.

There are many possible ways of handling multi-point data. Three of these were explored in detail during the workshop and in the data analysis exercise leading up to it. They are summarised below. A fourth variant (the Product-Ratio Method) was developed by Cole and Coward and introduced at the workshop. It gives the same answer as the simple log fit described below, but provides a better estimate of error and the most informative way of inspecting the raw data. It is described in full in Chapter 5.

__4.4.2.i Curve fitting to
multi-point data__

In order to obtain values for rate constants and intercepts (or volumes) using multi-point: data it is necessary to fit an appropriate curve to the data which describes the mean decay over time. Different fitting procedures attach more or less importance to individual points and the choice of the most appropriate treatment should ideally be based on knowledge of the system's error structure. At extreme ends of a range of possibilities are: a) that errors are proportional to the level of enrichment; or b) that the errors have a constant absolute value and are not dependent on enrichment. The first case might be represented by physiological variations such as random changes in water turnover. The second case could arise from constant analytical errors. In practice however it could be suggested that an ideal fitting procedure would be one that was appropriate to an error structure lying somewhere between the two extremes. A detailed discussion of this problem is given in Appendix 4, and further coverage appears in Chapter 5. However a few general statements can be made at this juncture and the following sections give a simple summary of 3 different data-fitting techniques and guidance as to which one is appropriate under different error conditions.

__4.4.2.ii Log transformation__

This procedure is the simplest that
can be used since any computer package or calculator programme will fit the best straight
line through log transformed data. The method assumes that errors are proportional; that
is to say that residuals (differences between observed points and the fitted enrichment
curve) become smaller as enrichment decreases or, in other words, are constant in size
relative to enrichment. This fitting procedure is thus appropriate for errors described in
case (a) above and has been used by most groups ^{8, 12-15}. The curve fitting
procedure described by Feldman ^{16} is used by the Cambridge group.

__4.4.2.iii Exponential fit__

Haggarty *et al* ^{17}
at the Rowett have estimated volumes and rate constants by fitting exponentials directly
to untransformed data. This approach was developed after inspecting residual plots which
indicated a constant error structure throughout the labelling period in the particular
group of subjects studied ^{18}. It was this process of scrutinising residual
plots which alerted these authors to the need for alternative fitting methods under
certain circumstances. In contrast to log transformation the exponential fit allows early
points (higher enrichment) to have a greater effect on the outcome of the fitting than
later points (lower enrichment). The curve fitting is not as readily available, but it is
found in a number of more sophisticated statistical packages for computers. These workers
use "Maximum Likelihood Program" (Numerical Algorithms Group, Oxford, UK) to fit
models to data.

__4.4.2.iv Poisson fit__

Franklin, also from the Rowett,
suggested a third "Poisson" model at the workshop which assumes that the error
structure lies somewhere between the constant: CV and constant SD situations covered
above. The simplest ways to achieve this are to use either a weighted linear regression
method or a 'generalised linear model' ^{19} with a logarithmic link function and
a Poisson error distribution coupled with a heterogeneity factor. Essentially this allows
the fitting of the log or exponential models with the standard deviation at each point
being proportional to the square root of the enrichment. "Maximum Likelihood
Program" also provides the facilities to carry out this type of analysis.

__4.4.2.v Choosing the
appropriate model__

Firstly, it should be re-emphasised
that with well-behaved data-sets it is virtually irrelevant as to which method of data
reduction is employed since they will yield very similar answers. The workshop therefore
recommended that users should adapt their software to calculate all data-sets in each of
the 3 possible ways. They can then predefine an acceptable level of agreement (say 3%). If
the 3 results agree to within this tolerance then the particular version favoured by that
laboratory can safely be used. If the results fail to satisfy this criteria then the data
must be carefully scrutinised.

1. Lifson N & McClintock R
(1966) Theory of the use of the turnover rates of body water for measuring energy
expenditure and material balance. **J Theoret Biol; 12:** 46-74

2. Shipley RA & Clark RE (1972) **Tracer
Methods for In Vivo Kinetics.** Academic Press, New York.

3. Schoeller DA, Ravussin E, Schutz
Y. Acheson KJ, Baertschi P & Jequier E. (1986) Energy expenditure by doubly-labelled
water: validation in humans and proposed calculations. **Am J Physiol; 250:** R823-R830

4. Roberts SB, Coward WA & Lucas
A (1987) Reply to letter by Wong, Butte, Garza and Klein. **Am J Clin Nutr; 45:**
1545-1547.

5. Culebras JM & Moore FD (1977)
Total body water and the exchangeable hydrogen in man. **Am J Physiol; 232:** R54-R59.

6. Coward WA (1988) The
doubly-labelled water (^{2}H_{2}^{18}O) method: principles and
practice. **Proc Nutr Soc; 47:** 209-218.

7. Coward WA, Cole TJ, Gerber H.
Roberts SB & Fleet, I (1982) Water turnover and the measurement of milk intake. **Pfleugers
Archiv; 393:** 344-347.

8. Coward WA, Prentice AM,
Murgatroyd PR et al (1984) Measurement of CO_{2} production rates in man using ^{2}H,
^{18}O-labelled H_{2}O: comparisons between calorimeter and isotope
values. In: Human energy metabolism: physical activity and energy expenditure measurements
in epidemiological research based upon direct and indirect calorimetry. **Euro-Nutr
Report no. 5** (A.J.H. van Es, editor). Den Haag: CIP-gegevens koninklijke, pp 126-128.

9. Roberts SB, Coward WA,
Schlingenseipen KH, Nohria VR & Lucas A (1986) Comparison of the doubly-labelled water
(^{2}H_{2}^{18}O) method with indirect calorimetry and a nutrient
balance study for simultaneous determination of energy expenditure, water intake and
metabolisable energy intake in pre-term infants. **Am J Clin Nutr; 44:** 315-322.

10. Wong WW, Butte NF, Smith EO,
Garza C & Klein PD (1989) Body composition of lactating women determined by
anthropometry and deuterium dilution. **Br J Nutr; 61:** 25-33.

11. Halliday D & Miller DG
(1977) Precise measurement of total body water using tracer quantities of deuterium oxide.
**Biomed Mass Spectrom; 4:** 82-87.

12. Feldman HA (1977) A numerical
method for fitting compartmental models directly to tracer data. **Am J Physiol; 233:**
R1 - R7.

13. Westerterp KR, Brouns F, Saris
WHM & Ten Hoor F (1988) Comparison of doubly labelled water with respirometry at low
and high-activity levels. **J Appl Physiol; 65:** 53-56.

14. Prentice AM, Coward WA, Davies
HL, Murgatroyd PR, Black AK, Goldberg GR, Ashford J, Sawyer M & Whitehead RG (1985)
Unexpectedly low levels of energy expenditure in healthy women. **Lancet; i:**
1419-1422.

15. Klein PD, James WPT, Wong WW,
Irving CS, Murgatroyd PR, Cabrera M, Dallosso HM, Klein ER & Nichols BL (1984)
Calorimetric validation of the doubly-labelled water method for determination of energy
expenditure in man. **Hum Nutr: Clin Nutr; 38C:** 95-106.

16. Roberts SB, Coward WA,
Schlingenseipen K-H, Nohria V & Lucas A (1986) Comparison of the doubly-labelled water
(^{2}H_{2}O-H_{2}^{18}O) method with indirect calorimetry
and a nutrient-balance study for simultaneous determinations of energy expenditure, water
intake and metabolisable energy intake in pre-term infants. **Am J Clin Nutr; 44:**
315-322.

17. Haggarty P, McGaw BA. &
Franklin MF (1988) Measurement of fractionated water loss and CO_{2} production
using triply labelled water. **J Theor Biol; 134:** 291-308.

18. James WPT, Haggarty P &
McGaw BA (1988) Recent progress in studies on energy expenditure: are new methods
providing answers to the old questions? **Proc Nutr Soc; 47:** 195-208.

19. Nelder JA & Wedderburn RWM
(1972) Generalised linear models. **J R Statist Soc A; 135:** 370-384.