- Email: [email protected]
Optimization of Release Kinetics from Sustained-Release Formulations using Model-Independent Pharmacokinetic Simulation
Optimization of Release Kinetics from Sustained-Release Formulations using Model-Independent Pharmacokinetic Simulation ´ CRISTINA MADERUELO,1 ARANZAZU ZARZUELO,1,2 JOSE´ M. LANAO2 1
Usala Laboratories, C/ Licenciado M´endez Nieto s/n 37007, Salamanca, Spain
2 Department of Pharmacy and Pharmaceutical Technology, Faculty of Pharmacy, University of Salamanca, C/ Licenciado M´endez Nieto s/n 37007, Salamanca, Spain
Received 24 December 2010; revised 2 March 2011; accepted 2 March 2011 Published online 11 April 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jps.22565 ABSTRACT: In the present work, a single model-independent approach was developed to optimize the release kinetics of drugs from sustained-release formulations, using stavudine (d4T) as a model drug. This approach is based on the pharmacokinetic simulation of drug plasma levels through a semiparametric approach of the input function and on convolution with an empirical polyexponential unit impulse response function. Input functions were evaluated using different zero-order and first-order release constants. Optimum drug release to obtain a specific pharmacokinetic profile was approached using target model-independent pharmacokinetic parameters such as Cmax SS , Cmin SS , tmax SS , and peak-trough fluctuations. A Monte Carlo simulation was performed to estimate the fractional attainment of d4T plasma concentrations over therapeutic d4T levels. Zero-order (K0 = 4 mg/h) and first-order (K1 = 0.05 h−1 ) release constants were optimal for the formulation of sustained-release d4T tablets, plasma concentrations within the therapeutic range being achieved. ©2011 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 100:3260–3267, 2011 Keywords: Sustained-release formulations; convolution; release rate; Monte Carlo simulation; preformulation; formulation; controlled release/delivery; pharmacokinetics; computer modeling; oral drug delivery
INTRODUCTION Sustained-release formulations (SRFs) are an alternative to immediate-release pharmaceuticals with undoubted advantages as regards to the therapeutic point of view, such as prolonging the therapeutic effect, increasing the dosage interval, or reducing side effects.1,2,3,4 From the technological point of view, there are different kinds of SRFs based, among others, on hydrophilic matrices, pellets, osmotic systems, or systems based on membranes to control diffusion. With these systems, the drug is released slowly, which allows the input of the drug to be controlled in the organism, leading to changes in the pharmacokinetic Correspondence to: Jos´e M. Lanao (Telephone: +34-923-294536; Fax: +34-923-294515; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 100, 3260–3267 (2011) © 2011 Wiley-Liss, Inc. and the American Pharmacists Association
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profile with therapeutic implications.5,6 With the different types of SRFs, it is possible to obtain different release rates, such as zero order, first order, or complex orders.7,8 Traditionally, model-independent pharmacokinetic analysis and, in particular, deconvolution methods have been used successfully to characterize the in vivo release kinetics of drugs from this type of formulation, once they have been developed and administered to animals or humans.9–12 However, there is a lack of information concerning the use of simulation methods and model-independent pharmacokinetic analysis for the optimal design of release rates in SRFs during the initial stages of pharmaceutical development. The aim of the present work is to optimize the drug release rate from SRFs using stavudine (d4T) as a model drug and model-independent methods based on convolution, in order to achieve a suitable pharmacokinetic profile of the drug of this type of pharmaceutical dosage form.
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MATERIAL AND METHODS Optimization of the Release Rate from SRFs Optimization of the release kinetics of a drug from SRFs to achieve a suitable pharmacokinetic profile requires different steps, as seen in the flowchart in Figure 1. These can be summarized as follows: 1. Information about the release rate of the drug from the sustained-release pharmaceutical form. In vitro dissolution assays using the United States Pharmacopeia (USP) methods are able to provide information about the type of release and the mechanisms involved.13 2. Prior knowledge about the pharmacokinetic behavior of the drug from essential pharmacokinetic parameters, such as bioavailability, the apparent distribution volume, and clearance, as well as about interindividual and residual variability. It is also possible to work with polyexponential equations that will define the pharmacokinetic profile of the drug. Previous studies addressing the population pharmacokinetics of the drug can provide this kind of information.14 3. Information about the therapeutic range of the drug or, alternatively, pharmacodynamic models that will quantify the relationship between the pharmacological response and the plasma concentrations of the drug. For a given drug, this type of information will allow the determination of the plasma concentration range and the pharmacokinetic profile associated with therapeutic
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efficiency and its relationship with the release profile of the drug from the SRFs. 4. Deterministic simulation. Simulation of the pharmacokinetics of the drug in the SRFs by means of model-dependent (e.g., compartmental analysis) or model-independent (e.g., convolution) methods. 5. Probabilistic simulation. Monte Carlo simulation is an excellent tool for estimating the probability distribution of target plasma concentrations using a specific dosage form (e.g., SRF) and dosage regimen.15 This is because of its flexibility and the fact that empirical distributions can be handled.16
With pharmacokinetic simulation, it is possible to establish correlations between changes in terms of the bioavailability of the drug in amount and rate from the SRFs and modifications in the plasma levels of the drug, reflected in important pharmacokinetic parameters such as the area under the curve, maximum plasma concentration (Cmax ), time to maximum concentration (tmax ), fluctuations in levels, and so on. According to the flowchart depicted in Figure 1, the simulation of the plasma levels using pharmacokinetic information of a drug incorporated in a drug delivery system can be accomplished with modeldependent methods, based on compartmental analysis, and model-independent methods, based on the convolution approach, which was the method followed here.
Figure 1. Flowchart with the steps to optimize the release kinetics of a drug from sustainedrelease formulations. DOI 10.1002/jps
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Pharmacokinetic Simulation Using Model-independent Methods Convolution is an interesting model-independent tool used in the field of pharmaceutical development for predicting plasma levels in type A in vivo/in vitro correlations (IVIVC), with implications at regulatory level,17–19 which can be used to optimize the release kinetics of the drug from different pharmaceutical dosage forms. Considering a SRF as a continuous-time, linear, and time-invariant system, drug plasma levels can be simulated from a time domain point of view using the following convolution integral: t I(t) F(t − J)dt
R (t) =
(1)
0
where R(t) represents the response function, which corresponds to the in vivo plasma levels curve after administration of the sustained-release dosage form; I(t) represents the input function, which corresponds to the in vivo drug release kinetics of the SRF; F(t) represents the unit impulse response, which corresponds to the in vivo plasma or serum levels curve after the administration of a unit impulse of the immediaterelease dosage form, and J is a constant. In an equivalent way, the system can be characterized in the frequency domain using Laplace transformation. Taking into account that the considered SRF can be considered as a linear system, the use of Laplace transforms facilitates the convolution operation transforming the integral in a simple multiplication operation of the Laplacian equations of the time functions. In the Laplace domain, the convolution integral can be expressed as: R(s) = I (s)F(s)
(2)
In a semiparametric approach, and depending on the kind of release kinetics considered, the input function may be expressed by the following algebraic expressions: K0 (3) I (s) = s FDK 1 (4) (s + K 1 ) The unit impulse response can be expressed as an empirical polyexponential function: I(s) =
F(s) =
n i=1
Ai (s + 8i )
(5)
where K0 is the zero-order release constant, K1 is the first-order release constant, F is the amount of bioavailability, D is the dose, s is the Laplace operaJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 8, AUGUST 2011
tor, Ai and λi are the coefficient and exponent of the exponential, and n is the number of exponentials. By using specific software, such as MULTI-FILT (Kyoto University, Kyoto, Japan), we can directly write the equations in the Laplace domain and solve it using the numerical inversion of the Laplace transformation.20 This program carries out the simulation of plasma concentration–time data by using Laplace transformed equations corresponding to the unit impulse response expressed as polyexponential function (Eq. 5) convoluted with the input function characterized by Eqs. 3 (zero-order release) or 4 (firstorder release). Then, Eqs. 2–5 can be used to simulate plasma level curves using pharmacokinetic information of a drug incorporated into an SRFs using different types of release and release rates. Release Optimization of d4t SRFs The antiretroviral agent d4T was chosen as a model drug in view of its good water solubility21,22 and its rapid and complete absorption after oral administration.23 Owing to its short elimination halflife, d4T needs to be dosed twice a day. Development of d4T SRFs can provide plasma d4T concentrations for 24 h with once-daily dosing. In the present work, pharmacokinetic simulation was used to estimate the optimal release kinetics of d4T from SRFs to achieve drug plasma levels within the therapeutic range for 24 h. Using the population pharmacokinetics of d4T in humans,14 shown in Table 1, an empirical polyexponential unit impulse response was generated, expressed as the plasma concentration (Cp = X p /Vd ), where Xp is the drug amount in plasma. Cp = 0.021e−0.408t − 0.021e−4.3t
(6)
This polyexponential equation expressed as Laplace transform (Eq. 5) was used as a unit impulse response in different simulations. Computer Table 1. Stavudine Population Pharmacokinetic Parameters and the Interindividual and Residual Variabilities Corresponding to Noncompartmental Pharmacokinetic Analysis 14 in Humans Receiving 40 mg Dose Orally (Schaad et al. ) Noncompartmental Kinetic Analysis
Vd (L/kg) CL [L/(h kg)] Cmax (ng/mL) AUC (ng h/mL) tmax (h) Fg2
Parameter
Variability CV (%)
0.743 0.303 1215 2087 0.600 –
20 32 32 39 17 17
Vd , apparent distribution volume; CL, plasma clearance; Cmax , maximum plasma concentration; AUC, area under plasma level curve; tmax , time to maximum concentration; Fg2 , residual variability; CV, coefficient of variation.
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simulations by convolution of d4T plasma levels in SRFs at a dose of 80 mg were used to analyze the changes in the release profile in plasma d4T concentrations under different release constants: first order (K1 = 0.05, 0.1, 0.15, and 0.2 h−1 ) and zero order (K0 = 4, 6, 8, 12, and 16 mg/h). Simulated plasma levels of d4T employing SRFs with different release rates were generated using Eqs. 1–5 through numerical inversion of the Laplace transform (FILT).20 Using the plasma level curves of d4T generated for the different release rates, model-independent pharmacokinetic parameters at steady state in a multiple dosage regimen as steady-state maximum plasma concentration (Cmax SS ), steady-state minimum plasma concentration (Cmin SS ), time to maximum concentration at steady state (tmax SS ), terminal slope of plasma level curve (λz ), plasma half-life (t1/2 ), and peak-trough fluctuation (PTF) were used for the evaluation. The percent PTF of plasma levels at steady state was calculated using the following equation24 : SS (CSS max − Cmin ) (7) Cav The average steady-state plasma concentration (Cav ) was calculated as the ratio between the area under plasma levels curve and the dosage interval (AUC/J). Previously published data from pharmacokinetic and pharmacodynamic studies and from work addressing the therapeutic efficiency of d4T in vitro/ in vivo were used to establish the relationship between the simulated plasma levels from the SRFs and the therapeutic efficiency potential of the drug when incorporated into SRFs.25 Pharmacokinetic analysis was completed using Monte Carlo simulations.15 This method considers the variability in pharmacokinetic behavior to determine the probability of a sustained target d4T
%PTF = 100
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plasma level being reached. On the basis of previous pharmacodynamic studies, a plasma level of d4T above 90 ng/mL was set as the target.25 The GoldSim Pro software version 10.1 (Goldsim Technology Group, Issaquah, WA, USA) was used for population simulations of subjects receiving d4T in different SRFs.26 The geometric mean and the geometric standard deviation of the pharmacokinetic parameter values (coefficients and exponents) corresponding to an empirical polyexponential function of the response function were employed. The parameters were sampled from a log-normal distribution. The choice of distribution was determined by the kind of statistical distribution of the original mean parameter vector and their variances. Monte Carlo sampling of stochastic elements was performed using a Latin Hypercube Sampling.27 Ten simulations were performed to generate 1000 simulated subjects d4T steady-state plasma level curves, each for dosages of 40 mg every 12 h incorporated into immediate-release formulations, and 80 mg every 24 h incorporated into SRFs with first-order release constants of 0.05, 0.1, 0.15, and 0.20 h−1 and zero-order release constants of 4, 6, 8, 12, and 16 mg/h. The first-order and zero-order constants were tested in independent simulations.
RESULTS Figures 2 and 3 show the simulated d4T concentration versus time profiles from SRFs obtained by numerical convolution for first-order release constant (K1 ) values between 0.05 and 0.2 h−1 and zero-order release constant (K0 ) values between 4 and 16 mg/h, respectively. Figure 4 shows the correlations established between the Cmax and the different first-order and zeroorder release constants used for the simulation. Table 2 shows the values obtained by pharmacokinetic simulation at steady state of Cmax SS , Cmin SS
Figure 2. Simulated d4T concentration versus time profiles from sustained-release formulations obtained by convolution for first-order release with K1 values between 0.05 and 0.2 h−1 . DOI 10.1002/jps
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Figure 3. Simulated d4T concentration versus time profiles from sustained-release formulations obtained by convolution for zero-order release with K0 values between 4 and 16 mg/h.
tmax SS , λz , t1/2 , and the PTF of d4T for first-order and zero-order release kinetics. Figure 5 shows the simulated probability density to steady-state plasma concentrations of d4T at 24 h, after one simulation with 1000 patients considered as replications using Monte Carlo simulation for a SRF with a first-order release constant of 0.05 h−1 administered through the oral route at a dose of 80 mg every 24 h.
The results of the analyses of the probability of d4T plasma concentrations being reached through Monte Carlo simulation using immediate-release formulations and SRFs are shown in Figure 6. The overall fractional attainments of achieving a plasma concentration greater than 90 ng/mL with d4T SRFs (80 mg every 24 h) at 12 and 24 h with a first-order rate constant of 0.05 h−1 were 95% and 30%, respectively. These overall fractions at 12 and 24 h with a
Figure 4. Linear correlations established between Cmax SS obtained by simulation for firstorder kinetics and the different K1 tested (a) and between Cmax SS obtained by simulation for zero-order kinetics and the different K0 tested (b). JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 8, AUGUST 2011
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Table 2. Values Obtained by Simulation of Pharmacokinetic Parameters of Stavudine at Steady State for First-Order and Zero-Order Release Kinetics by Using an 80 mg/day Dosage regimen First-Order Release Constant [Kl (h−1 )] 0.05 λz (h−1 ) t1/2 (h) tmax SS (h) Cmax SS (ng/mL) Cmin SS (ng/mL) PTF (%)
0.055 12.60 5.46 189.07 87.51 64.59
0.10
0.15
0.20
0.11 6.30 4.72 250.54 50.22 127.40
0.16 4.30 4.14 305.89 22.20 178.51
0.21 3.30 3.66 374.72 13.21 229.92
Zero-Order Release Constant [K0 (mg/h)] 4
6
8
0.40 1.73 20.00 188.51 37.40 96.11
0.40 1.73 13.33 283.66 3.97 177.88
0.40 1.73 10.00 370.13 1.37 234.53
12 0.40 1.73 6.67 528.14 0.52 335.57
16 0.40 1.73 5.00 645.94 0.32 410.62
λz , terminal slope of plasma level curve; t1/2 , plasma half-life; tmax SS , time to maximum concentration at steady state; Cmax SS , steady-state maximum plasma concentration; Cmin SS , steady-state minimum plasma concentration; PTF, peak-trough fluctuation.
zero-order rate constant of 4 mg/h were 99.9% and 1.5%, respectively. The optimal d4T release constant from drug delivery devices is 0.05 h−1 (first order) and 4 mg/h (zero order), which ensures that the plasma concentrations and the predicted patient variability will be within the therapeutic range (100–1000 ng/mL) close to a 24 h period.
DISCUSSION In SRF, the rate of bioavailability is conditioned by the release rate, which in turn governs the pharmacokinetic profile of the drug, prolonging the pharmacological response and allowing the dosage interval to be increased.
Within the field of the development of this type of formulation, it is of crucial interest to have a priori knowledge about the optimal profile of the drug in vivo that will allow a pharmacokinetic profile appropriate for guaranteeing therapeutic efficiency to be obtained. Prior knowledge of the plasma levels associated with a suitable pharmacological response, together with pharmacokinetic simulation methods, will allow the optimal kinetics of the release of this type of formulation to be established, depending on the type of release kinetics. Here, we developed a pharmacokinetic simulation methodology that allowed this aim to be achieved, following the flowchart depicted in Figure 1. d4T was used as the model drug; it is an antiretroviral that is highly soluble in water, with an elimination
Figure 5. Simulated probability density to steady-state plasma concentrations of stavudine at 24 h, using Monte Carlo simulation for an 80 mg/day oral sustained-release formulation (one simulation with 1000 subjects) with a first-order release constant of 0.05 h−1 . DOI 10.1002/jps
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Figure 6. Probability of exceeding the stavudine steady-state plasma concentrations with different release constants: (a) first-order release at 12 h, (b) first-order release at 24 h, (c) zero-order release at 12 h, and (d) zero-order release at 24 h.
half-life of about 2 h, and hence a good candidate for being incorporated into SRFs.28 As may be seen in Figures 2 and 3, a progressive change occurs in the kinetic profile of d4T as the release constant is modified, with both zero-order and first-order profiles. These changes are reflected in modifications in the plasma levels; in the fluctuation of levels; and in the Cmax SS , Cmin SS , and tmax SS values, as seen in Figure 4 and Table 2. One crucial aspect is to know the doses of d4T associated with clinical efficiency and the plasma levels of the drug associated with a suitable clinical and pharmacological response. Clinical studies have shown that the minimum efficient and safe dose of d4T is 0.5 mg/(kg day), administered with dosage intervals of 12 h.29 Later studies addressing the pharmacokinetics and efficiency of d4T, using an in vitro model, confirmed the minimum efficient dose of the drug proposed by Anderson et al.,29 together with the pharmacological efficiency of d4T when sustained drug levels are achieved by continuous perfusion. With this model, the plasma levels of d4T administered intravenously (i.v.) under different administration conditions were simulated and its antiretroviral efficiency was explored by studying the inhibition of de novo replication of HIV by measurement of reverse transcrip-
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tase through quantitative analysis of unintegrated species of HIV-1 DNA by PCR25 . In that study, the authors demonstrated that a dose of d4T of 1 mg/ (kg day), administered both i.v., with dosage intervals of 12 h, and with continuous perfusion, achieved effective suppression of the virus. In the same work, they showed that administration by continuous perfusion of a dose of d4T of 1 mg/(kg day) over 24 h achieved antiretroviral efficacy, sustained plasma levels of around 100 ng/mL being maintained. The dose of 80 mg used in our study to simulate the plasma level curves using pharmacokinetic information of d4T administered orally in SRFs can be considered equivalent to the dose of 1 mg/(kg day) i.v. used by Bilello et al.25 As may be deduced from these results, sustained levels of d4T above 90 ng/mL for 24 h are best achieved with first-order release, with first-order constants of 0.05 and 0.1 h−1 . In the case of using SRFs with zero-order kinetics to guarantee d4T sustained drug levels over 20 h a 4 mg/h release constant may be used. As may be observed in the Figure 6, through Monte Carlo simulation, the use of SRFs with first-order release constant of 0.05 h−1 and zero-order release constant of 4 mg/h means that there is a robust likelihood of d4T target plasma concentrations close to a 24-h period.
DOI 10.1002/jps
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CONCLUSIONS The convolution approach and Monte Carlo simulation are useful tools in drug preformulation studies in order to optimize critical pharmacokinetic parameters of the formulation. This pharmacokinetic analysis provides a breakthrough in the in silico prediction of drug release profiles from sustained-release dosage forms. Pharmacokinetic simulation using convolution techniques predicts that the first-order release constant (K1 = 0.05 h−1 ) and the zero-order release constant (K0 = 4 mg/h) are optimal for formulating d4T hydrophilic matrices, achieving plasma concentrations within the therapeutic range.
13.
14.
15.
16.
17.
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JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 8, AUGUST 2011
Usala Laboratories, C/ Licenciado M´endez Nieto s/n 37007, Salamanca, Spain
2 Department of Pharmacy and Pharmaceutical Technology, Faculty of Pharmacy, University of Salamanca, C/ Licenciado M´endez Nieto s/n 37007, Salamanca, Spain
Received 24 December 2010; revised 2 March 2011; accepted 2 March 2011 Published online 11 April 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jps.22565 ABSTRACT: In the present work, a single model-independent approach was developed to optimize the release kinetics of drugs from sustained-release formulations, using stavudine (d4T) as a model drug. This approach is based on the pharmacokinetic simulation of drug plasma levels through a semiparametric approach of the input function and on convolution with an empirical polyexponential unit impulse response function. Input functions were evaluated using different zero-order and first-order release constants. Optimum drug release to obtain a specific pharmacokinetic profile was approached using target model-independent pharmacokinetic parameters such as Cmax SS , Cmin SS , tmax SS , and peak-trough fluctuations. A Monte Carlo simulation was performed to estimate the fractional attainment of d4T plasma concentrations over therapeutic d4T levels. Zero-order (K0 = 4 mg/h) and first-order (K1 = 0.05 h−1 ) release constants were optimal for the formulation of sustained-release d4T tablets, plasma concentrations within the therapeutic range being achieved. ©2011 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 100:3260–3267, 2011 Keywords: Sustained-release formulations; convolution; release rate; Monte Carlo simulation; preformulation; formulation; controlled release/delivery; pharmacokinetics; computer modeling; oral drug delivery
INTRODUCTION Sustained-release formulations (SRFs) are an alternative to immediate-release pharmaceuticals with undoubted advantages as regards to the therapeutic point of view, such as prolonging the therapeutic effect, increasing the dosage interval, or reducing side effects.1,2,3,4 From the technological point of view, there are different kinds of SRFs based, among others, on hydrophilic matrices, pellets, osmotic systems, or systems based on membranes to control diffusion. With these systems, the drug is released slowly, which allows the input of the drug to be controlled in the organism, leading to changes in the pharmacokinetic Correspondence to: Jos´e M. Lanao (Telephone: +34-923-294536; Fax: +34-923-294515; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 100, 3260–3267 (2011) © 2011 Wiley-Liss, Inc. and the American Pharmacists Association
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profile with therapeutic implications.5,6 With the different types of SRFs, it is possible to obtain different release rates, such as zero order, first order, or complex orders.7,8 Traditionally, model-independent pharmacokinetic analysis and, in particular, deconvolution methods have been used successfully to characterize the in vivo release kinetics of drugs from this type of formulation, once they have been developed and administered to animals or humans.9–12 However, there is a lack of information concerning the use of simulation methods and model-independent pharmacokinetic analysis for the optimal design of release rates in SRFs during the initial stages of pharmaceutical development. The aim of the present work is to optimize the drug release rate from SRFs using stavudine (d4T) as a model drug and model-independent methods based on convolution, in order to achieve a suitable pharmacokinetic profile of the drug of this type of pharmaceutical dosage form.
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OPTIMIZATION OF SUSTAINED RELEASE KINETICS
MATERIAL AND METHODS Optimization of the Release Rate from SRFs Optimization of the release kinetics of a drug from SRFs to achieve a suitable pharmacokinetic profile requires different steps, as seen in the flowchart in Figure 1. These can be summarized as follows: 1. Information about the release rate of the drug from the sustained-release pharmaceutical form. In vitro dissolution assays using the United States Pharmacopeia (USP) methods are able to provide information about the type of release and the mechanisms involved.13 2. Prior knowledge about the pharmacokinetic behavior of the drug from essential pharmacokinetic parameters, such as bioavailability, the apparent distribution volume, and clearance, as well as about interindividual and residual variability. It is also possible to work with polyexponential equations that will define the pharmacokinetic profile of the drug. Previous studies addressing the population pharmacokinetics of the drug can provide this kind of information.14 3. Information about the therapeutic range of the drug or, alternatively, pharmacodynamic models that will quantify the relationship between the pharmacological response and the plasma concentrations of the drug. For a given drug, this type of information will allow the determination of the plasma concentration range and the pharmacokinetic profile associated with therapeutic
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efficiency and its relationship with the release profile of the drug from the SRFs. 4. Deterministic simulation. Simulation of the pharmacokinetics of the drug in the SRFs by means of model-dependent (e.g., compartmental analysis) or model-independent (e.g., convolution) methods. 5. Probabilistic simulation. Monte Carlo simulation is an excellent tool for estimating the probability distribution of target plasma concentrations using a specific dosage form (e.g., SRF) and dosage regimen.15 This is because of its flexibility and the fact that empirical distributions can be handled.16
With pharmacokinetic simulation, it is possible to establish correlations between changes in terms of the bioavailability of the drug in amount and rate from the SRFs and modifications in the plasma levels of the drug, reflected in important pharmacokinetic parameters such as the area under the curve, maximum plasma concentration (Cmax ), time to maximum concentration (tmax ), fluctuations in levels, and so on. According to the flowchart depicted in Figure 1, the simulation of the plasma levels using pharmacokinetic information of a drug incorporated in a drug delivery system can be accomplished with modeldependent methods, based on compartmental analysis, and model-independent methods, based on the convolution approach, which was the method followed here.
Figure 1. Flowchart with the steps to optimize the release kinetics of a drug from sustainedrelease formulations. DOI 10.1002/jps
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Pharmacokinetic Simulation Using Model-independent Methods Convolution is an interesting model-independent tool used in the field of pharmaceutical development for predicting plasma levels in type A in vivo/in vitro correlations (IVIVC), with implications at regulatory level,17–19 which can be used to optimize the release kinetics of the drug from different pharmaceutical dosage forms. Considering a SRF as a continuous-time, linear, and time-invariant system, drug plasma levels can be simulated from a time domain point of view using the following convolution integral: t I(t) F(t − J)dt
R (t) =
(1)
0
where R(t) represents the response function, which corresponds to the in vivo plasma levels curve after administration of the sustained-release dosage form; I(t) represents the input function, which corresponds to the in vivo drug release kinetics of the SRF; F(t) represents the unit impulse response, which corresponds to the in vivo plasma or serum levels curve after the administration of a unit impulse of the immediaterelease dosage form, and J is a constant. In an equivalent way, the system can be characterized in the frequency domain using Laplace transformation. Taking into account that the considered SRF can be considered as a linear system, the use of Laplace transforms facilitates the convolution operation transforming the integral in a simple multiplication operation of the Laplacian equations of the time functions. In the Laplace domain, the convolution integral can be expressed as: R(s) = I (s)F(s)
(2)
In a semiparametric approach, and depending on the kind of release kinetics considered, the input function may be expressed by the following algebraic expressions: K0 (3) I (s) = s FDK 1 (4) (s + K 1 ) The unit impulse response can be expressed as an empirical polyexponential function: I(s) =
F(s) =
n i=1
Ai (s + 8i )
(5)
where K0 is the zero-order release constant, K1 is the first-order release constant, F is the amount of bioavailability, D is the dose, s is the Laplace operaJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 8, AUGUST 2011
tor, Ai and λi are the coefficient and exponent of the exponential, and n is the number of exponentials. By using specific software, such as MULTI-FILT (Kyoto University, Kyoto, Japan), we can directly write the equations in the Laplace domain and solve it using the numerical inversion of the Laplace transformation.20 This program carries out the simulation of plasma concentration–time data by using Laplace transformed equations corresponding to the unit impulse response expressed as polyexponential function (Eq. 5) convoluted with the input function characterized by Eqs. 3 (zero-order release) or 4 (firstorder release). Then, Eqs. 2–5 can be used to simulate plasma level curves using pharmacokinetic information of a drug incorporated into an SRFs using different types of release and release rates. Release Optimization of d4t SRFs The antiretroviral agent d4T was chosen as a model drug in view of its good water solubility21,22 and its rapid and complete absorption after oral administration.23 Owing to its short elimination halflife, d4T needs to be dosed twice a day. Development of d4T SRFs can provide plasma d4T concentrations for 24 h with once-daily dosing. In the present work, pharmacokinetic simulation was used to estimate the optimal release kinetics of d4T from SRFs to achieve drug plasma levels within the therapeutic range for 24 h. Using the population pharmacokinetics of d4T in humans,14 shown in Table 1, an empirical polyexponential unit impulse response was generated, expressed as the plasma concentration (Cp = X p /Vd ), where Xp is the drug amount in plasma. Cp = 0.021e−0.408t − 0.021e−4.3t
(6)
This polyexponential equation expressed as Laplace transform (Eq. 5) was used as a unit impulse response in different simulations. Computer Table 1. Stavudine Population Pharmacokinetic Parameters and the Interindividual and Residual Variabilities Corresponding to Noncompartmental Pharmacokinetic Analysis 14 in Humans Receiving 40 mg Dose Orally (Schaad et al. ) Noncompartmental Kinetic Analysis
Vd (L/kg) CL [L/(h kg)] Cmax (ng/mL) AUC (ng h/mL) tmax (h) Fg2
Parameter
Variability CV (%)
0.743 0.303 1215 2087 0.600 –
20 32 32 39 17 17
Vd , apparent distribution volume; CL, plasma clearance; Cmax , maximum plasma concentration; AUC, area under plasma level curve; tmax , time to maximum concentration; Fg2 , residual variability; CV, coefficient of variation.
DOI 10.1002/jps
OPTIMIZATION OF SUSTAINED RELEASE KINETICS
simulations by convolution of d4T plasma levels in SRFs at a dose of 80 mg were used to analyze the changes in the release profile in plasma d4T concentrations under different release constants: first order (K1 = 0.05, 0.1, 0.15, and 0.2 h−1 ) and zero order (K0 = 4, 6, 8, 12, and 16 mg/h). Simulated plasma levels of d4T employing SRFs with different release rates were generated using Eqs. 1–5 through numerical inversion of the Laplace transform (FILT).20 Using the plasma level curves of d4T generated for the different release rates, model-independent pharmacokinetic parameters at steady state in a multiple dosage regimen as steady-state maximum plasma concentration (Cmax SS ), steady-state minimum plasma concentration (Cmin SS ), time to maximum concentration at steady state (tmax SS ), terminal slope of plasma level curve (λz ), plasma half-life (t1/2 ), and peak-trough fluctuation (PTF) were used for the evaluation. The percent PTF of plasma levels at steady state was calculated using the following equation24 : SS (CSS max − Cmin ) (7) Cav The average steady-state plasma concentration (Cav ) was calculated as the ratio between the area under plasma levels curve and the dosage interval (AUC/J). Previously published data from pharmacokinetic and pharmacodynamic studies and from work addressing the therapeutic efficiency of d4T in vitro/ in vivo were used to establish the relationship between the simulated plasma levels from the SRFs and the therapeutic efficiency potential of the drug when incorporated into SRFs.25 Pharmacokinetic analysis was completed using Monte Carlo simulations.15 This method considers the variability in pharmacokinetic behavior to determine the probability of a sustained target d4T
%PTF = 100
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plasma level being reached. On the basis of previous pharmacodynamic studies, a plasma level of d4T above 90 ng/mL was set as the target.25 The GoldSim Pro software version 10.1 (Goldsim Technology Group, Issaquah, WA, USA) was used for population simulations of subjects receiving d4T in different SRFs.26 The geometric mean and the geometric standard deviation of the pharmacokinetic parameter values (coefficients and exponents) corresponding to an empirical polyexponential function of the response function were employed. The parameters were sampled from a log-normal distribution. The choice of distribution was determined by the kind of statistical distribution of the original mean parameter vector and their variances. Monte Carlo sampling of stochastic elements was performed using a Latin Hypercube Sampling.27 Ten simulations were performed to generate 1000 simulated subjects d4T steady-state plasma level curves, each for dosages of 40 mg every 12 h incorporated into immediate-release formulations, and 80 mg every 24 h incorporated into SRFs with first-order release constants of 0.05, 0.1, 0.15, and 0.20 h−1 and zero-order release constants of 4, 6, 8, 12, and 16 mg/h. The first-order and zero-order constants were tested in independent simulations.
RESULTS Figures 2 and 3 show the simulated d4T concentration versus time profiles from SRFs obtained by numerical convolution for first-order release constant (K1 ) values between 0.05 and 0.2 h−1 and zero-order release constant (K0 ) values between 4 and 16 mg/h, respectively. Figure 4 shows the correlations established between the Cmax and the different first-order and zeroorder release constants used for the simulation. Table 2 shows the values obtained by pharmacokinetic simulation at steady state of Cmax SS , Cmin SS
Figure 2. Simulated d4T concentration versus time profiles from sustained-release formulations obtained by convolution for first-order release with K1 values between 0.05 and 0.2 h−1 . DOI 10.1002/jps
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Figure 3. Simulated d4T concentration versus time profiles from sustained-release formulations obtained by convolution for zero-order release with K0 values between 4 and 16 mg/h.
tmax SS , λz , t1/2 , and the PTF of d4T for first-order and zero-order release kinetics. Figure 5 shows the simulated probability density to steady-state plasma concentrations of d4T at 24 h, after one simulation with 1000 patients considered as replications using Monte Carlo simulation for a SRF with a first-order release constant of 0.05 h−1 administered through the oral route at a dose of 80 mg every 24 h.
The results of the analyses of the probability of d4T plasma concentrations being reached through Monte Carlo simulation using immediate-release formulations and SRFs are shown in Figure 6. The overall fractional attainments of achieving a plasma concentration greater than 90 ng/mL with d4T SRFs (80 mg every 24 h) at 12 and 24 h with a first-order rate constant of 0.05 h−1 were 95% and 30%, respectively. These overall fractions at 12 and 24 h with a
Figure 4. Linear correlations established between Cmax SS obtained by simulation for firstorder kinetics and the different K1 tested (a) and between Cmax SS obtained by simulation for zero-order kinetics and the different K0 tested (b). JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 100, NO. 8, AUGUST 2011
DOI 10.1002/jps
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Table 2. Values Obtained by Simulation of Pharmacokinetic Parameters of Stavudine at Steady State for First-Order and Zero-Order Release Kinetics by Using an 80 mg/day Dosage regimen First-Order Release Constant [Kl (h−1 )] 0.05 λz (h−1 ) t1/2 (h) tmax SS (h) Cmax SS (ng/mL) Cmin SS (ng/mL) PTF (%)
0.055 12.60 5.46 189.07 87.51 64.59
0.10
0.15
0.20
0.11 6.30 4.72 250.54 50.22 127.40
0.16 4.30 4.14 305.89 22.20 178.51
0.21 3.30 3.66 374.72 13.21 229.92
Zero-Order Release Constant [K0 (mg/h)] 4
6
8
0.40 1.73 20.00 188.51 37.40 96.11
0.40 1.73 13.33 283.66 3.97 177.88
0.40 1.73 10.00 370.13 1.37 234.53
12 0.40 1.73 6.67 528.14 0.52 335.57
16 0.40 1.73 5.00 645.94 0.32 410.62
λz , terminal slope of plasma level curve; t1/2 , plasma half-life; tmax SS , time to maximum concentration at steady state; Cmax SS , steady-state maximum plasma concentration; Cmin SS , steady-state minimum plasma concentration; PTF, peak-trough fluctuation.
zero-order rate constant of 4 mg/h were 99.9% and 1.5%, respectively. The optimal d4T release constant from drug delivery devices is 0.05 h−1 (first order) and 4 mg/h (zero order), which ensures that the plasma concentrations and the predicted patient variability will be within the therapeutic range (100–1000 ng/mL) close to a 24 h period.
DISCUSSION In SRF, the rate of bioavailability is conditioned by the release rate, which in turn governs the pharmacokinetic profile of the drug, prolonging the pharmacological response and allowing the dosage interval to be increased.
Within the field of the development of this type of formulation, it is of crucial interest to have a priori knowledge about the optimal profile of the drug in vivo that will allow a pharmacokinetic profile appropriate for guaranteeing therapeutic efficiency to be obtained. Prior knowledge of the plasma levels associated with a suitable pharmacological response, together with pharmacokinetic simulation methods, will allow the optimal kinetics of the release of this type of formulation to be established, depending on the type of release kinetics. Here, we developed a pharmacokinetic simulation methodology that allowed this aim to be achieved, following the flowchart depicted in Figure 1. d4T was used as the model drug; it is an antiretroviral that is highly soluble in water, with an elimination
Figure 5. Simulated probability density to steady-state plasma concentrations of stavudine at 24 h, using Monte Carlo simulation for an 80 mg/day oral sustained-release formulation (one simulation with 1000 subjects) with a first-order release constant of 0.05 h−1 . DOI 10.1002/jps
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Figure 6. Probability of exceeding the stavudine steady-state plasma concentrations with different release constants: (a) first-order release at 12 h, (b) first-order release at 24 h, (c) zero-order release at 12 h, and (d) zero-order release at 24 h.
half-life of about 2 h, and hence a good candidate for being incorporated into SRFs.28 As may be seen in Figures 2 and 3, a progressive change occurs in the kinetic profile of d4T as the release constant is modified, with both zero-order and first-order profiles. These changes are reflected in modifications in the plasma levels; in the fluctuation of levels; and in the Cmax SS , Cmin SS , and tmax SS values, as seen in Figure 4 and Table 2. One crucial aspect is to know the doses of d4T associated with clinical efficiency and the plasma levels of the drug associated with a suitable clinical and pharmacological response. Clinical studies have shown that the minimum efficient and safe dose of d4T is 0.5 mg/(kg day), administered with dosage intervals of 12 h.29 Later studies addressing the pharmacokinetics and efficiency of d4T, using an in vitro model, confirmed the minimum efficient dose of the drug proposed by Anderson et al.,29 together with the pharmacological efficiency of d4T when sustained drug levels are achieved by continuous perfusion. With this model, the plasma levels of d4T administered intravenously (i.v.) under different administration conditions were simulated and its antiretroviral efficiency was explored by studying the inhibition of de novo replication of HIV by measurement of reverse transcrip-
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tase through quantitative analysis of unintegrated species of HIV-1 DNA by PCR25 . In that study, the authors demonstrated that a dose of d4T of 1 mg/ (kg day), administered both i.v., with dosage intervals of 12 h, and with continuous perfusion, achieved effective suppression of the virus. In the same work, they showed that administration by continuous perfusion of a dose of d4T of 1 mg/(kg day) over 24 h achieved antiretroviral efficacy, sustained plasma levels of around 100 ng/mL being maintained. The dose of 80 mg used in our study to simulate the plasma level curves using pharmacokinetic information of d4T administered orally in SRFs can be considered equivalent to the dose of 1 mg/(kg day) i.v. used by Bilello et al.25 As may be deduced from these results, sustained levels of d4T above 90 ng/mL for 24 h are best achieved with first-order release, with first-order constants of 0.05 and 0.1 h−1 . In the case of using SRFs with zero-order kinetics to guarantee d4T sustained drug levels over 20 h a 4 mg/h release constant may be used. As may be observed in the Figure 6, through Monte Carlo simulation, the use of SRFs with first-order release constant of 0.05 h−1 and zero-order release constant of 4 mg/h means that there is a robust likelihood of d4T target plasma concentrations close to a 24-h period.
DOI 10.1002/jps
OPTIMIZATION OF SUSTAINED RELEASE KINETICS
CONCLUSIONS The convolution approach and Monte Carlo simulation are useful tools in drug preformulation studies in order to optimize critical pharmacokinetic parameters of the formulation. This pharmacokinetic analysis provides a breakthrough in the in silico prediction of drug release profiles from sustained-release dosage forms. Pharmacokinetic simulation using convolution techniques predicts that the first-order release constant (K1 = 0.05 h−1 ) and the zero-order release constant (K0 = 4 mg/h) are optimal for formulating d4T hydrophilic matrices, achieving plasma concentrations within the therapeutic range.
13.
14.
15.
16.
17.
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