Monte Carlo simulations and fractional kinetics considerations for the Higuchi equation

We highlight some physical and mathematical aspects relevant to the derivation and use of the Higuchi equation. More specifically, the application of the Higuchi equation to different geometries is discussed and Monte Carlo simulations to verify the validity of Higuchi law in one and two dimensions, as well as the derivation of the Higuchi equation under alternative boundary conditions making use of fractional calculus, are presented.     

On Monte Carlo estimation of relative power

This paper derives standard errors for Monte Carlo (MC) estimators of (relative) power of tests when the critical values under the null have also been estimated. This situation is common, for example, in unit root and cointegration (CI) tests. The associated issue of MC design is discussed. The results are illustrated on likelihood‐based tests for CI rank determination.     

Free energies of hydration for organic molecules from Monte Carlo simulations

Summary Methods for the computation of free energies of solvation are reviewed, with emphasis on aqueous systems and discrete representation of the solvent molecules in Monte Carlo (MC) or molecular dynamics simulations. Precise results for small organic solutes can now be obtained routinely by MC/free energy perturbation calculations. The average error for computed free energies of hydration is 1.0 kcal/mol with the use of partial charges for the solutes from either OPLS unitedatom potential functions or from fitting to the electrostatic potential surface from 6-31G* ab initio calculations. A less rigorous, computationally more rapid method based on linear response theory is also found to be valuable and reduces the average error for the test set of 16 solutes to 0.6 kcal/mol.     

Input estimation for drug discovery using optimal control and Markov chain Monte Carlo approaches

Input estimation is employed in cases where it is desirable to recover the form of an input function which cannot be directly observed and for which there is no model for the generating process. In pharmacokinetic and pharmacodynamic modelling, input estimation in linear systems (deconvolution) is well established, while the nonlinear case is largely unexplored. In this paper, a rigorous definition of the input-estimation problem is given, and the choices involved in terms of modelling assumptions and estimation algorithms are discussed. In particular, the paper covers Maximum a Posteriori estimates using techniques from optimal control theory, and full Bayesian estimation using Markov Chain Monte Carlo (MCMC) approaches. These techniques are implemented using the optimisation software CasADi, and applied to two example problems: one where the oral absorption rate and bioavailability of the drug eflornithine are estimated using pharmacokinetic data from rats, and one where energy intake is estimated from body-mass measurements of mice exposed to monoclonal antibodies targeting the fibroblast growth factor receptor (FGFR) 1c. The results from the analysis are used to highlight the strengths and weaknesses of the methods used when applied to sparsely sampled data. The presented methods for optimal control are fast and robust, and can be recommended for use in drug discovery. The MCMC-based methods can have long running times and require more expertise from the user. The rigorous definition together with the illustrative examples and suggestions for software serve as a highly promising starting point for application of input-estimation methods to problems in drug discovery.     

Modeling and Monte Carlo simulations in oral drug absorption

Drug dissolution, release and uptake are the principal components of oral drug absorption. All these processes take place in the complex milieu of the gastrointestinal tract and they are influenced by physiological (e.g. intestinal pH, transit time) and physicochemical factors (e.g. dose, particle size, solubility, permeability). Due to the enormous complexity issues involved, the models developed for drug dissolution and release attempt to capture their heterogeneous features. Hence, Monte Carlo simulations and population methods have been utilized since both dissolution and release processes are considered as time evolution of a population of drug molecules moving irreversibly from the solid state to the solution. Additionally, mathematical models have been proposed to determine the effect of the physicochemical properties, solubility/dose ratio and permeability on the extent of absorption for regulatory purposes, e.g. biopharmaceutics classification. The regulatory oriented approaches are based on the tube model of the intestinal lumen and apart from the drug's physicochemical properties, take into account the formulation parameters the dose and the particle size.     

Proposed occupational exposure limits for select ethylene glycol ethers using PBPK models and Monte Carlo simulations.

Methoxyethanol (ethylene glycol monomethyl ether, EGME), ethoxyethanol (ethylene glycol monoethyl ether, EGEE), and ethoxyethyl acetate (ethylene glycol monoethyl ether acetate, EGEEA) are all developmental toxicants in laboratory animals. Due to the imprecise nature of the exposure data in epidemiology studies of these chemicals, we relied on human and animal pharmacokinetic data, as well as animal toxicity data, to derive 3 occupational exposure limits (OELs). Physiologically based pharmacokinetic (PBPK) models for EGME, EGEE, and EGEEA in pregnant rats and humans have been developed (M. L. Gargas et al., 2000, Toxicol. Appl. Pharmacol. 165, 53-62; M. L. Gargas et al., 2000, Toxicol. Appl. Pharmacol. 165, 63-73). These models were used to calculate estimated human-equivalent no adverse effect levels (NAELs), based upon internal concentrations in rats exposed to no observed effect levels (NOELs) for developmental toxicity. Estimated NAEL values of 25 ppm for EGEEA and EGEE and 12 ppm for EGME were derived using average values for physiological, thermodynamic, and metabolic parameters in the PBPK model. The uncertainties in the point estimates for the NOELs and NAELs were estimated from the distribution of internal dose estimates obtained by varying key parameter values over expected ranges and probability distributions. Key parameters were identified through sensitivity analysis. Distributions of the values of these parameters were sampled using Monte Carlo techniques and appropriate dose metrics calculated for 1600 parameter sets. The 95th percentile values were used to calculate interindividual pharmacokinetic uncertainty factors (UFs) to account for variability among humans (UF(h,pk)). These values of 1.8 for EGEEA/EGEE and 1.7 for EGME are less than the default value of 3 for this area of uncertainty. The estimated human equivalent NAELs were divided by UF(h,pk) and the default UFs for pharmacodynamic variability among animals and among humans to calculate the proposed OELs. This methodology indicates that OELs (8-h time-weighted average) that should protect workers from the most sensitive adverse effects of these chemicals are 2 ppm EGEEA and EGEE (11 mg/m(3) EGEEA, 7 mg/m(3) EGEE) and 0.9 ppm (3 mg/m(3)) EGME. These recommendations assume that dermal exposure will be minimal or nonexistent.     

Propagation of uncertainty in nasal spray in vitro performance models using Monte Carlo simulation: Part I. model prediction using Monte Carlo Simulation

Design of experiment (DOE) methodology can provide a complete evaluation of the influences of nasal spray activation and formulation properties on delivery performance which makes it a powerful tool for product design purposes. Product performance models are computed from complex expressions containing multiple factor terms and response terms. Uncertainty in the regression model can be propagated using Monte Carlo simulation. In this study, four input factors, actuation stroke length, actuation velocity, concentration of gelling agent, and concentration of surfactant were investigated for their influences on measured responses of spray pattern, plume width, droplet size distribution (DSD), and impaction force. Quadratic models were calculated and optimized using a Box–Behnken experimental design to describe the relationship between factors and responses. Assuming that the models perfectly represent the relationship between input variables and the measured responses, the propagation of uncertainty in both input variables and response measurements on model prediction was performed using Monte Carlo simulations. The Monte Carlo simulations presented in this article illustrate the propagation of uncertainty in model predictions. The most influential input variable variances on the product performance variance were identified, which could help prioritize input variables in terms of importance during continuous improvement of nasal spray product design. This work extends recent Monte Carlo simulations of process models to the realm of product development models. © 2009 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 99: 2114–2122, 2010     

Accelerating Monte Carlo power studies through parametric power estimation

Estimating the power for a non-linear mixed-effects model-based analysis is challenging due to the lack of a closed form analytic expression. Often, computationally intensive Monte Carlo studies need to be employed to evaluate the power of a planned experiment. This is especially time consuming if full power versus sample size curves are to be obtained. A novel parametric power estimation (PPE) algorithm utilizing the theoretical distribution of the alternative hypothesis is presented in this work. The PPE algorithm estimates the unknown non-centrality parameter in the theoretical distribution from a limited number of Monte Carlo simulation and estimations. The estimated parameter linearly scales with study size allowing a quick generation of the full power versus study size curve. A comparison of the PPE with the classical, purely Monte Carlo-based power estimation (MCPE) algorithm for five diverse pharmacometric models showed an excellent agreement between both algorithms, with a low bias of less than 1.2 % and higher precision for the PPE. The power extrapolated from a specific study size was in a very good agreement with power curves obtained with the MCPE algorithm. PPE represents a promising approach to accelerate the power calculation for non-linear mixed effect models.     

Exploiting substrate recognition for selective inhibition of protein kinases

Protein kinases are potential targets of drugs to treat many human diseases. Intensive efforts have been made to develop protein kinase inhibitors, but a major challenge is achieving specificity. Exploiting regulatory elements outside the ATP binding pocket, such as the substrate binding site, may provide an alternative that allows generation of competitive inhibitors with improved selectivity. In-depth understanding of substrate recognition by protein kinase is essential for design and refinement of competitive inhibitors. Here we described strategies for specifically targeting protein kinases and highlight our current progress in the development of substrate competitive inhibitors for glycogen synthase kinase-3 (GSK-3).     

Prediction on the efficacy of fluoroquinolones for anaerobic infections using Monte Carlo simulation

Various analysis based on PK-PD (pharmacokinetics-pharmacodynamics) theory were performed for prediction on the efficacy of antimicrobial agents for infectious diseases. However, there have been few reports when causative organisms would be anaerobic bacteria. Although the antimicrobial activity of fluoroquinolones against anaerobes has been weak, gatifloxacin (GFLX), moxifloxacin (MFLX), and garenoxacin (GRNX) are considered as anti-anaerobic quinolones in recent days. Therefore, we investigated prediction on the efficacy of fluoroquinolones for anaerobic infections using Monte Carlo Simulation. Although the target attainment of fluoroquinolones (levofloxacin, GFLX, and MFLX) on AUC/MIC value has not established yet, we have assumed 40 and 125 as the target attainment of AUC/MIC value. GFLX and MFLX would be effective against anaerobic infections caused by Prevotella intermedia, Fusobacterium nucleatum, Micromonas micros, which are the causative organisms for respiratory tract infections. From these results, GFLX and MFLX would be effective against anaerobic infections in respiratory tract as anti-anaerobic fluoroquinolones.     

DNA cross-linking by azinomycin B: Monte Carlo simulations in the evaluation of sequence selectivity

A new set of charges specifically developed for biologically relevant N7-alkylated purine adducts have been implemented in the AMBER force field of the MacroModel package and applied to the conformational search of azinomycin B-DNA interactions. To perform a sequence dependent reactivity relationship study, four DNA triplets known to interact differently with the drug, 5'-GCT-3', 5'-GCC-3', 5'-GTC-3', and 5'-GTT-3', have been modeled in B-form and intercalative conformations. Monte Carlo simulations of all possible monoadducts and intercalative complexes have been carried out and analyzed using a filtering criterion that estimates the probability of covalent bond formation and covalent cross-linking. We observed a good correlation between existing experimental data and our computational estimations that validate the approach. The comparison of the conformational properties of the drug-DNA monoadducts and complexes confirms the most probable mechanism of action involving an initial aziridine and subsequent epoxide alkylation. The different hydrogen bond network in the monoadducts and in the intercalative complexes between the drug and the three base-pair receptor is the primary reason for the different cross-linking reactivity. In addition, steric hindrance of the major groove exposed methyl group of central thymine-based triplets plays an important role in the lack of the reactivity of these sequences. Synthetic work on the azinomycins and the information coming from this computational study will be important for the design of more potent or DNA sequence-selective agents based on the azinomycin skeleton.     

Bioavailability estimation by semisimultaneous drug administration: a Monte Carlo simulation study

The performance of a novel method for determination of drug absorption characteristics was evaluated by Monte Carlo simulations. In bioavailability studies with use of this method, the test and the reference doses are administered within a time interval of hours. Estimates of bioavailability are obtained by fitting an appropriate model to the concentration-time profile, which in its terminal portion is thus the summed concentration of the two doses. Drugs with different properties, mimicked by varying the kinetic rate constants (ka, lambda 1 and lambda 2), and experimental designs with different sets of conditions regarding the interval between doses, dose ratio, dose order, and duration of sampling, were simulated to determine what factors govern parameter estimation. The absorption characteristics of the simulated drugs could be adequately determined in experiments lasting for 12 hr or less, provided that a proper design was used. Fitting of a simpler or a more complex disposition model produced estimates with similar accuracy and precision to those noted with the true model. For some conditions the use of an improper absorption model resulted in slightly reduced accuracy, but as these fits were poor there was a clear need to try other models. In another set of simulations the use of the proposed method to assess the relative availability of two extravascular doses was evaluated. The relative rate and extent of absorption could be estimated with good precision for two formulations exhibiting a rapid to moderate rate of absorption.     

Bioavailability estimation by semisimultaneous drug administration: A Monte Carlo simulation study

The performance of a novel method for determination of drug absorption characteristics was evaluated by Monte Carlo simulations. In bioavailability studies with use of this method, the test and the reference doses are administered within a time interval of hours. Estimates of bioavilability are obtained by fitting an appropriate model to the concentrationtime profile, which in its terminal portion is thus the summed concentration of the two doses. Drugs with different properties, mimicked by varying the kinetic rate constants (k_a, ₁,and ₂),and experimental designs with different sets of conditions regarding the interval between doses, dose ratio, dose order, and duration of sampling, were simulated to determine what factors govern parameter estimation. The absorption characteristics of the simulated drugs could be adequately determined in experiments lasting for 12 hr or less, provided that a proper design was used. Fitting of a simpler or a more complex disposition model produced estimates with similar accuracy and precision to those noted with the true model. For some conditions the use of an improper absorption model resulted in slightly reduced accuracy, but as these fits were poor there was a clear need to try other models. In another set of simulations the use of the proposed method to assess the relative availability of two extravascular doses was evaluated. The relative rate and extent of absorption could be estimated with good precision for two formulations exhibiting a rapid to moderate rate of absorption.     

Profile likelihood-based confidence intervals using Monte Carlo integration for population pharmacokinetic parameters

Population pharmacokinetic (PPK) analysis usually employs nonlinear mixed effects models using first-order linearization methods. It is well known that linearization methods do not always perform well in actual situations. To avoid linearization, the Monte Carlo integration method has been proposed. Moreover, we generally utilize asymptotic confidence intervals for PPK parameters based on Fisher information. It is known that likelihood-based confidence intervals are more accurate than those from the usual asymptotic confidence intervals. We propose profile likelihood-based confidence intervals using Monte Carlo integration. We have evaluated the performance of the proposed method through a simulation study, and analyzed the erythropoietin concentration data set by the method.     

Selective inhibition of eukaryote protein kinases by anti-inflammatory triterpenoids

The ursane triterpenoid alpha-amyrin and the lupane triterpenoid lupeol are potent inhibitors of the catalytic subunit (cAK) of rat liver cyclic AMP-dependent protein kinase (PKA) with IC50 values of 8 and 5 microM, respectively. The palmitate and linoleate esters of alpha-amyrin and lupeol are also potent inhibitors of cAK (IC50 values in the range of 4-9 microM). alpha-Amyrin, lupeol and lupeol linoleate are much less potent as inhibitors of rat brain Ca(2+)- and phospholipid-dependent protein kinase (PKC) (IC50 values 32, 82 and 35 microM; respectively) and alpha-amyrin linoleate and the palmitate esters of lupeol and alpha-amyrin are ineffective or very poor inhibitors of this protein kinase. These compounds are very poor or ineffective as inhibitors of chicken gizzard calmodulin-dependent myosin light chain kinase (MLCK). alpha-Amyrin inhibits plant Ca(2+)-dependent protein kinase (CDPK) (IC50 52 microM) but lupeol and the triterpenoid esters tested are ineffective. alpha-Amyrin and the linoleate and palmitate esters of alpha-amyrin and lupeol inhibit cAK in a fashion that is competitive with respect to both peptide substrate and ATP (Ki values in the range 2-6 microM). However, while lupeol is competitive with respect to ATP it is apparently non-competitive with respect to peptide substrate. alpha-Amyrin also inhibits CDPK competitively and alpha-amyrin, lupeol and lupeol linoleate are competitive inhibitors of PKC. alpha-Amyrin and the palmitate esters of lupeol and alpha-amyrin are competitive inhibitors of the potato high affinity cyclic AMP-binding phosphatase (Pase) but lupeol inhibits the Pase non-competitively. These hydrophobic triterpenoids are further examples of anti-inflammatory triterpenoids that are cAK inhibitors.