Clinical Science and Molecular Medicine (1977) 52,97-101.
The analysis of decay curves
H. H. LAFFERTY, A. E. B. G I D D I N G S A N D D. MANGNALL
Department of Applied Mathematics and Computing Science, University of Shefield, The Royal Infirmary,
Bristol, and The Wellcome Laboratory, University Surgical Unit, Northern General Hospital, Shefield, U.K.
(Received 12 May 1976; accepted 8 July 1976)
Methods
SY
1. The limitations inherent in the conventional
treatment of glucose decay curves as first-order
rate systems are described.
2. The conventionally derived K value is a
rate constant and should not be confused with a
rate.
3. First-order systems are described by this
rate constant and the initial concentration of
substance studied. They cannot be described by
either factor alone.
4. Two parallel curves cannot both result
from first-order systems.
5. If K is conventionally calculated for two
parallel curves, then the value obtained for the
upper curve must be smaller than the value for
the lower.
Eighteen otherwise healthy men with normal
fasting levels of plasma glucose were studied
before and after abdominal surgery. Each
underwent a glucose infusion test on the
morning before and the morning after operation.
The subjects were starved overnight before the
pre-operative test, and only received sodium
chloride solution (150 mmol/l) between the
operation and the post-operative test. Each
initially received 2.78 mmol (0.5 g) of glucose/
kg body weight, intravenously over 3 min,
followed by a constant infusion of 0.11 mmol
(20 mg) of glucoselkg body weight per min for
the next 42 min. Infusions were given into an
arm vein and venous samples were taken from
the other arm at 0,3,6,10,20,30,45,60,75 and
90 min into ice-cold polystyrene tubes containing lithium-heparin beads. Plasma was separated
by centrifugation within 10 min.
Plasma glucose concentration was determined
enzymatically by hexokinase and glucose 6phosphate dehydrogenase (Boehringer Corp.,
London). Replicate determinations lay within
4%.
The rate constant, K, was conventionally
derived as 100 times the slope of the best
straight-line fit through a plot of log. (glucose
concentration) against time, for the period
45-90 min. This assumes a first-order system.
Key words: decay curve analysis, disappearance
rate, first-order rate constant, K value, plasma
glucose.
Introduction
A variety of interpretations in the analysis of
glucose decay curves are currently offered. This
study arose from investigation of glucose
tolerance in pre- and post-operative surgical
patients. The mathematical basis of first-order
systems is defined with illustrations of the
limitations which result from the treatment of
glucose decay CUNM as a first-order system.
Results and discussion
The mean changes in plasma glucose concentration during glucose infusion tests for the
eighteen patients (Fig. l), show that the post-
Correspondence: Mr H. H. Lafferty, Department of
Applied Mathematics and Computing Science,
University of Sheffield, Sheffield S10 2TN U.K.
H
97
H. H. Laferty, A. E. B. Giddings and D. Mangnall
98
Infusion
I
c
30
I
60
I
90
Time (min)
FIG.1 . Changes in plasma glucose concentration during
a glucose infusion test, 1 day before and 1 day after an
operation. The infusion consisted of 2.78 mmol(O.5 g) of
glucose/kg body weight over the first 3 min, followed by
0.1 1 mmol(20 mg)/kg body weight per min for a further
42 min.
operative plasma glucose curve was similar in
shape to that pre-operatively, but at a higher
value throughout. The K values calculated for
45-90 min were K,,,.,,.= 2.05 (range 1.303.03) and KpO,l-OP.
= 1.36 (range 0.69-1.79).
Analysis of decay curves frequently assumes
I
0
1
0
'
the system to follow first-order kinetics, where
the disappearance rate is proportional to the
amount of substance present. This appears to
apply to the decay of radioactive isotopes, and
for some biological reactions such as the oxidation of reduced cytochrome c by the mitochondrial cytochrome oxidase (Smith, 1955).This has
also been the conventional model for analysis of
glucose decay curves, assuming that the disappearance of glucose depends only upon the
prevailing glucose concentration (Amatuzio,
Stutzman, Vanderbilt & Nesbitt, 1953; Ikkos &
Luft, 1957; Wright, Henderson & Johnston,
1974; Allison, Hinton & Chamberlain, 1968;
Marks & Marrack, 1962; Franckson, Malaise,
Arnould, Rasio, Ooms, Balasse, Conard &
Bastenie, 1966; Heard & Henry, 1969; Hamilton & Stein, 1942). However, to be treated as
first-order, decay curves must fulfil two requirements. First, the experimental data should yield
a straight line semi-logarithmic plot, and
secondly, there should be no physiological
indications that the system is not first-order. We
do not find that glucose curves satisfy these
criteria. Accurate plotting of the logarithm of
the glucose concentration against time frequently yields a curve rather than a straight line.
Furthermore there is ample physiological
evidence that the rate of glucose disappearance
does not depend solely upon the prevailing
glucose concentration, as the amount of insulin,
the presence or absence of other hormones, the
number of insulin-binding sites, as well as the
nutritional state of the subject, all influence the
rate of glucose disappearance (Heath & Corney,
1973; Long, Spencer, Kinney & Geiger, 1971).
The assumption of first-order kinetics grossly
simplifies this complex system.
Nonetheless this simple approach, regarding
decay curves as first-order, has been widely
employed, but one should be aware of the
consequences of such a treatment, as illustrated
below.
A system which decays with first-order
kinetics obeys the law 'the rate of change of the
substance is proportional to the amount of the
substance present', i.e.
dG/dt = - K . G
Time
FIG. 2. Two parallel decay curves that cannot both be
first-order but which are frequently treated as such. For
details see the text and Appendix.
(1)
where G is the amount of the substance being
studied, and K is a constant of proportionality,
with the dimensions time-', which has been
used as a characteristic of glucose decay curves
Franckson et al.
(1966)
Amatuzio et al.
(1953)
See note
Reference
Comment
0 3 3 g/kg i.v.
Mean 1.58
25 g i.v.
340+4*84
Dose
Range of K values
Normal subjects
25 g i.v.
Mean 1.72
096+3.4+
Lundbaek (1962)
(1953)
See note
Amatuzio et al.
25 g i.v.
0*93+246
Mild
Lundbaek (1962)
Amatuzio et 01.
(1953)
See note
25 g i.v.
Mean 0.63
Unspecified
25 g i.v.
0*15+ 1.87
Severe
Diabetic patients
produce a lower curve parallel to the original, and claim that the lower curve follows first-order rate kinetics, whereas most other workers consider the upper
curve to follow first-order rate kinetics. The Appendix shows that both curves cannot follow first-order rate kinetics.
TABLE.
1. Summary of glucose tolerance test data taken from the literature showing the variation in quoted K values
Note. Amatuzio. Stutzman. Vanderbilt & Nesbitt (1953) calculate Kin terms of glucose excess. They subtract the glucose fasting value from all values, and thus
100
H. H. Lafferty, A . E. B. Giddings and D. Mangnall
(see Appendix). Since G decreases with time
and K is a constant, the rate of change of G is
also time-dependent.
Many authors regard K as a rate of disappearance, assimilation or utilization (Amatuzio et al., 1953; Ikkos & Luft, 1957; Allison et
al., 1968; Wright et al., 1974; Marks & Marrack, 1962; Franckson et al., 1966; Heard &
Henry, 1969), but this conflicts with eqn. (l), by
which K is defined as a rate constant and not a
rate. Precision in the definition of K is important
for some proposed treatment schedules have
been based, at least in part, upon the misconception that K is the rate and not the rate
constant (Wright, 1973; Allison et al., 1968).
The solution to eqn. (1) is
G = Goe-"
(2)
where Go is the value of G at time t = 0. Eqn.
(2) thus shows that K alone cannot define G
since Go must also be known. K alone gives no
information about the shape of the curve and
Table 1 shows that K is an unreliable guide with
which to compare patients under different
conditions. K alone would serve as a useful
factor only if the curves to be considered had the
same initial glucose concentration, which is
unusual in clinical situations.
The decay curves that we obtained before and
after surgery are approximately parallel. We
consider the limitations of the conventional
analysis of two such curves which are separated
by a constant vertical distance. From the
mathematical analysis presented in the Appendix: (i) two parallel curves cannot both result
from first-order systems, (ii) if K is calculated for
each such curve, then the two K values cannot
be equal, and the K value for the upper curve
will be smaller than that for the lower. Hence
statements that K for one curve is less than K
for another may merely reflect the higher
values of glucose for the first, and do not
necessarily indicate a lower rate of glucose
disappearance.
We have described the basic mathematics of
first-order systems, but we have not discussed
the many biological variations which may
affect the clinical interpretation of tests using
disappearance curves. Such factors as distribution space, renal losses, incomplete mixing and
the interpretation of metabolic clearance, which
may themselves occasion other errors, are
outside the scope of this paper. However, the
confusion that exists in the literature about the
precise meaning of the K value justifies the
restatement of the principles from which K is
derived, notably that K is a rate constant and
not a rate.
APPENDIX
(a) The calculation of K
A first-order rate system is defined by
dGldt = - K . G
(1)
whose solution is
G
=
Goe-",
(2)
where Gois the value of G at time t
logarithms of eqn. (2) we have
=
0. Taking
InG = lnCo - Kt
(3)
Evaluating eqn. (3) at two times, t l and t z , gives
InG,, = InGo-Ktl
(4)
InG,,
(5)
=
lnGo -Kt2
Hence, subtracting eqn. (5) from eqn. (4):
InG,, -InC,,
=
-K(tl -tz)
(6)
and thus
K =
InC, - InG,,
tz-tl
(7)
Hence K can be calculated from a plot of InG
against time.
In practice, it is usual to work in logarithms
to the base 10, and multiply by 1 0 0 to derive
K'. The arguments below apply equally to K or
K'.
(b) The curves of two first-order rate systems
cannot be parallel
By definition the distance between two parallel curves is independent of t. From eqn. (2)
the equations would be
G8 - G.O e-K.1
(8)
Gb = Gb. e-Kbr
(9)
The vertical distance between these curves is
C:
C = G., e-'8'-Gbo
e-Kbr
(10)
This equation is dependent upon t unless
G,, = Gboand K. = Kb, in which case c = 0.
Analysis of decay curves
Thus if two curves are parallel they cannot
both describe first-order rate systems. Nonetheless, if one persists in trying to analyse parallel
curves on the assumption that both follow firstorder kinetics, then it can be shown that the K
values cannot be equal and that the upper curve
must, of necessity, have a smaller K value.
A formal proof is available on request from
the authors.
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