Population pharmacokinetics and pharmacodynamics of warfarin in healthy young adults

European Journal of Pharmaceutical Sciences, 1 (1993) 151-157 0928-0987/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

151

PHASCI 21

Population pharmacokinetics and pharmacodynamics of warfarin in healthy young adults M a r i a Pitsiu, E v a M . P a r k e r , L e o n A a r o n s , M a l c o l m R o w l a n d Pharmacy Department, University of Manchester, UK (Received 16 February 1993; accepted 28 May 1993)

Abstract The population pharmacokinetics and pharmacological response - - prothrombin complex activity and factor VII activity - - were studied in a group of 48 normal, healthy young volunteers. Population parameter estimates were obtained using a standard two-stage method, a nonlinear mixed effect model (NONMEM) and a two-stage Bayesian method (EM algorithm). A modified sigmoid-Imax model was used to relate the concentration of s-warfarin to the rate of clotting factor synthesis. The three methods produced similar estimates of the population pharmacokinetic parameters, although the standard two-stage method overestimated the contribution of the pharmacokinetic parameters to the interindividual variability. It was not possible to partition the interindividual variability in response between the pharmacodynamic parameters with the NONMEM procedure: the estimates obtained from the EM algorithm were generally in good agreement with those obtained using the standard two-stage approach. The variability in the warfarin concentration contributed at most only 40% of the observed variability in the pharmacological response, and then only for times greater than 96 h after the dose. Most of the variability in the pharmacodynamics was due to interindividual differences in the clotting factor degradation rate constant and Cso,s, the s-warfarin concentration causing a 50% decrease in synthesis rate. Keywords: Population pharmacokinetics; Population pharmacodynamics; Warfarin; NONMEM; EM algorithm; Nonlinear regression

Introduction Warfarin exerts its anticoagulant effect by inhibition o f the synthesis o f the vitamin K-dependent clotting factors. There is considerable interindividual variability in the response to warfarin (Breckenridge, 1977) which is partly pharmacokinetic and partly p h a r m a c o d y n a m i c in origin. Various models have been proposed to describe the relationship between warfarin concentration and its anticoagulant effect (Holford, 1986) and several o f these models have been implemented in computerized prediction algorithms (Lee et al., 1987; McAleer and Chrystyn, 1990). The measures o f anticoagulant response and warfarin concentration which have been m o s t frequently employed in these prediction algorithms are the p r o t h r o m b i n time and total (racemic) warfarin concentration, respectively, even t h o u g h s-warfarin, which has the shorter halflife o f the two enantiomers, is responsible for the

majority o f the anticoagulant activity (Bell and Ren, 1981). Computerized dose prediction algorithms based on Bayesian updating (Sheiner and Beal, 1982) require population parameter estimates. As a prelude to the investigation o f the prediction o f warfarin dosing requirements based on clotting factor activity measurements, we have evaluated the population pharmacokinetics and p h a r m a c o d y n a m i c s o f warfarin in a group o f y o u n g healthy volunteers following a single 25 m g racemic dose o f warfarin. The concentration o f b o t h r and s enantiomers, p r o t h r o m b i n time and factor VII activity (using a chromogenic assay) were measured. T w o different algorithms - - N O N M E M (Beal and Sheiner, 1982) and an E M m e t h o d (Racine-Poon, 1985) - - were used for the estimation o f the population parameters.

Methods Correspondence to: L. Aarons, Pharmacy Department, University of Manchester, Manchester, MI3 9PL, UK. Tel: (0)61-275-2357; Fax: (0)61-275-2396.

Data We analyzed data from the placebo phase o f five single dose warfarin drug interaction studies

152

M. Pitsiu et al./Populationpharmacokinetics and pharrnacodynamics ~/ wa(/arin

described in detail elsewere (Toon et al., 1987a,b; Pitsiu et al., 1992). Forty-eight healthy non-smoking male volunteers took part in the studies (ages 20 27 years, weights 66 75 kg, heights 170 185 cm). Three pre-warfarin blood baseline samples were taken and the protocol involved blood sampling for 7 days after oral administration of 25 mg (5 x 5 mg) racemic warfarin, following an overnight fast (sampling times: 2, 4, 6, 12, 24, 36, 48, 60, 72, 96, 120, 144, 168 h). All studies were subject to ethical review and each subject gave informed written consent before participating. Assays

The concentrations of the two enantiomers of warfarin were measured using stereoselective HPLC (Banfield and Rowland, 1984). Prothrombin time measurements were made by Quick's one-stage test, using the Manchester Comparative Thromboplastin (National (UK) Laboratory for Anticoagulant Reagents and Control). The prothrombin time measurements were converted into prothrombin complex activity using a standard curve of diluted normal plasma (the CV for a PCA of 100% was 5% and for a PCA of 25% it was 10%). Factor VII measurements were performed using a chromogenic amidolytic assay in heparinized plasma (25 #1), based on the method of Seligsohn et al. (1978). The assay involves activation of factor X by factor VII, tissue thromboplastin, and calcium ions via the extrinsic coagulation mechanism. Activated factor X (factor Xa) cleaves a factor Xa-specific chromogenic substrate (S-2765) releasing p-nitroaniline. The hydrolysis reaction is terminated by addition of citrate buffer, pH 3.5. The factor Xa generated in the first step, and thus the absorbance of the produced nitroaniline, is proportional to the concentration of factor VII in the original sample ('end-point' method). The plot of absorbance versus factor VII activity (FA) was curvilinear over the range of interest and was described by the following relationship Absorbance = [(a- FAb)/(c + FAb)] + d

(1)

where a, b, c and d are constants. Despite the nonlinearity, the coefficient of variation for the same batch of reagents was small, 4.6% for the lowest calibrator (at 6.25% of factor VII activity) and 3% for 100% factor VII. Between batches the CVs for the lowest and highest calibrators were 20 and 17% respectively. Citrated blood was used for the measurement of prothrombin time and plasma from heparinized blood for H P L C assays of warfarin enantiomers. Experiments with citrated and heparinized

controls and samples showed that neither citrate nor heparin interfered with the chromogenic assay. D a t a analysis

The concentration-time profile for both enantiomers was modelled with the following monoexponential function C(t) = ( D / V )

*e ~

(2)

where V and k are the volume of distribution and elimination rate constant, respectively, and D, the dose, is half the racemic dose (12.5 mg). As the first blood sample was not taken until 2 h after the dose, it was not possible to model the absorption of warfarin. However, warfarin is rapidly and completely absorbed (Breckenridge and Orme, 1973) and so any bias in the estimation of V should be quite small. The effect of warfarin on the synthesis of the clotting factors was modelled using a modified sigmoidImax model (Holford, 1986). The time profile of clotting factor activity is determined from the following differential equation dCA dt

kd *

- - -75-

(3)

Ik +-d-Z 50,s

where C A is clotting factor activity, CAnorm is the clotting factor activity in the absence of warfarin (fixed to the mean of three pre-dose measurements), kd is the degradation rate constant for the clotting factor activity, Cs is the concentration of s-warfarin, Cso,s is the concentration of s-warfarin which reduces the synthesis rate by 50% and "7 is a shape parameter. In addition, a lag time was incorporated into both the pharmacokinetic and pharmacodynamic models to allow for the observed time delay in the onset of the effect after warfarin administration, r-Warfarin was assumed to have a negligible effect on clotting factor activity. The pharmacokinetic and pharmacodynamic models were fitted to individual data sets by nonlinear least squares (Metzler et al., 1974) using both homoscedastic and heteroscedastic (weighting proportional to the reciprocal of the predicted value and to the square of the predicted value) weighting schemes. Two methods of obtaining estimates of the population parameters were investigated. Initially we used the N O N M E M programme (Version I Level 1) developed by Beal and Sheiner (1982). N O N M E M uses a first order Taylor series expansion in order to produce a linear random effects model. Estimation of the

M. Pitsiu et al./Population pharmacokinetics and pharmacodynamics of warfarin

population parameters is achieved through maximization of the associated likelihood objective function. The individual parameters are assumed to arise from a distribution (either normal or log normal) characterized by a population mean and interindividual covariance matrix. The model describing the ith measurement in the j t h individual, Cij (where C is either the measured warfarin concentration or the clotting factor activity), at time tij, is shown in Eq. 4. Cij( t ) = f ( pj, tij ) + rlj -I- £ij

(4)

where f is the predicted value and pj are the parameters of the j th individual. ~Tj describes the departure of the response of the j t h individual from the population response and eij represents the residual departure of the model from the observations, e was assumed to be a random Gaussian variable with a mean of zero and either constant variance (additive model) or variance proportional to f2 (multiplicative model). ~7 can be further partitioned amongst the parameters of the model.

Op~ .~jk

(5)

where the ~jk are the contributions from the kth parameter to the interindividual variability and were assumed to be random variables with mean values of zero and either additive or multiplicative variances. An alternative approach for estimating population parameters and their variability in nonlinear random effects models has been proposed by Racine-Poon (1985). The procedure (hereafter called EM) is Bayesian in nature and follows a method suggested by Lindley and Smith (1972). In essence the method utilizes the least squares parameter estimates and covariance matrix of each individual, with very weak assumptions about the prior distribution of the population parameters, to calculate a posterior density function from which the population means and covariance matrix can be obtained. The posterior density function is calculated by an iterative method suggested by Dempster et al. (1977). The steps in the EM algorithm are summarized by the following equations. At the ruth iteration E-step N

0 (m) = D (m) Z ( V j * + s(m-'))-'O;

(6)

j=l

D(m) =

O~m) = (V; -1 + ~'](m-1)-')-I (V;-10; + ~a(m-l)-lo(m) ) (8) M-step N R + E(0J

~(m) =

*

)-1

(7)

- o!m )(OJ

- O!m ) T

j=l

(9) N - nparm

where 0, ~ and Oj are the estimates of the population parameters, the population covariance matrix and the posterior individual parameters. 0. denotes the mean of the Oj. N is the number of individuals and 0] and V] are the least squares estimates of the parameters and asymptotic covariance matrix obtained from fitting the individuals. The starting values for the E-step were N

0(°) = N - I Z

0;

(10)

j=l

and N

R + E(0;

nparm ,q~c tJJ iJ

oj = ~

153

~-](0) =

- 0(°/)(0; - 0(°/) T

j=l

(11)

N - nparm where R was taken as 61 (I is the identity matrix of dimension nparm), reflecting very vague prior information about the population covariance matrix: being chosen small enough so as not to influence the posterior estimates. The kinetic and effect models were fitted sequentially. The estimated individual values for k s and Vs were used to generate Cs(t) in Eq. 3, for both the individual and population fitting of the clotting factor activity data.

Results and discussion

The parameter estimates obtained after fitting the pharmacokinetic and pharmacodynamic models to the individual data sets by nonlinear regression are shown in Table 1. The parameter estimates obtained with the several weighting schemes that were examined were very similar. Only results for the pharmacokinetics with weights proportional to the reciprocal of the square of the predicted value and for the pharmacodynamics with weights proportional to the reciprocal of the predicted value are shown. For the majority of fits, no systematic trends were observed in the plots of weighted residuals versus both time and predicted concentration for the respective weighting schemes. A typical set of pharmacokinetic and pharmacodynamic data together with the fitted

M. Pitsiu et al./ Population pharmacokinetics and pharmacodynamics of warfarin

154

Table 1 Individual pharmacokinetic and pharmacodynamic parameter estimates obtained by nonlinear least squares (a) Pharmacokinetics I

Mean Variance

k~ (h -~)

V~ (1)

k~ (h ')

Vs (1)

0.019 2.5 x 10-s

11.5 6.3

0.027 8.1 x 10 -s

12.6 9.0

Cso,s (mg/l)

ka (h-l)

7

tlag (h)

0.040 0.047

0.125 0.0051

1.03 0.19

8.58 7.53

Cso,s (mg/l)

kd (h -t )

"y

t lag (h)

O.31 0.026

0.28 0.27

2.63 6.91

6.91 8.83

(b) Pharmacodynamics: PCA data 2

Mean Variance

(c) Pharmacodynamics: factor VII data 2

Mean Variance

1Weighting proportional to the reciprocal of the square of the predicted value. 2Weighting proportional to the reciprocal of the predicted value.

models for one of the individuals is displayed in Fig. 1. The half-life of the r-enantiomer was about 40% longer than the s-enantiomer, which is consistent with earlier reports (O'Reilly, 1982). As expected, the degradation rate constant for factor VII activity Concentration ( m g / t )

% Activit'

0

0

1.0

100

0.8

80

60

0.6 ©

0.4

40

0.2

20

0.0 0

I

:

I

I

I

I

I

20

40

60

80

100

120

140

Hours Concentration

o

% Activity

Fig. 1. Concentration-time profile of s-warfarin and factor VII activity following a single 25 mg racemic dose of warfarin to a normal, healthy volunteer.

was larger than that for PCA, as the latter represents the overall activity of the vitamin K-dependent clotting factors and the half-lives of the other vitamin K-dependent clotting factors, II, IX and X, are considerably longer than factor VII. It is difficult to explain the difference in the shape parameter, 7, between PCA and factor VII as this parameter has no clear physiological interpretation. The results from the N O N M E M analysis are given in Table 2. The population estimates for the pharmacokinetic parameters were in good agreement with the means obtained from individual fitting. However, the interindividual variances were smaller than the variances obtained from the individual fits (only results for the additive error model are shown; the results from the multiplicative model were very similar). This is to be expected as the variances obtained from individual fitting contain a contribution from the error of estimation associated with each individual fit and unless the mean estimation error is relatively small the estimates of interindividual variance can be seriously upwards biased (Beal and Sheiner, 1982). The N O N M E M estimates are also broadly in agreement with those obtained by Mungall et al. (1985) from a multiple dose study of racemic warfarin in a patient population. The interindividual variabilities in the parameter estimates from the present normal, young population were less than those seen in the patient population, which is to be expected, particularly since the analysis of Mungall et al. involved sparse data.

155

M. Pitsiu et al./Population pharmacokinetics and pharmacodynamics of warfarin Table 2 Population pharmacokinetic and pharmacodynamic parameter estimates obtained by NONMEM analysist (a) Pharmacokinetics2 k, (h ~)

V~(1)

ks (h-~)

V~(1)

Estimate

0.0193 (0.0011)

10.5 (0.37)

0.0254 (0.0012)

11.8 (0.39)

Variance

2.37 × 10-5 (7.9 x 10 6)

4.12 (0.80)

4.40 x 10-5 (1.1 × 10-5)

5.61 (1.24)

Intraindividual variance

a2(r) = 0.0011 + 0.019 , f a

a2(s) = 0.015 + 0.026 , f 2

(b) Pharmacodynamics: PCA data 3

Cso,s (mg/l)

kd (h -1)

"7

Estimate

0.394 (0.032)

0.094 (0.004)

1.00 (0.04)

Interindividual variance Intraindividual variance

~r2 = 78.6 a 2 = 41.9

C5o,~(mg/l)

k d (h I)

7

Estimate

0.252 (0.020)

0.080 (0.003)

1.38 (0.08)

Interindividual variance Intraindividual variance

a 2 = 99.4 a 2 = 249

(c) Pharmacodynamics: factor VII data 3

1Parameter estimates with standard error of estimation in parentheses. 2Additive model used for interindividual variance components. 3Additive model for both intra- and interindividual variance models. In the case o f the response data it was not possible to partition the interindividual variance between the parameters (viz Eq. 5 - - the p r o g r a m m e failed to converge) and only a global estimate o f the interindividual variance was obtained with the N O N M E M analysis. In addition, it was not possible to estimate the lag time for the onset o f the anticoagulant response, and this parameter was set to the mean value obtained f r o m the individual analyses (Table 1). A n u m b e r o f variance models were tried with the constant variance (additive) model for b o t h the intraand interindividual variances, reported in Table 2, giving the lowest objective function and a satisfactory pattern o f residuals. The population parameter estimates obtained f r o m the P C A data are in g o o d agreement with the means obtained from the individual fits; However, there were some discrepancies for the factor V I I data. The q/2 for factor V I I corresponding to the kd estimate obtained f r o m the individual fits (2.5 h) was smaller than that previously reported whereas the N O N M E M estimate (8.7 h) approaches the literature value (Zolton, 1986). The estimate o f ' y was also over-

estimated f r o m the individual fits: the reported value being close to 1 for P C A (Holford, 1986). The Cso,s values were smaller than that reported by C h a n et al. (1984) based on P C A measurements. The estimated intraindividual variance for the factor V I I measurements was considerably greater than that for P C A , reflecting either greater imprecision in the factor VII assay or a p o o r e r fit o f the sigmoid-Imax model to the data. The results from the E M analysis are given in Table 3. The population estimates for the p h a r m a c o kinetic parameters are in g o o d agreement with both the means obtained from the individual fits and the N O N M E M analysis. The interindividual variances estimated by the E M algorithm were in g o o d agreement with those calculated by N O N M E M and, as noted before, are smaller than those obtained from the individual fits. There was g o o d agreement between the N O N M E M and E M estimates for b o t h the P C A and factor VII population parameter estimates. In addition it was possible to obtain estimates o f the lag time for onset

M. Pitsiu et al./Population pharmacokinetics and pharmacodynamics of warfarin

156

Table 3 Population pharmacokinetic and pharmacodynamic parameter estimates obtained by EM analysis (a) Pharmacokinetics I

Estimate Variance

k, (h l)

P,. (1)

k~ (h

V,.(1)

0.019 1.6 × 10 5

11.3 3.9

0.026 3.3 × 10 5

12.4 4.0

Cbo,s (mg/l)

ka (h - l )

"7

t lag (h)

0.379 0.038

0.087 1.7 × 10 5

1.05 0.08

8.44 4.45

Cbo,s (mg/1)

ka (h-l)

3'

t lag (h)

0.298 0.023

0.123 1.5 x 10 -3

1.66 0.15

6.76 7.31

')

(b) Pharmacodynamics: PCA data 2

Estimate Variance

(c) Pharmacodynamics: factor VII data 2

Estimate Variance

l Nonlinear least squares estimates obtained with weights proportional to the reciprocal of the predicted value squared. 2Nonlinear least squares estimates obtained with weights proportional to the reciprocal of the predicted value.

of anticoagulant response with the EM algorithm, which were in good agreement with the values obtained from the individual fits. The estimates of the interindividual variability in the parameter estimates were again somewhat smaller than those obtained from the individual fits. No comparison with N O N M E M was possible because, as noted above, N O N M E M failed to produce estimates of the interindividual variances associated with the pharmacodynamic parameters. The population mean response and variability for factor VII activity is shown in Fig. 2. The variability was estimated from the population model as shown by FVII

activity(%)

120

100 i

60 ~

4 0 -,

LI

~

"~

\

20 ! 0----0

~ 20

40

60

80

Time

100

120

i

i

140

160

(hr)

Fig. 2. Population mean response (solid line) and variability (broken line) for factor VII activity in a group of normal, healthy volunteers after receiving a 25 m g racemic dose of warfarin. See text for details.

the following equation

I po,m,, O/,2

¢0S7.

:

where c~is an estimate of the standard deviation of the response about its mean due to the interindividual variability (as expressed by the terms c~) in the pharmacodynamic parameters (estimated from the EM algorithm) and the warfarin concentration (cr2), and the intraindividual variability (or,2: estimated as 15% from the sum of squared residuals from the individual fits). The contribution of the various interindividual terms to the observed variability in the factor VII response is displayed in Fig. 3. The importance of the different terms changes with time. Most of the variability during the decay phase is, as expected, associated with the interindividual variability in k a, whereas the variabilty at 36 h, around the minimum in the activity profile, is associated with C5o:. Later on in time the term associated with the warfarin concentration becomes more important as the decline in the warfarin level dictates the return of the factor activity to its baseline. However, even at 96 h the interindividual variability in warfarin concentration only contributes 40% to the total interindividual variability. At steady state, for a factor VII clotting factor activity of 25% of normal, an interindividual variability of -4-18% would be predicted from these results, approximately 20% of which is due to the interindividual

M. Pitsiu et al./Population pharmacokinetics and pharmacodynamics of warfarin 1.0~

o8i I 0.67 0.4-

0.2!

D.Oi -

12

24

36

48

60

72

84

96

Time(hr) [[~

C50

~

kd

~

gamma~

conc

Fig. 3. Contribution of the various interindividual factors to the observed interindividual variability in factor VII response as a function of time. The ordinate scale is expressed as a fraction of the total interindividual variability.

variability in warfarin concentration and 80% due to the interindividual variability in Cso,s. It was not obvious why NONMEM failed to estimate the interindividual variability components for the response data and it would be dangerous to generalize from this one analysis. The disadvantage of the EM procedure is that it is a two-stage method and requires that each individual subject is modelled (successfully) before proceeding to the next stage. In contrast, NONMEM was designed to be used with sparse data. In data-rich situations like the present one, two-stage procedures, like the EM algorithm, may be more robust.

Acknowledgements We would like to thank Amy Racine-Poon for helpful discussions and Kabi Ltd, Sweden, for the supply of all reagents and the method for the factor VII chromogenic assay. We would also like to thank Medeval Ltd, University of Manchester, for providing the warfarin data. One of us (MP) thanks the Greek government and another (EMP) the University of Manchester for financial support.

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