4.1 Introduction

4.2 The basic model

4.3 Calculation of pool sizes

4.4 Calculation of flux rates

4.5 References

Contributor: Andy Coward

This chapter will discuss the basic
theory underlying the isotope kinetic models employed in DLW studies, and will summaries
the various methods available for calculating pool spaces, disappearance rates and hence
CO_{2} production rates. It is written in the expectation that the reader will be
familiar with Lifson and McClintock's early work ^{1}, and the many publications
derivative of it (see Chapter 1). Anyone new to the general field of kinetic studies with
isotopes is also advised to have close to hand a text-book that explains; the theory and
techniques of compartmental analysis ². A brief introduction is also given in Appendix 1.

The public conception may be that there is considerable controversy about the appropriate methods of calculation and treatment of DLW data. Fortunately, close scrutiny of the subject indicates that this really is not the case and in fact there is a reasonable amount of common ground between the protagonists of two-point and multi-point methodologies. Identifying the common ground enables us to highlight the significance of areas of disagreement.

The first point to make is that with non-invasive tracer techniques we can only deal with what we observe and although such observations may lead us to the conclusion that a certain model provides an appropriate basis for the treatment of the data the observations do not prove that the model is a valid one. The model has to make sense from the physiological point of view and if the model is inappropriate the answers will be incorrect, although apparently precise.

A simple example will illustrate
this point. Suppose you are asked to calculate the rate of water-output from the only
apparent exit in a water tank. Common sense would suggest that the best way in which to do
this would be to directly measure the rate (r_{out}) at which water flows through
that exit and this will give the correct result. If on the other hand this measurement
cannot be made, an alternative method is to add a known amount of tracer to the tank,
measure the tank volume (N) from the instantaneous dilution of tracer, and the rate
constant for tracer exit (k) and calculate the outflow as Nk. If the tank has only one
exit r_{out} will equal Nk, but if there is an exit in the form of a leak, Nk will
be greater than r_{out} and if the presence of a leak is unsuspected r_{out}
will be incorrectly estimated because the model was wrong although it fitted the data! The
analogy should be evident. Direct measurement of r_{out} and Nk are both available
when respirometry is combined with a doubly-labelled water study (as, for instance, during
cross-validation studies), but in field applications of DLW the equivalence of these two
values can never be checked. Our models and treatment of data must therefore be secure for
all circumstances or, as if seems likely, there are points of insecurity these should be
identified and their consequences understood.

With these reservations we can now
return to the assumptions originally made in the doubly-labelled water method ^{1}
and summarised in Section 1.8.

With these assumptions we can write:

_{}

..........1_{}

but there is the immediate
difficulty that many observations show us that the size of N is estimated differently by ^{2}H
and ^{18}O, with N_{D} being about 1.03 x N_{O}, and N_{O}
being closer to true body-water than N_{D}.

Thus an equation for CO_{2}
production could be written as:

_{}

..........2

or

_{}

..........3

or

_{}

..........4

Equations 2 and 3 produce results
that are different by 3% but for typical values of k_{O} (e.g. 0.130) and k_{D}
(e.g. 0.105) the result from Equation 4 will be 13% less than that from Equation 3.

The solution to the dilemma of which
equation to chose for studies in man emerges from both practical and theoretical work.
Firstly, validations using equations respecting differences between N_{D} and N_{O}
appear to work better than those that do not 3 and secondly theoretical treatment of the
system suggests that this is the correct approach ^{4}. We can assume that ^{2}H
rapidly equilibrates with both body water and other exchangeable hydrogen ^{5},
but this secondary pool cannot be a pool into which ^{2}H migrates never to return
to body water because if this was the case initial dilutions of ^{2}H and ^{18}O
in body water (the first diluting compartment) would be identical because body water would
not be able to 'see' the secondary pool. Apparently different body-water volumes can only
be explained by a rapidly exchanging secondary pool. In these circumstances it is not
strictly speaking correct to use the product N_{D}k_{D} to calculate water
outflow from body-water. A correct solution in compartmental analysis is to be calculated
from the slopes and intercepts of the exponentials that add together to produce a ^{2}H
disappearance curve. By using the product N_{D}k_{D} to calculate output,
the two pools are being lumped together as one. When a secondary pool exchanges slowly
with total body water large errors will be produced by treating two pools as if they were
one, but if we imagine that rates of exchange increase to the extent that double
exponential curves cannot or can rarely be observed the simple treatment of data becomes
more acceptable. The reader is referred to the comments of Roberts et al 4 for a fuller
treatment of these concepts.

Unfortunately there is very little
experimental data that can be drawn on to test the view that total body water and a small
subsidiary pool can be combined. Such data could be obtained by repeatedly sampling total
body water in the first few hours after dose administration (preferably, dose
administration by an intravenous route) and examination of these early parts of the curve
for an exponential slope that is different from the terminal exponential. Coward has
investigated this problem by oral administration of isotope and the collection of a large
number of samples on the first day ^{6}. Outflow by one- and two-compartmental
procedures (see Table 4.1) was then calculated. In these circumstances differences in
outflow calculated by these methods were trivial even when no early data was used for the
one compartment solution.