Modelling circadian variation in the pharmacokinetics of non-steroidal anti-inflammatory drugs. J. K. Aronson I , M.J. Chappell 2, K. R. Godfrey 2, and M. K. Yew 2.
Eur J Clin Pharmacol (1993) 45:357-361 EuropeanJ..... ,of ~ ( ~ ( ~ r J
J lbQ[ss sc]¢el]e®g @ Springer-Verlag 1993
Modelling circadian variation in the pharmacokinetics of non-steroidal anti-inflammatory drugs J. K. Aronson I, M.J. Chappell 2, K. R. Godfrey 2, and M. K. Yew 2 1University Department of Clinical Pharmacology, Radcliffe Infirmary, Oxford, UK 2Department of Engineering, University of Warwick, Coventry, UK Received: November 5, 1992/Accepted in revised form: July 8,1993
Summary. A o n e - c o m p a r t m e n t m o d e l with first-order absorption has p r o v i d e d g o o d fits to five sets of i n d o m e t h acin data and four sets of k e t o p r o f e n data taken at different times of day. T h e r e was substantial variation in the m o d e l p a r a m e t e r s with time of administration and most of the features of this variation applied equally to b o t h drugs. F r o m the data examined, the source of variation appears to be mainly in the absorption phase and this was confirmed using a chronokinetic analysis, in which simultaneous fits were o b t a i n e d with time-variant rate p a r a m e ters. However, there m a y also be circadian variation in protein binding. T h e d a n g e r of quoting p a r a m e t e r values for either of these two drugs based on administration at a single time of day has b e e n illustrated, and this m a y well be true for other drugs.
Key words: Circadian rhythms, I n d o m e t h a c i n , Ketoprofen; pharmacokinetics, time-varying models, nonsteroidal anti-inflammatory drugs
The p h a r m a c o k i n e t i c s of m a n y drugs vary with the time of day of administration [1], although p h a r m a c o k i n e t i c variables are traditionally r e p o r t e d with the underlying assumption that t h e y are time invariant. O n e of us has previously published a theoretical chronokinetic m o d e l for analysing data collected at different times of the day [2, 3]. Because of the practical difficulties in collecting such data, there have b e e n comparatively few reports in which plasm a c o n c e n t r a t i o n responses to doses administered at different times of the day have b e e n detailed. We have therefore taken the o p p o r t u n i t y to analyse two such published studies, in o r d e r to test the usefulness of the m o d e l in practice. T h e first of these published studies p r o v i d e d a set of five plasma c o n c e n t r a t i o n responses to oral doses of indomethacin given at 07.00, 11.00, 15.00, 19.00, or 23.00 h [4] and the second p r o v i d e d a similar set of four responses to oral doses of k e t o p r o f e n administered at 07.00, 13.00, 19.00, or 01.00 h [5]. To each of the nine sets of data, we have fitted a chronokinetic model, involving cosine wave time-variant
parameters, and a simple mathematical model, consisting of a single c o m p a r t m e n t with first-order absorption and an initial time lag. To do this, we have used the Harwell p a c k a g e Facsimile [6]. O u r objective was to see h o w the m o d e l p a r a m e t e r s vary with time of administration and to determine w h e t h e r there are similar features of variation b e t w e e n the i n d o m e t h a c i n responses and the k e t o p r o f e n responses.
Subjects and methods Data analysed Indomethacin data. The indomethacin mean data were shown graphically by Clench et al. [4], and Dr. Alain Reinberg supplied a list of the mean and individual data. There were 9 subjects, aged 19 to 29 y, 7 men and 2 women. They each took five single 100 mg oral doses of indomethacin, with an interval of i week between doses. One dose was given at 07.00 h, one at 11.00, one at 15.00, one at 19.00, and one at 23.00, and the order of timing was random. Plasma drug concentrations were measured at 0, 0.33, 0.67, 1.0, 1.5, 2.0, 4.0, 6.0, 8.0, and 10.0 h after ingestion. The mean plasma concentrations are shown in Fig. 1 a; samples were missing for two of the subjects for the 07.00 h administration and for four of the subjects for the 11.00 h administration.
Ketoprofen dam. The mean ketoprofen data were reported by Ollagnier et al. [5]. There were 8 healthy male volunteers, mean age 27 y. They each took four single 100 mg oral doses of ketoprofen with an interval of a month between doses. One dose was given at 07.00 h, one at 13.00, one at 19.00, and one at 01.00, and the order of timing was random. Plasma drug concentrations were measured at 0, 0.25, 0.5, 0.75,1.0, 1.5, 2.0, 3.0, 4.0, 6.0, 8.0, 10.0, and 12.0 h after ingestion. The mean plasma concentrations are shown in Fig. 1 b. No mean plasma concentration value was available at 6 h after the 07.00 administration.
Data analysis First we fitted a chronokinetic model involving drug disposition in one compartment with first-order absorption and elimination, and the facility for imposing time variation in the absorption rate constant (k~), the elimination rate constant (Z~), or both. The model is given by:
358 dA~ . dt (t) - ko(t) A~(t); A~(t = O) = f O
(1)
dA~ dt (t) = k~(t) A~(t) - Az(t) A~(t); A~(t = 0) = 0
(2)
k(t) = k(1 + y. cos(o)t + ~o))
with concentration values given by C(t) = AI(t)/V~.
(3)
In these equations, A, and A ~are the amounts of the drug in the "absorption compartment" and compartment 1 respectively, D is the
6-
where k is either k~ or 2z. Here, 7 is a constant between 0 and 1, both limits ensuring non-negativity of k(t). Since we are considering circadian variations, co is fixed at 2~/24- 0.262 rad h-1. We applied this model in four different ways:
Inspection of the concentration-time curves suggested that we should incorporate a time lag (pure time delay, t~,g) into the model, but with this included it was not possible to obtain satisfactory parameter estimates with ko and Zz varying in the above manner. Hence, we next fitted the same one-compartment model with time lag and with k~ and 2~ fixed for each response. For this model, if t-t~g is denoted by t', the differential equations are, for t' > O,
'iii I
0
"~
(4)
1. Time invariant (i. e. 7 = 0), with k, and 2z estimated. 2. Time variance in both k,(t) and 2~(t), with k~, Zz, ?;, andg, estimated. 3. Time variance in ko(t) only, with k~, Zz, 7, and~o estimated. 4. Time variance in Z~(0 only, with k~, Zz, 7, ands0 estimated.
a
£
dose given at time t = 0, f is the fractional systemic availability, and V~ is the apparent volume of distribution of compartment 1. The general form of the chronokinetic model for this variation was:
4
2
I
3
I
I
1
4 ; 6 Time (h)
7
8
9
dA~ dt (() = - k~ A~(£); Aa(t' = 0) = f D
(5)
dA~ (t') = k~ A~(t') - Az A~(t'); A~(t' = 0) = 0 dt
(6)
18 with concentration given by
12-
C(t') = A~(t')/Vv
10-
These equations have solution, for ka ~ 2z, fD
ka
8-
c ( o = V1 k o -
6-
and, for ka = Z~,
4-
C(t') = @ k / e I/1
~
(7)
(e - ~ / - e - / : / )
(8)
2
20
0
2
I
4
6
8
1~}
1'2
Time (h) Fig. 1 a,b. Mean plasma drug concenration versus time data. a Indomethacin • ....... • 07.00 h, © ....... O 11.00 h, • ....... • 15.00h, [] ....... [] 19.00h, • ....... • 23.00h; bKetoprofen • ....... • 07.00h, • ....... • 13.00h, [] ....... [] 19.00 h, • ....... • 23.00 h
k/
(9)
Both forms of solution apply for t' _>0. With the administrations of drug well spaced, it is reasonable to assume that concentration is zero immediately before the drug is administered and since a time lag is being assumed, C(t') = 0 for t" < O. The model parameters t~ag,K (=-fD/V1), ka, and 2~ were fitted to the five sets ofindomethacin data and the four sets of ketoprofen data using the Harwell package Facsimile [6]. The differential equation formulation [equations (5), (6), and (7)] was used in Facsimile; virtually identical results were obtained using the algebraic formulation [equation (8) or equation (9)]. In the first attempt at fitting the data, t~o.~ was taken outside the parameter fitting routine and incremented at
Table 1. Optimum parameter values, with 5 % and 95 % confidence limits for k~ and 2z
Time of administration (h)
flag (h)
K (mg. 1 1)
k~, (h 1)
Zz (h- i)
Sum of squared errors (mg-1 1)2
Indomethacin
07.00 11.00 15.00 19.00 23.00
0.60 0.45 0.85 0.45 0.75
17 8 6 10 8
1.185 2.409 1.477 0.587 0.660
(1.048-1.338) (2.201-2.639) (1.384-1.576) (0.492-0.700) (0.611-0.713)
0.856 0.491 0.407 0.541 0.484
(0.75043.976 (0.459~3.525 (0.387~3.428 (0.448~3.652 (0.4544).517
0.741 1.049 0.139 1.085 0.157
Ketoprofen
07.00 13.00 19.00 01.00
O.20 0.20 0.20 0.65
18 7 9 19
1.892 4.212 1.820 0.752
(1.697-2.110) (3.545-5.004) (1.585-2.090) (0.697-0.812)
0.704 0.336 0.410 0.457
(0.65443.757 (0.308-0.366 (0.373-0.451 (0.418-0.500
3.224 1.059 1.780 1.906
359 nTnnh
61
1100h
',
!
.C
v
N
ii
"k
iI 0
5
10
0
5
I500h
"~11%
10
1900h I
I
07.00
11.00
I
15.00
I
I
19.00
23.00
I
07.00
Time of administration(h)
0
5
10
0
5
I0
Time (h)
4 2300h
!
A
"7 "
/
\
I I
2
\\
/
3
,g 2e
\
Z
4
i
\
\
I
o= %)
•
0
5
/
10
Time (h)
Fig.2. Indomethacin data and the corresponding model responses
I
I
F
I
07.00
13.00
19.00
01.00
t
07.00
Time of administration (h) 0700h
15
g
1300h
Fig. 4 a,b. Variation of/ca and 2~ with time of administration, a Indomethacin; b Ketoprofen. • ....... • ka; © ....... © >~
10'
o=
U
0
5
0
5
10
15
10
15
5
10
15
10
15
g E 8
distribution. This distribution is used within the package, rather than the normal distribution, to ensure the non-negativity of the estimated parameters. It is not possible to calculate similar confidence limits for ~ag and K, because both of these parameters were taken outside the parameter fitting routine. Since K is unknown a priori, the values of/ca and lz are interchangeable (the so-called flip-flop model), but in all the curve fits ka has been taken as the larger of the two values, on the grounds that absorption is usually assumed to be at a faster rate than elimination for both indomethacin and ketoprofen.
L)
Time (h)
0
5 Time (h)
Fig.3. Ketoprofen data and the corresponding model responses
intervals of 0.05 h and K, k~, and 2z were fitted to minimize the sum of squared errors. However, it was not possible to fit all the sets of data satisfactorily with this procedure. In the method adopted, Kwas also taken outside the parameter fitting routine and incremented at intervals of 1 mg. 1-* with values of k~ and Z~ only being fitted. The global minimum was then sought over a two:dimensional grid of tt~,g(at increments of 0.05 h) and K (at increments of i mg- 1- ~). A slightly lower sum of squares might have been obtained if a finer grid had been used, but this was not considered worth while. The results are tabulated in Table 1, along with the 5 % and 95 % confidence limits (i. e. 90 % confidence interval) for k~ and 2~ also indicated. The confidence limits presented in Table 1 are calculated within Facsimile from the Covariance Matrix, with the underlying assumption that the parameters to be estimated follow a log-normal
Results The time-variant approach yielded fits to the data sets which were similar to those obtained by the fits involving tlag but without simultaneous time-variant fitting (see below). However, the sums of squares were generally larger. The smallest sum of squares using the time-variant approach was obtained with the fits which involved time variance of k~ but not 2,. This was consistent with the results obtained with the fits involving ttag. The results of the fits involving t/~eare shown in Table 1. The five indomethacin data sets and the corresponding model responses are shown in Fig. 2, while the four ketoprofen data sets and corresponding model responses are shown in Fig. 3. The variations of ka and 2z with time of administration are shown in Fig.4a (indomethacin) and Fig.4b (ketoprofen); the corresponding variations of K
360
20
0.8 /i
15
\
\
/ \
/
Ae- 0.6
/
\
v
,,
/ "~
//
\
E
\
//
\
/
-~ ~
/ \
".g
/
,e 10
\\
/
~,
0.4
/
0.2 I
07.00
I
11.00
I
I
15.00 19.00 23.00 Time of administration(h)
I
I
I
07.00
07.00
11.00
/ \
15
I
19.00
I
I
23.00
07 O0
Time of administration (h)
20
J,,
I
15.00
0.8
- --.A
I \
/ \ \
'r
\
/
\\
/
E 10
/
0.6
/ I
\
,t
v
/
/
\
I
\
/
\ \
/
---0.4
\
/ I
0.2
I
07.00
I
I
I
13.00
19.00
01.00
I
=... . . . .
I
07.00
07.00
Time of a d m i n i s t r a t i o n (h)
..=_ . . . .
I
13.00
.='
',=
I
I
19.00
01.00
I
07.00
Time of administration ( h )
Fig.6a, b. Variation of ttagwith time of administration. a Indomethacin; b Ketoprofen
and tlagare shown in Figs. 5 and 6. In both cases there was substantial variation in k~ and K, but less substantial variation in 2z and no consistent variation in tlag.
Ketoprofen
The variations of parameter values with time of administration for the two drugs show a number of similarities. For both, the absorption rate constant was high for administration around mid-day (11.00 h for indomethacin and 13.00 h for ketoprofen) and low for administration around midnight (23.00 h for indomethacin and 01.00 h for ketoprofen). The variation in ka was considerably greater than the variation in Az for both drugs, suggesting that the main source of variability of the concentration-time responses is the absorption process. Values of K were high for the 07.00 h administration and lower for administrations around the middle of the day for both drugs. Recalling that K =fD/V1 and that the dose size D is fixed, the variation in K could be due to variation in the systemic availability f o r the apparent volume of distribution V1, or both. Without corresponding responses after intravenous administration it is not possible to determine the main source of variation of K; however, some conclusions are possible.
\ \
Hg.Sa, b. Variation of Kwith time of administration. a Indomethacin; b Ketoprofen
Discussion
\
First, there is clear evidence that there is a circadian variation in the rate of absorption of ketoprofen, which is faster during the day. Circadian variations in gastrointestinal function have been reviewed [7] and some of those variations could have contributed to the circadian variation in the absorption rate of ketoprofen. We believe that the circadian variation in the absorption rate of ketoprofen is most likely to have been due to variation in intestinal motility, which is greater during the day. This would also have accounted for the higher values of Cm~ and the lower values of tmaxseen during the day (see Fig. 1). Gastric emptying rate is also increased during the day, and that would have been expected to have led to time variance in both the rate of absorption and t~,g;we found no consistent variation in t~ag,but the data may have been inadequate to demonstrate such variation, and future studies should address that question more carefully. Alternatively, it may be that gastric emptying does not vary for all types of meal, since it has been shown to vary for solids but not liquids. With the same data that we have analysed here, O1lagnier et al. [5] did find circadian variation in tla~,but they used three exponentials in their analysis, and the data are not robust enough for such treatment.
361 profen). It is clear that these parameters do in fact vary with time. It may therefore have been possible to obtain improved fits to the data by imposing some form of circadian variation, for example a cosine function of time [2], to the parameters. Cosine variation of ka and ,;z would appear quite reasonable from examination of Fig.4. The variations of K and ttag (Figs. 5 and 6) are less obviously sinusoidal, and more data would be needed to establish the form of their variation with any confidence. Applying a circadian variation to ka and 2z would, however, probably achieve comparatively little improvement in the data fits. While ka varied substantially, its effect diminishes quite rapidly in many of the responses, i. e. the absorption phase is relatively short. Thus, any circadian variation would not give much variation in ka over the duration of the absorption phase. Conversely, while 2z is generally smaller, its variability was much less for both drugs. From Figs. 2 and 3 it may be seen that for several of the data sets the tail-end of the data is not fitted very closely. Adding a further compartment to the model should, in theory, improve this part of the fit. However, when this was attempted, it was not possible to obtain a satisfactory fit to the parameters of this more complex model for any of the concentration-time curves.
Gastric acid secretion is greater at night than during the day and is therefore unlikely to have contributed to these findings, since one would have expected gastric acid to have caused a higher rate of capsule dissolution at night along with a greater rate of absorption of ketoprofen, a weak acid. It is also unlikely that the circadian variation in absorption rate was due to variations in food intake, since that was standardized in both studies relative to the time of drug administration. Second, we found variation in the composite variable K ( = fD/V1), which implies circadian variation in either f o r 1/1. Ollagnier et al. [5] showed that the A UC for ketoprofen had significant circadian variation, being lower during the day, and this implies variation in either the systemic availability, f, or clearance, CL. If the variation in A UC was due to variation in CL, then CL must be higher during the day. However, since ,~z did not vary at times when A U C was varying (i. e. at times other than 07.00 h), this presumed variation in CL would imply that the apparent volume of distribution, V1, must also be higher during the day, and that would explain the variation we found in K (i. e. fD/V1), which was lower during the day. This is also consistent with the observation of circadian variation in the mean plasma concentrations of ketoprofen after intravenous administration to man [8], which suggested increased clearance during the day. Of course, we cannot rule out the possibility that circadian variation in the systemic availability of ketoprofen explains the variations in A UC and K. However, if that were s o f w o u l d be lower during the day, which is unlikely, in view of the fact that at that time absorption was faster.
Acknowledgements. We are most grateful to Dr. Alain Reinberg of the Fondation A-de-Rothschild, Paris, for supplying a listing of the original indomethacin data. JKA gratefully acknowledges the support of the Wellcome Trust. The contributions of KRG, MJC, and MKY were financed by a grant "Identifiability and other properties of physically-based system models" from the Science and Engineering Research Council.
Indomethacin
References
The arguments we have applied above to ketoprofen can also be applied to indomethacin in relation to the circadian variation in absorption rate. If A UC also varied similarly then the arguments about variable protein binding would also apply. Although Clench et al. [4] found a lower A U C in the morning, they found no significant circadian variation; however, this may have been because of missing data points, and circadian variation in the A U C of indomethacin has been reported, both in rats [9] and man [10]. If there is no circadian variation in the A UC of indomethacin, the data cannot be explained satisfactorily. If the variation in K is explained only by variation in 1/1, this in turn implies variation in CL (because of invariance in 2z at all times except 07.00 h), which in turn implies variation in A UC, a reductio ad absurdum. Alternatively, if the variation in K is explained only by variation in f V1 and CL being invariant, this similarly implies variation inA UC. Finally, if the variation in K is explained by disproportionate variations in f and V1, this implies disproportionate variations i n l a n d CL, which again implies variation inA UC. We believe therefore that there is circadian variation in the A U C of indomethacin [9, 10] and that the same arguments apply to indomethacin as to ketoprofen. In the model fitting, the values of K, ka, and 2z have been kept constant over the time span of each concentration-time curve (10 h for indomethacin and 12 h for keto-
1. Reinberg A, Smolensky MH (1982) Circadian changes of drug disposition in man. Clin Pharmacokinet 7:401-420 2. Godfrey KR (1988) The effects of circadian rhythms on pharmacokinetic quantities. Biomed Meas Infor Contr 2:14-24 3. Godfrey KR (1989) Chronopharmacology and its application to the development of theophylline treatment schedules for asthma. Eur J Clin Pharmaco136:103-109 4. Clench J, Reinberg A, Dziewanowska Z, Ghata J, Smolensky MH (1981) Circadian changes in the bioavailability and effects of indomethacin in healthy subjects. Eur J Clin Pharmaco120:359-369 5. Ollagnier M, Decousus H, Cherrah Y, Levi F, Mechkouri M, Queneau R Reinberg A (1987) Circadian changes in the pharmacokinetics of oral ketoprofen. Clin Pharmacokinet 12:367-378 6. Curtis AR, Sweetenham WP (1987) Facsimile/Chekmat Users' Manual. AERE Harwell Report No. 12805 7. Vener KJ, Moore JG (1987) Chronobiologic properties of the alimentary canal affecting xenobiotic absorption. Ann Rev Chronopharmacot 4:257-281 8. Decousus H, Ollagnier M, Cherrah Y, Perpoint B, Hocquart J, Queneau P (1986) Chronokinetics of ketoprofen infused intravenously at a constant rate. Ann Rev Chronopharmacol 3:321-322 9. Cuisinaud G, Guissou R Sassard J (1984) Chronopharmacokinetic study ofindomethacin. Ann Rev Chronopharmacol 1:341-344 10. Guissou R Cuisinaud G, Llorca G, Leleune E, Sassard J (1983) Chronokinetic study of a prolonged release form of indomethacin. Eur J Clin Pharmaco124:667-670 Prof. K. R. Godfrey Department of Engineering, University of Warwick Coventry CV4 7AL, UK
