# Chapter 4: Calculation of pool sizes and flux rates

Contributor: Andy Coward

## 4.1 Introduction

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 CO2 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 (rout) 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 rout will equal Nk, but if there is an exit in the form of a leak, Nk will be greater than rout and if the presence of a leak is unsuspected rout will be incorrectly estimated because the model was wrong although it fitted the data! The analogy should be evident. Direct measurement of rout 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.

## 4.2 The basic model

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 2H and 18O, with ND being about 1.03 x NO, and NO being closer to true body-water than ND.

Thus an equation for CO2 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 kO (e.g. 0.130) and kD (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 ND and NO 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 2H rapidly equilibrates with both body water and other exchangeable hydrogen 5, but this secondary pool cannot be a pool into which 2H migrates never to return to body water because if this was the case initial dilutions of 2H and 18O 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 NDkD 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 2H disappearance curve. By using the product NDkD 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.