# Appendix 3: Derivation of a general equation to cope with changes in pool space

Contributor: Mike Franklin

## App 3.1: Basic derivation

In this appendix we outline some of the mathematics underlying the DLW technique, we will attempt to keep the presentation simple and refer readers to the original work of Lifson and McClintock 1 and Coward et al ² for a more formal development.

A useful starting point is to assume that all the Lifson and McClintock assumptions hold except that neither the flow rates nor the pool sizes are necessarily constant. The change of deuterium tracer enrichment during any interval is affected by the relative quantity of water leaving the pool, the enrichment and any change in pool size. The relationship can be expressed by the differential equation:

.........1

where N(t) denotes the pool size at time t, rH2O(t) and d (t) denote the water flow rate and enrichment; d'(t), N'(t) denote the rates of change of d (t) and N(t) respectively (ie d (t) denotes dd (t)/dt); the suffix D denotes we are refering to the deuterium pool. A useful way to reformulate Equation 1 is to write it as:

.........2

which when all the assumptions hold simplifies to

.........2a

This differential equation can be solved to give:

NDlogdD(t) = NDlogdD - rH2Ot

or:

dD(t) = dDe-kt

where ED is the enrichment at time 0 and k = rH2O/ND is the rate constant.

Similar equations can be derived for the change in oxygen-18 enrichment by noting it is lost through both water and CO2.

.........3

where rCO2(t) denotes the flow rate of CO2 and the suffix O denotes oxygen. As before we can reformulate this equation as:

.........4

which when all assumptions hold simplifies to:

.........4a

from Equations 2 and 4 we get:

.........5

Note that dlogdD(t) is the slope of the loge transformed enrichment curve at time t. Thus we can obtain the flow rate (ie production rate of CO2) at any instant if we know the oxygen and deuterium pool sizes and the slopes of the loge enrichment curves. By integrating 5 between the two times t1, t2 say we can derive the total CO2 production over the period:

.........6

which if all assumptions hold simplifies to:

2rCO2 = NOlog(dO1/dO2) - NDlog(dD1/dD2)

.........6a

The final term in Equation 6 involves the changes in the two pool sizes and is equal to zero if the pool sizes are equal. The problem we usually encounter is the evaluation of the two integrals. Integration is a simple matter in certain special cases (eg when pool sizes are constant) but this does not help us understand what happens in general. To do this we look at Equation 2 and see what happens under conditions which will be approximately true at most times.

## App 3.2: A general solution to equation 2

Consider a short time period t1, t2 say, in which the water output flow is in constant ratio with the input flow and both are directly related to the pool size. We may express this fairly generally as:

fi(t) = xi(y N(t))²

fo(t) = xo(y N(t))²

where fi(t), fo(t) denote the input and output flows and xi, xo and z are constants, xi, x0 and y are greater than 0, but no constraint is placed on z. It can be shown that if Equation 7 is true then:

.........8

where r is the total flux in the period t1, t2; d1 and d2 are the enrichments at times t1, t2. This is the equation originally derived by Coward et al 1.

The wide range of conditions for which 8 holds true make it a very important equation underlying DLW work. We may consider three special cases:

(i) z = 0, here the pool size varies but the flow rates are constant.

(ii) z = 1, here the relative flow rates are constant although the pool sizes and flow rates vary.

(iii) d = +¥, to understand this write q = z-1 and write:

N(t) = y-1(Fi(t)/xi)q = y-1(F0(t)/x0)q

and let q tend to zero. Thus we see the equation hold if the flow rates vary and the pool size is constant. In this case Equation 6 further simplifies to:

r = Nlog(d1/d2) = N(logd1- logd2)

.........9

where N denotes the constant pool size.

The importance of Equation 8 lies not only in its generality but in its simplicity. All that needs to be known are the pool sizes and the enrichments at the beginning and end of the period. In the event that even greater simplicity is required then the expression (N1-N2)/log(N1/N2) is equal to the mean pool size when the pool size N has changed exponentially over the period (t1, t2). It can be approximated by the mean of N1 and N2 but the geometric mean provides an even better approximation, differing from N by less than 1% for a 20% difference in N1 and N2.

## App 3.3: Estimating CO2 production

We return to Equation 6. Firstly we observe that the third term will normally contribute very little for the two pools are of very similar size and any change will have virtually identical effects on both. In quantitative terms let the two pool sizes be in constant ratio such that p = ND(t)/NO(t) and let the oxygen pool increase from NO to NO(1+m) then the third term reduces to yield a change equal to (p-1)mNO. Typically p = 1.03 and in a period with 5% growth m = 0.05 yielding a change equal to 0.0015 NO. The problem then reduces to determining the difference between the two integrals.

## App 3.4: The shape of the enrichment curve

If the enrichment of deuterium in the urine is studied from the time of introducing the tracer, there is a short stabilisation period up to time t0, say, followed by a period in which the enrichment decays to zero. As an illustration we may seek the shape of the curve for the general case described in Section App 3.2. We can show that:

.........10

where a and a-1 ¹ 0 so that the slope of the log enrichment curve is constant only in certain special cases. The principal special case is when K(t) = F(t)/N(t) is constant. Note from Equation 8 that when the enrichment curve is not straight the principal parameters of interest are the values at the extremes of the range of interest.

## App 3.5: References

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

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