App 3.1: Basic derivation

App 3.2: A general solution to equation 2

App 3.3: Estimating CO_{2}production

App 3.4: The shape of the enrichment curve

App 3.5: References

Contributor: Mike Franklin

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, r_{H2O}(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:

N_{D}logd_{D}(t) = N_{D}logd_{D} - r_{H2O}t

or:

d_{D}(t)
= d_{D}e^{-kt}

where E_{D} is the
enrichment at time 0 and k = r_{H2O}/N_{D} 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 CO_{2}.

_{}

.........3

where r_{CO2}(t) denotes the
flow rate of CO_{2} 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 dlogd_{D}(t) is the slope
of the loge transformed enrichment curve at time t. Thus we can obtain the flow rate (ie
production rate of CO_{2}) at any instant if we know the oxygen and deuterium pool
sizes and the slopes of the log_{e} enrichment curves. By integrating 5 between
the two times t_{1}, t_{2} say we can derive the total CO_{2}
production over the period:

_{}

.........6

which if all assumptions hold simplifies to:

2r_{CO2} = N_{O}log(d_{O}1/d_{O}2) - N_{D}log(d_{D}1/d_{D}2)

.........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.

Consider a short time period t_{1},
t_{2} 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:

f_{i}(t) = x_{i}(y
N(t))²

f_{o}(t) = x_{o}(y
N(t))²

where f_{i}(t), f_{o}(t)
denote the input and output flows and x_{i}, x_{o} and z are constants, x_{i},
x_{0} 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 t_{1}, t_{2}; d_{1} and d_{2} are the enrichments at times t_{1}, t_{2}.
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}(F_{i}(t)/x_{i})^{q}= y^{-1}(F_{0}(t)/x_{0})^{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(d_{1}/d_{2}) = N(logd_{1}- logd_{2})

.........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 (N_{1}-N_{2})/log(N_{1}/N_{2})
is equal to the mean pool size when the pool size N has changed exponentially over the
period (t_{1}, t_{2}). It can be approximated by the mean of N_{1}
and N_{2} but the geometric mean provides an even better approximation, differing
from N by less than 1% for a 20% difference in N_{1} and N_{2}.

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 = N_{D}(t)/N_{O}(t)
and let the oxygen pool increase from N_{O} to N_{O}(1+m) then the third
term reduces to yield a change equal to (p-1)mN_{O}. Typically p = 1.03 and in a
period with 5% growth m = 0.05 yielding a change equal to 0.0015 N_{O}. The
problem then reduces to determining the difference between the two integrals.

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 t_{0}, 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.

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.