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.
References ALLISON, S.P., HINTON,P. & CHAMBERLAIN, M.J. (I 968) Intravenous glucose tolerance, insulin and free fatty acid levels in burned patients. Lancet, ii, 11 13-1 116. AMATUZIO, D.S., STUTZMAN, F.L., VANDERBILT, M.J. & NESBI~,S. (1953) Interpretation of the rapid intravenous glucose tolerance test in normal individuals and in mild diabetes mellitus. Journal of Clinical Investigation, 32,428-435. FRANCKSON.J.R.M.. MALAISE,W., ARNOULD,Y., Rwro, E., OOMS,H.A., BALASSE, E., CONARD,V. & BASTENIE, P.A. (1966) Glucose kinetics in human obesity. Diabetologia, 2, 96-103. B. &STEIN,A.F. (1942) The measurement of HAMILTON, intravenous blood sugar curves. Journal of Laboratory and Clinical Medicine, 27, 491-497.
101
HEARD,C.R.C. & HENRY, P.A.J. (1969) Glucose tolerance and insulin sensitivity. Clinical Science, 37, 3744. HEATH,D.F. & CORNEY,P.L. (1973) The effects of starvation, environmental temperature and injury on the rate of disposal of glucose by the rat. Biochemical Journal, 136,519-530. IKKOS, D. & LUFT,R. (1957) On the intravenous glucose tolerance test. Acta Endocrinologica, 25, 312-334. J.L.. KINNEY.J.M. & GEIGER, LONG,C.L., SPENCER, J.W. (1971) Carbohydrate metabolism in man: effect of elective operations and major injury. Journal of Applied Physiology, 31, 110-1 16. LLINDBAEK, K. (1962) Intravenous glucose tolerance as a tool in definition and diagnosis of diabetes mellitus. British Medical Journal, i, 1507-1 5 13. MARKS, V. & MARRACK. D. (1962) Glucose assimilation in hyperinsulinism. A critical evaluation of the intravenous glucose tolerance test. Clinical Science, 23, 103-113. SMITH, L. (1955) Spectrophotometric assay of cytochrome c oxidase. Methods of Biochemical Analysis, 2,427-434. WRIGHT,P.D. (1973) Glucose utilisation and insulin secretion during surgery and their clinical significance. Journal of the Royal College of Surgeons, Edinburgh, 15,284-289. WRIQHT,P.D., HENDERSON, K. & JOHNSTON, I.D.A. (1974) Glucose utilisation and insulin secretion during surgery in man. British Journal of Surgery, 61.5-8.
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.
References ALLISON, S.P., HINTON,P. & CHAMBERLAIN, M.J. (I 968) Intravenous glucose tolerance, insulin and free fatty acid levels in burned patients. Lancet, ii, 11 13-1 116. AMATUZIO, D.S., STUTZMAN, F.L., VANDERBILT, M.J. & NESBI~,S. (1953) Interpretation of the rapid intravenous glucose tolerance test in normal individuals and in mild diabetes mellitus. Journal of Clinical Investigation, 32,428-435. FRANCKSON.J.R.M.. MALAISE,W., ARNOULD,Y., Rwro, E., OOMS,H.A., BALASSE, E., CONARD,V. & BASTENIE, P.A. (1966) Glucose kinetics in human obesity. Diabetologia, 2, 96-103. B. &STEIN,A.F. (1942) The measurement of HAMILTON, intravenous blood sugar curves. Journal of Laboratory and Clinical Medicine, 27, 491-497.
101
HEARD,C.R.C. & HENRY, P.A.J. (1969) Glucose tolerance and insulin sensitivity. Clinical Science, 37, 3744. HEATH,D.F. & CORNEY,P.L. (1973) The effects of starvation, environmental temperature and injury on the rate of disposal of glucose by the rat. Biochemical Journal, 136,519-530. IKKOS, D. & LUFT,R. (1957) On the intravenous glucose tolerance test. Acta Endocrinologica, 25, 312-334. J.L.. KINNEY.J.M. & GEIGER, LONG,C.L., SPENCER, J.W. (1971) Carbohydrate metabolism in man: effect of elective operations and major injury. Journal of Applied Physiology, 31, 110-1 16. LLINDBAEK, K. (1962) Intravenous glucose tolerance as a tool in definition and diagnosis of diabetes mellitus. British Medical Journal, i, 1507-1 5 13. MARKS, V. & MARRACK. D. (1962) Glucose assimilation in hyperinsulinism. A critical evaluation of the intravenous glucose tolerance test. Clinical Science, 23, 103-113. SMITH, L. (1955) Spectrophotometric assay of cytochrome c oxidase. Methods of Biochemical Analysis, 2,427-434. WRIGHT,P.D. (1973) Glucose utilisation and insulin secretion during surgery and their clinical significance. Journal of the Royal College of Surgeons, Edinburgh, 15,284-289. WRIQHT,P.D., HENDERSON, K. & JOHNSTON, I.D.A. (1974) Glucose utilisation and insulin secretion during surgery in man. British Journal of Surgery, 61.5-8.
