PHARMACOKINETIC-PHARMACODYNAMIC MODELLING OF SIDE EFFECTS OF NITRENDIPINE I. Locatelli, I. Grabnar, A. Belič, A. Mrhar, R. Karba, University of Ljubljana, Slovenia Corresponding Author: I. Grabnar Faculty of Pharmacy, University of Ljubljana Aškerčeva 7, 1000 Ljubljana, Slovenia Phone: +386-1-4258-031, Fax: +386-1-4769-543 E-mail:
[email protected]
Abstract. The modelling aim in this study was to explore the relationship between plasma concentrations of a cardiovascular drug nitrendipine and occurrences of its side effects. The data were taken from a bioequivalence study in 40 men with a 20 mg single dose of nitrendipine. During the study occurrences of side effects, their severity, onset time and duration were recorded. Overall, side effects occurred in 26 applications. A two compartment pharmacokinetic model with lag time and first order absorption was fitted to plasma concentrations. For a pharmacodynamic part fixed effect model was applied. An indirect link model based on effect compartment was used to connect the pharmacokinetic and pharmacodynamic data. Estimated model parameters of a group with side effects and a group without them were compared. Increased extent and rate of absorption raised the probability for occurrence of side effect. Incidence of side effect was also determined by increased sensitivity to nitrendipine at the site of action. 1.
Introduction
Pharmacokinetics (PK) describes time courses of drug concentration in body fluids (mainly in plasma or blood) resulting from a certain drug dose and pharmacodynamics (PD) describes the observed effects resulting from a certain drug concentration [3]. Linking these two aspects results in pharmacokineticpharmacodynamic (PK/PD) modelling, which evaluates dose-concentration-response and facilitates description and prediction of the time course of drug effects resulting from a certain dosing regimen [3]. PK-PD models consist of 3 components: - a pharmacokinetic model; characterising time courses of plasma concentrations of the active drug and metabolite, - a pharmacodynamic model; characterising the relationship between drug concentration at site of action and effect, - a link model, which serves to account for the often-observed delay (especially under non steady-state conditions) of the effect relative to the plasma concentration [2]. The observed delay can be a consequence of a much slower distribution of the active drug towards the site of action - biophase. Another kind of delay that can be observed for some drugs is a delay in pharmacological reaction of a certain drug (e.g. warfarin). In this study PK/PD modelling was focused on a safety aspect of a cardiovascular drug nitrendipine. Our aim was to explore the relationship between its plasma concentration and a possibility for occurrence of side effect. In PK/PD modelling of side effects some uncommon problems can be met. Side effects are mainly represented as discrete random variables either in order of magnitude (e.g. slight, moderate, severe pain) or as dichotomous, quantal variable (e.g. pain or no pain) [1]. Another characteristic that has to be addressed is the fact that they are usually observed continuously. This can present many difficulties, since the majority of the PK/PD modelling software is designed to estimate the parameters by fitting a predicted curve to experimental data points and not to an experimental side effect-time curve. To get an insight into nitrendipine plasma concentration – side effect relationship we first developed a multiple logistic model to define the most influential demographic, pharmacokinetic, haematologic, haemodynamic and biochemical parameters t for the incidence of side effects. In the second stage a dynamic PK/PD model was developed. 2.
Nitrendipine
Nitrendipine is a potent calcium entry blocker. It inhibits movement of calcium ions through the voltagesensitive channels of vascular smooth muscles and consequentially causes vasodilatation of large and small arteries [5]. Its main therapeutic effect is a decrease in peripheral vascular resistance with subsequent
reduction of blood pressure [5]. It was demonstrated that its pharmacokinetics is very variable due to an extensive first pass metabolism and binding to plasma proteins (98 %). Because of first pass metabolism its bioavailability is low (F = 16 ± 6 %), despite almost complete absorption after peroral application due to high lipophilicity. In the range of therapeutic doses nitrendipine pharmacokinetics is linear. It does not accumulate after multiple dosing with once daily dosing regimen. Nitrendipine is eliminated mainly through liver by biotransformation into inactive and more polar metabolites. The terminal half-life is 10 to 24 hours. Using less sensitive assays the terminal half-life was estimated at lower values (2 to 10 h) [5]. It has been shown that antihypertensive effect of nitrendipine was presented even at plasma concentrations below the limit of quantification. Appearance of its pharmacological effect is in close relationship with its activity at the calcium channels rather than to its plasma concentrations. Nitrendipine causes some mild but quite frequent adverse effects, which usually disappear during continued treatment. The most important and the most frequent adverse effect is headache, which is caused by vasodilatation in the brain. Other common side effects are flushing, palpitations, ankle oedema and dizziness as a consequence of peripheral vasodilatation and increased baroreflex feedback control [5]. 3.
Database
The database for this study was taken from a blind, randomised, four-way cross over bioequivalence study with a single dose of 20 mg nitrendipine tablet. Forty healthy young male volunteers aged from 18 to 30 years were involved in the study. In each period 16 blood samples were taken from each volunteer at the following times (in hours): 0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 6, 8, 10, 12, 16, 24, 36 and 48. Three hours after drug administration blood pressure was measured in each volunteer. During the period of 48 hours after administration the volunteers were observed for eventual occurrences of side effects. The onset time, duration and severity of side effect was reported. Headache, flush and vertigo occurred in 26 out of 160 applications, headache in 24 applications, flush in 4 applications and vertigo in 1 application. In some applications more than one side effect occurred. The average duration of side effect was 3.3 ± 2.7 h for headache and 0.5 ± 0.1 h for flush and vertigo (mean ± standard deviation). Each volunteer was physically examined and complete laboratory tests were performed. The following parameters were evaluated: age, weight, height, heart rate, systolic (SBP0), and diastolic (DBP 0) blood pressure. Among haematological tests counts of erythrocytes, differential leukocytes (neutrophils, eosinophils, basophils, lymphocytes and monocytes) and thrombocytes were assessed, haemoglobin and haematocrite was determined. Among biochemical tests plasma concentrations of chloride, sodium and potassium ions, total proteins, albumin, urea, creatinine, total bilirubin, glucose and total cholesterol were estimated. Plasma levels of alkaline phosphatase (ALP), aspartate transaminase (AST), γglutamyltransferase (GGT), and lactate dehydrogenase (LDH) were also determined. All the values of laboratory examinations were in the normal range, but differed among the subjects. Nitrendipine concentrations in plasma samples were determined by GLC-MS. The linear area ranged from 0.1 ng/ml to 20 ng/ml. The limit of quantification was 0.1 ng/ml. 4.
Pharmacokinetic analysis
Noncompartmental and model dependent pharmacokinetic analysis of individual nitrendipine plasma concentration-time profiles was performed using WinNonLin software (Standard Edition, Version 2.1 Pharsight Corporation). The following noncompartmental PK parameters were identified (mean ± standard deviation): - terminal half-life (t1/2λz = 9.1 ± 6.5 h), - area under plasma concentration-time curve, by linear trapezoidal method (AUC0→∞ = 40 ± 31 h·ng/ml), - observed maximum plasma concentration (CMAX = 10 ± 7.2 ng/ml), - time when maximum concentration was observed (tMAX = 1.7 ± 0.9 h), - mean residence time (MRT = 7.1 ± 3.1 h), - systemic plasma clearance divided by absolute bioavailability (CLP/F = 14 ± 10 L/min). Obtained parameters are in agreement to the literature data [5]. Large variability in AUC0→∞ and CLP/F demonstrates that bioavailability is very variable even in relatively homogenous group of healthy men. Nitrendipine pharmacokinetics after oral administration was assessed also using model-dependent approach. Four linear compartment models with first order absorption (one and two-compartment, with and
without lag time) were fitted to individual plasma concentration-time profiles. Model parameters were estimated by iterative Gauss-Newton algorithm with Levenberg and Hartley modification. Residual sum of squares was used as a criterion function in the optimisation procedure. Uniform data weighting scheme was applied. Model parameters are summarised in Table 1. Table 1: Estimated pharmacokinetic model parameters (mean ± standard deviation), absorption rate constant kA, elimination rate constant kEL, distribution rate constant from central to peripheral compartment k12, distribution rate constant from peripheral to central compartment k21, volume of distribution VD, absolute bioavailability F and absorption lag time tLAG. PK model
VD/F [L]
kA [h-1]
kEL [h-1]
k12 [h-1]
k21 [h-1]
tLAG [h]
one-compartment one-compartment with tLAG two-compartment two compartment with tLAG
2200±1940
1.1±1.4
0.50±0.20
/
/
/
2360±1940
6.2±6.7
0.48±0.28
/
/
0.54±0.43
1850±1490
1.1±2 .7
0.48±0.16
0.17±0.23
0.10±0.24
/
2020±1790
4.7±5.6
0.48±0.23
0.35±0.46
0.33±0.50
0.55±0.50
The results in Table 1 demonstrate that PK variability of nitrendipine was very high, especially in the absorption rate constant kA. This could be partially attributed to a small number of observations in the absorption phase (in some applications the first data point, t = 0.5 h, was also the concentration maximum). Therefore a bias in kA estimation was taken into account. Validity of each PK model was investigated through determination of Akaike Information Criterion (AIC), Schwarz Criterion (SC) and parameters’ coefficients of variation (CV). The two-compartment model with lag time of absorption was found as the most adequate, as it had the lowest average values of AIC, SC and CVs. The graphical analysis confirmed this evaluation. This finding coincided with data from the literature [5, 7], where biphasic nitrendipine disposition was proposed [7]. 5.
Statistical evaluation
SPSS 11.0 (SPSS Inc.) was used for statistical analysis. All data from 160 drug applications were divided into two groups; those, in which side effects occurred (26 applications) and those without side effects (134 applications). Summary statistics of several variables including compartmental and noncompartmental PK parameters were computed. Inter-group comparisons were done using independent samples t-test and Levens’ F-test was applied for evaluation of homoscedascity. A two-tailed p-value less than 0.05 was considered to be significant. Results of the noncompartmental PK analysis and parameters of the twocompartment model with lag time are presented in Table 2. Table 2: Descriptive statistics of noncompartmental pharmacokinetic parameters and parameters of the twocompartment model with lag time in absorption. noncompartmental PK analysis no side effect side effects p value model-dependent PK analysis no side effect side effects
AUC0→∞ [h·ng/ml]
CMAX [ng/ml]
t1/2λz [h]
CLP/F [L/h]
tMAX [h]
35.1 ± 28.2
9.0 ± 6.9
8.5 ± 6.6
0.9 ± 0.6
1.8 ± 0.9
56.6 ± 33.6 13.8 ± 7.2
11.4 ± 5.7
0.5 ± 0.4
1.6 ± 0.7
0.001 * AUC0→∞ [h·ng/ml]
0.002 * CMAX [ng/ml]
0.040 * CMAX /(tMAX - tLAG) [ng/ml·h]
36.6 ± 31.8
9.2 ± 7.5
21.1 ± 28.2
0.26 ± 0.13
1.0 ± 0.7
54.7 ± 31.5 14.2 ± 7.8
35.9 ± 36.6
0.28 ± 0.15
0.9 ± 0.7
0.010 *
0.253
0.242
p value 0.004 * 0.001 * * significant difference (p < 0.05)
0.000 * 0.500 CMAX /AUC0→∞ tMAX–tLAG [h-1] [h]
MRT [h] 6.9 ± 3.1 8.0 ± 2.7 0.080 kA [h-1] 4.4 ± 5.4 5.8 ± 6.8 0.136
Values of AUC0→∞ and CMAX were found to be significantly higher in the group with side effects using noncompartmental and model dependent PK analysis. This finding indicates that increased extent of absorption raised the probability for occurrence of side effects. Parameter CMAX /(tMAX - tLAG) was used as a measure of the absorption rate. Although it was significantly higher in the group with side effects, other absorption rate parameters (tMAX, tMAX–tLAG and kA) did not differ significantly. This can be attributed to few observations in the absorption phase. Rate of absorption therefore was not reliably estimated. It is therefore possible that also increased absorption rate raises the probability for occurrence of side effects. Comparison of the parameters of the physical examination, laboratory tests and blood pressure measurements revealed significant differences in body height, it was smaller in the group with side effects, p = 0.04; diastolic blood pressure drop 3 hours after drug administration (DBPdiff), it was bigger in the group with side effects, p = 0.04; number of neutrophils, it was smaller in the group with side effects, p = 0.02; number of monocytes, it was smaller in the group with side effects, p = 0.02; number or lymphocytes;it was bigger in the group with side effects, p = 0.04 and total plasma protein concentration, it was smaller in the group with side effects, p = 0.04. Body height is related to the volume of distribution, therefore increased drug exposure in smaller subjects is expected. Since nitrendipine is highly bound to plasma proteins, total plasma protein concentration affects the concentration of unbound drug and its distribution to the site of action. The influence of diastolic blood pressure drop can be explained by baroreflex feedback mechanism, which is controlling blood pressure. Rapid drop of blood pressure results in incresed heart rate. 6.
Multiple logistic regression
To be able to predict the occurrence of side effects multiple logistic regression models were built, based on a set of predictor variables of the physical examination laboratory tests and noncompartmental PK parameters. Models were estimated by block entry of variables using the SPSS method ENTER and a stepwise forward conditional method (FSC). In the second method - FSC (p1, p 2) each predictor variable was included in the regression model only if its starting significance level was less than p1 and if during regression it did not exceed p2. Six logistic regression models were built. They included observed systolic and diastolic blood pressures (BP) or drops in systolic and diastolic blood pressures (BPdiff): - Enter, [BP], - Enter, [Bpdiff], - FSC (0.1, 0.2), [BP], - FSC (0.1, 0.2), [BPdiff], - FSC (0.2, 0.4), [BP], - FSC (0.2, 0.4), [BPdiff]. This way the most influential predictor variables for occurrence of side effects were identified. Results are summarised in Table 3. Table 3: Learning errors (L.E.) of the six different logistic regression models and a list of the most influential parameters. logistic models L.E.
Enter [BP] 9.4 % tMAX (–),
influential AUC0→∞ (+), parameters SBP 3 (+), DBP 3 (+)
Enter [Bpdiff] 10.1 %
FSC (0.1, 0.2) FSC (0.1, 0.2) FSC (0.2, 0.4) [BP] [BPdiff] [BP] 13.8 % 13.2 % 13.8 %
AUC0→∞ (+), tMAX (–), K+ (–), Na+ (+), AUC0→∞ (+), total prot. (–), SBPdiff (+), lymphocyt. (+), DBPdiff (–) DBP 0 (+)
FSC (0.2, 0.4) [BPdiff] 11.3 %
AUC0→∞ (+), AUC0→∞ (+), height (–), AUC0→∞ (+), K+ (–), Na+ (+), Na+ (+), LDH (–), height (–), total prot. (–), tot. prot. (–), total prot. (–), lymphocyt. (+), lymph. (+), lymphocyt. (+) DBP 0 (+) gluc. (–), erytr. (+), DBPdiff (–)
(+) and (–) denotes a positive or negative value of a regression coefficient Again higher value of AUC0→∞ indicated higher probability for the occurrence side effect. The influence of increased absorption rate (smaller tMAX) was also observed. Again the effect of total plasma protein concentration, body height, and count of lymphocytes was evident. Moreover logistic regression revealed
the importance of plasma concentrations of sodium and potassium ions. These ions are the main body electrolytes and act on cell’s resting membrane potential. Higher sodium and a lower potassium plasma concentration results in increased sensitivity of target cells to nitrendipine. To test the predictive capability of the logistic regression models the database was split to a learning set of randomly selected 112 applications (18 with side effect and 94 without) and a test set of 48 applications (8 with side effect and 40 without). This cross validation procedure was repeated three times (group I to III) choosing three different learning and testing sets. Learning (L.E.) and prediction (P.E.) errors for all the models are summarised in Table 4. Table 4: Learning and prediction errors for multiple logistic regression models. Enter [BP] L.E. for I. group 0% L.E. for II. group 0% L.E. for III. group 5% mean 2% P.E. for I. group 42 % P.E. for II. group 38 % P.E. for III. group 23 % mean 34 % * no convergence was achieved logistic models
Enter [Bpdiff] /* 5% 5% 5% /* 31 % 21 % 26 %
FSC (0.1, 0.2) [BP] 13 % 14 % 14 % 14 % 27 % 12 % 15 % 18 %
FSC (0.1, 0.2) [Bpdiff] 13 % 14 % 14 % 14 % 27 % 17 % 17 % 20 %
FSC (0.2, 0.4) [BP] 9% 10 % 11 % 11 % 31 % 17 % 25 % 24 %
FSC (0.2, 0.4) [BPdiff] 9% 12 % 11 % 11 % 31 % 19 % 25 % 25 %
Learning errors were between 0 and 14 %, while prediction errors were between 12 and 42 %. Models based on block entry of variables (Enter) had the lowest learning errors and the biggest prediction errors. The best model for prediction of side effects was FSC (0.1, 0.2) [BP]. 7.
PK/PD link model
For the PK part of the link model already established two-compartment PK model with lag time (2-comp.+ tLAG) was used. Fixed effect (logistic) model was used to relate nitrendipine concentration to the probability of side effect. This model is based on a side effect threshold concentration. The probability of the side effect (p) is then calculated as p = cN/(ec50N + cN) – equation 1, where c is a concentration of a drug and ec50 is the concentration that produces a 50 % probability that the effect occurred. Parameter N denotes a slope of probability versus concentration curve [6]. In our case a time delay between maximum plasma concentration and appearance of side effect was observed (Figure 1), which resulted in a hysteresis in a side effect versus plasma concentration plot. For the side effects flush and vertigo little or no hysteresis was observed. Furthermore the duration and the onset of the headache differed from the other two side effects considerably. Therefore PK/PD model parameters were estimated only for the headache time courses. Due to the observed hysteresis indirect link model was established. This link model included additional, hypothetical compartment; an effect compartment (EC). The drug concentrations in effect compartment (CE) represented the drug concentrations at the effect site, which were related to the probability of side effect. Time delay between side effect occurrence and plasma drug concentration was modelled as a first order distribution of a drug from plasma to the effect compartment (rate constant kEO). A PK/PD model is schematically presented in Figure 2. Six PK parameters (VD/F, kA, kEL, k12, k21, and tLAG) that were already estimated were set as individual constraints. Two different procedures were used to fit the three PK/PD model parameters (kEO, ec50, and N) to the observed headache-time courses in each of the 24 applications with side effect. Firstly the parameters were estimated using WinNonLin. Continuously observed headache-time data were transformed to equally spaced discrete data at time intervals of 0.1 h. The estimation of N depended on initial values and it tended to higher values to get a better fit. In a second procedure a custom spreadsheet in Excel was developed, where the N was set to 300, as this was the highest possible value for which the expression (CE)N could be computed. Parameter kEO was evaluated by solving the equation CE(tSTART) = CE(tEND). The concentration in the EC (CE) at the onset of headache is equal to the concentration in EC at the termination of headache in order to get the best fit. Moreover these concentrations also equalled ec50 (figure 1). Since the values for N were high in order to get steep and momentary onset of probability of headache (left bottom graph in figure 2), population value for N (n’) was calculated as n’ ≈ 4/SQRT(var(lnec50)·2π). The expression var(lnec50) denotes a variance of lognormal distribution of ec50 and n’ denotes a slope of a cumulative curve of
lognormal distribution of individual ec50. These estimates were comparable to those obtained by WinNonLin (Table 5). 2
ABSORPTION
1.0 15 0.8
10
0.6
0.4 5 0.2
0
k21
k12
CENTRAL 20 mg kA 1 COMPARTMENT nitrendi- t LAG lnCP pine VD/F
1.2
Probability of headache
Nitrendipine concentration [ng/ml]
20
PERIPHERAL COMPARTMENT
kEL
t
kEO kEO
CE(t) EFFECT COMPARTMENT Ν
ec 50
p
E L I M I N A T I O N
headache 1
0.0 0
4
8
12
16
20
24
CE
Time [h]
Figure 1. PK/PD model response (dash-dotted curve) for a representative application with headache. Pharmacokinetic model (2-comp.+ tLAG) response (solid curve), concentration in the effect compartment (dashed curve). The observed plasma concentrations (dots), observed headache (grey area). The PK model response was obtained by WinNonLin. The presented PK/PD model response and CE were calculated in Excel.
t
Figure 2. A diagram of the overall PK/PD model. The upper part represents a PK model with parameters VD/F, kA, kEL, k12, k21, and tLAG. In central compartment the observed plasma concentrations (CP) are drawn. The lower part presents PD model parameters (kEO, ec50, and N) that were estimated. In right bottom graph pharmacodynamic data are drawn. The aim of the PK/PD model was to get the concentrations in EC (CE) and to get the pharmacodynamic curve (left bottom graph), where p is a probability for occurrence of headache.
Table 5: Means and standard deviations of the pharmacodynamic parameters estimated by WinNonLin and an Excel spreadsheet. Estimates in WinNonLin Estimates in EXCEL
n’ 2.2 2.0
kEO [h-1] 0.15 ± 0.12 0.21 ± 0.19
ec50 [ng/ml] 2.6 ± 1.9 2.9 ± 2.2
Population values for each of the three PK/PD model parameters were estimated using nonlinear regression in SPSS. The PK/PD model was fitted to discrete headache-time courses (intervals of 0.05 h) for all of 158 applications in a single step. The two applications in which only flush and vertigo occurred were not included. Parameters of the subpopulation of 24 applications with headache were also evaluated in order to compare them with the average values of the previously estimated parameters. The subpopulation value of N estimated in SPSS is compared to previously determined n’. The results are shown in Figure 3.
p
1
0.8
0.6
0.4
0.2
0 0.1
1
10
100
1000
concentration in the effect compartment [ng/ml]
Figure 3. Vertical lines denote individual pharmacodynamic curves (estimated in MS EXCEL) for each of all 24 applications that had the headache. A thick curve represents the subpopulation curve calculated on equation 1 using average value of the ec50 (estimated by MS EXCEL) and using n’ as N. The thick curve also represents a cumulative lognormal distribution of individual values of ec50. The other two curves were also calculated by equation 1 using subpopulation (dashed curve) and population (thin curve) values for N and ec50 that were estimated with the SPSS. The thick and the dashed curves were expected to overlap as they both represent a subpopulation with headache. This discrepancy can be explained by a strong positive correlation between N and ec50. A higher value of N that was set in Excel calculations resulted in lower values of ec50 and movement of the thick curve to the left. Even though considerable difference between population and subpopulation curves was found (the dashed and the thin curve). Subpopulation and population values for ec50 were 6.62 ± 0.22 ng/ml and 39.8 ± 2.5 ng/ml, respectively. Here it must not be forgotten that our data is based on healthy young male population and the absolute values should not be extrapolated. As the difference was significant we concluded that the headache was occurred also due to the individual’s increased sensitivity (lower ec50) to nitrendipine concentration at the site of action. 8.
Conclusions
The aim of the modelling was to relate nitrendipine side effects to the pharmacokinetic and pharmacodynamic properties of this drug. This was done in two mayor steps. First only the presence of side effects was taken into account. Here the reasons for presence of side effects were established using some statistical tools. An increased extent and rate of absorption as well as an increased sensitivity to nitrendipine at the site of action were found to expand the probability for side effects (headache, flush and vertigo). A PK/PD model was fitted to side-effects-time courses. A two-compartment PK model with lag time and fixed effect PD model were indirectly linked into PK/PD model. Individual and population values of the PD parameters were evaluated. Differences in population and subpopulation values of ec50 confirmed that side effects partly occurred due to increased sensitivity to nitrendipine. The developed methodology could supply useful criteria for the design of optimal drug dosage regimen, moreover it offers possibility for selecting individuals with high probability for adverse drug reactions. 9.
References
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3. Derendorf, H. and Meibohm, B., Modeling of Pharmacokinetic/Pharmacodynamic (PK/PD) Relationships: Concepts and Perspectives. Pharmaceutical Research, 16 (1999), 176 – 185. 4. Gabrielsson, J. and Weiner, D., Pharmacokinetic and Pharmacodynamic Data Analysis; Concepts and Applications. 3rd Ed., Swedish Pharmaceutical Society, Stockholm, 2000, 261-317. 5. Goa, K.L. and Sorkin, E.M., Nitrendipine A Review of its Pharmacodynamic and Pharmacokinetic Properties, and Therapeutic Efficacy in the Treatment of Hypertension. Drugs, 33 (1987), 123 – 125. 6. Hung, O.R., et al., Thiopental Pharmacodynamics II. Quantitation of Clinical and Electroencephalographic Depth of Anesthesia. Anesthesiology, 77 (1992), 237 – 244. 7. Raemsch, K.D. and Sommer, J., Pharmacokinetics and metabolism of nitrendipine. In: Nitrendipine, (Eds.: Scriabne et al), Urban and Schwarzenberg, 1984, 409 – 421. 8. Welling, P.G., Pharmacokinetics: Processes, Mathematics, and Applications. 2nd Ed., American Chemical Society, Washington, DC, 1997. 9. WinNonLin User’s Guide, Scientific Consulting, Inc., 1997.
