Mathematical Formalism and Characteristics of Four Basic Models of Indirect Pharmacodynamic Responses for Drug Infusions

Indirect response models require differential equations to describe the nonlinear inhibition or stimulation of the production or loss (k_out ) of the response variable. Partially integrated solutions for these models developed previously for iv bolus or biphasic pharmacokinetics were extended to consider drug infusions for limited or extended durations. Qualitative examination was made of the role of infusion rate and duration, type and rate of drug disposition, I_max or S_max capacity factors, IC₅₀ or SC₅₀ sensitivity factors, and k_out values. Properties of the response curves characterized include curve shapes, maximum or minimum response, onset rate, steady-state, and return to baseline. Some comparisons were made with behavior of iv bolus doses. These relationships provide both a formal and practical basis for better understanding of the time-course of basic indirect response models.     

Bioavailability Assessment from Pharmacologic Data: Method and Clinical Evaluation

A novel method is described for assessing drug bioavailability from pharmacologic data. The method is based upon a generalized model for the relationship between the observed effect (E) and the input rate (f): E = (c_e * f), where * denotes convolution, c_e is effect site unit impulse response (amount of drug at the effect site resulting from the instantaneous input of a unit amount of drug) and is transduction function (relates amount of drug at the effect site to E). The functions and c_e are expressed as cubic splines for maximum versatility. Pharmacologic data collected after the administration of two different doses by iv infusion are analyzed simultaneously to estimate the function parameters. This experimental design addresses the fact that and c_e cannot be uniquely estimated from the results of a single dose experiment. The unknown f from a test treatment is then estimated by applying an implicit deconvolution method to the pharmacologic data collected during that treatment. The method was tested with simulated data. The method and the model were further evaluated by application to a clinical study of verapamil (V) pharmacodynamics in 6 healthy volunteers. Simulations showed that the method is accurate and precise in the presence of a high degree of measurement error, but large intrasubject variability in the model functions can result in biased estimates of the amount absorbed. The method produced reasonably accurate estimates of the V input rate and systemic availability (F) in the 6 human volunteers though there was a trend towards underestimation (estimated total F%=93.6±14 vs. the true F% of 100).     

The effect of gender on the pharmacokinetics of verapamil and norverapamil in human

The effects of gender on the pharmacokinetics of verapamil and its active metabolite, norverapamil, following single oral dose (80 mg, Isoptin) to 12 healthy male (mean age: 25.75+/-2.42 years, mean body weight: 70.59+/-9.94 kg) and 12 healthy female subjects (mean age: 24.08+/-2.84 years, mean body weight: 56.67+/-5.23 kg) were investigated in the present study. Plasma concentrations of verapamil and norverapamil were analysed using a modified high-pressure liquid chromatography method. Pharmacokinetic parameters were calculated by non-compartmental analysis for each subject. For verapamil the half-life (t1/2) and mean residence time (MRT) were significantly shorter in women than men (p<0.01 and p<0.05, respectively). For other pharmacokinetic parameters of verapamil there were no significant differences between males and females. For norverapamil, t1/2, MRT and time to reach to the maximum plasma concentration (Tmax) showed statistically significant differences between the two genders. The AUC(0-24) and AUC(0-infinity) ratios of norverapamil to verapamil were also calculated. The ratios were significantly higher in women compared with men. These observations indicate that the elimination rate of verapamil is faster in women than men which may be attributed to the higher activity of CYP3A4 or lower activity of P-glycoprotein in women compared with men. A contribution of both factors in the appearance of gender differences in verapamil pharmacokinetics is also possible.     

Mathematical Formalism for the Properties of Four Basic Models of Indirect Pharmacodynamic Responses

Four basic models for characterizing indirect pharmacodynamic responses were proposed previously and applied using differential equations. These models consider inhibition or stimulation by drug of the production or loss of mediators or response variables. This report develops partially integrated solutions for these models which allow more detailed examination of the roles of model parameters and pharmacokinetic functions in affecting the time course of drug effects. Because of the nonlinear Hill function, the solutions are represented by means of definite integrals containing kinetic and dynamic functions. These solutions allow a qualitative examination, using calculus, of how response is controlled by Dose. IC₅₀ or SC₅₀, I_max or S_max, and k_out for drugs exhibiting monotonic or biphasic disposition. Characteristics of the response curves that were identified include shape, maximum or minimum, and changes with the above parameters and time. These relationships, together with simulation studies, provide a fundamental basis for understanding the temporal aspects of the basic indirect response models.     

广义线性混合效应模型在临床疗效评价中的应用

目的:探讨临床疗效评价中分类重复测量资料的广义线性混合效应模型(GLMMs)及SAS8.0的GLIMMIX宏实现。方法:利用GLIMMIX宏ERROR和LINK语句来指示疗效指标的分布及连接函数,通过REPEATED和RANDOM语句的TYPE选项选择合适方差-协方差结构矩阵来模拟不同时间疗效指标的相关性,采用基于线性的伪似然函数进行模型参数估计。结果:广义线性混合效应模型允许临床疗效评价指标是指数家族中任意分布(如:连续分布包括正态分布、beta分布、卡方分布等;离散分布包括二项分布、泊松分布、负二项分布等),可以通过连接函数将疗效指标的均数向量与模型参数建立线性关系,简化运算过程。结论:广义线性混合效应模型建模灵活,可为临床疗效评价提供更丰富的信息。     

Assessment of Basic Indirect Pharmacodynamic Response Models with Physiological Limits

Many physiological factors are regulated by homeostatic mechanisms to maintain normal body function. Empirical lower R _l (Model I and IV) or upper R _h limits (Model II and III) were included in current basic indirect response (IDR) models to account for the additional role of physiological limits (IDRPL). Various characteristics of these models were evaluated with simulations and explicit equations. The simulations reveal that the expanded models exhibit most properties of basic indirect response models, such as slow response initiation, lag time between the kinetic and dynamic peaks, a large dose plateau, and shift in T _max with dose. The proposed models always produce lesser net responses than predicted by basic IDR models. Simulations demonstrate that addition of a parameter limit which is close to the baseline has a great influence on the overall and maximum responses and fitted model parameters. Only stimulatory IDRPL Models III and IV allow resolution of all model parameters in the absence of clear indications or predetermined values of the lower or upper limits. However, all four models are able to resolve model parameters when subgroups with different baselines are simultaneously fitted. These models create new interpretations of the determinants of baseline conditions which can be important in assessing inter-subject variability in responses. The applicability of IDRPL models is demonstrated using several examples from the published literature. Indirect response models with physiological limits will be useful in characterizing drug responses for turnover systems which are maintained within a certain range. Supported by Grant GM 57980 from the National Institute of General Medical Sciences, National Institutes of Health.     

Pharmacokinetic/Pharmacodynamic Evaluation of Deflazacort in Comparison to Methylprednisolone and Prednisolone

Purpose. The pharmacokinetics and pharmacodynamics of deflazacort after oral administration (30 mg) to healthy volunteers were determined and compared with those of 20 mg of methylprednisolone and 25 mg of prednisolone. Methods. Methylprednisolone, prednisolone and the active metabolite of deflazacort, 21-desacetyldeflazacort, were measured in plasma using HPLC. For the assessment of pharmacodynamics, differential white blood cell counts were obtained over 24 hours. An integrated pharmacokinetic-pharmacodynamic (PK-PD) model was applied to link corticosteroid concentrations to the effect on lymphocytes and granulocytes. Results. Deflazacort is an inactive prodrug which is converted rapidly to the active metabolite 21-desacetyldeflazacort. Maximum concentrations of 21-desacetyldeflazacort averaged 116 ng/ml and were observed after 1.3 h. The average area under the curve was 280 ng/ml · h, and the terminal half-life was 1.3 h. 21-Desacetyldeflazacort was cleared significantly faster than both methylprednisolone and prednisolone. The PK-PD-model was suitable to describe time course and magnitude of the observed effects. The results were consistent with reported values for glucocorticoid receptor binding affinities for the investigated compounds. Conclusions. Due to the short pharmacokinetic half-life of its active metabolite, pharmacodynamic effects of deflazacort are of shorter duration than those of methylprednisolone and prednisolone. The PK-PD model allows good prediction of pharmacodynamic effects based on pharmacokinetic and receptor binding data.     

Pharmacokinetics of controlled-release verapamil in healthy volunteers and patients with hypertension or angina

AIMS: To study the dose-ranging population pharmacokinetics of controlled release verapamil in healthy subjects and patients with angina or hypertension. To characterize the pharmacodynamics of controlled-release verapamil in patients with hypertension. METHODS: Dose-ranging studies were conducted in healthy volunteers and patients with hypertension and angina. Subjects received doses of 120, 180, 360, or 540 mg racemic verapamil as an osmotic controlled-release formulation. A population pharmacokinetic model involving zero-order release of verapamil into the gastrointestinal tract with first-order absorption and elimination was used to describe the steady-state plasma concentration profile for R- and S-verapamil. A population sigmoid E(max) pharmacodynamic model was used to describe the effect of R- and S-verapamil on mean arterial blood pressure. RESULTS: S-verapamil had an approximate 4-fold greater apparent clearance than R-verapamil in both healthy volunteers and patients. The apparent plasma clearance of R- and S-verapamil in healthy volunteers decreased over the dose range of 120-540 mg. A similar dose-dependent decrease in apparent plasma clearance was also noted in patients. None of the patient demographic variables examined (age, total body weight, lean body weight, body mass index, and height) explained the variability in verapamil pharmacokinetics. The pharmacodynamic model describing the relationship between verapamil plasma concentration and mean arterial blood pressure indicated that the S-verapamil had a 3.6-fold lower estimated EC(50) compared to R-verapamil. CONCLUSIONS: The results from this dose-ranging pharmacokinetic investigation in healthy volunteers and patients are consistent with previous reports in healthy subjects. S-verapamil is cleared more rapidly than R-verapamil and the estimated EC(50) for S-verapamil was 3.6-fold lower than for R-verapamil. Estimated EC(50) values for R- and S-verapamil decreased with increasing age and decreasing weight.     

Population Pharmacokinetic/Pharmacodynamic Modeling of Systemic Corticosteroid Inhibition of Whole Blood Lymphocytes: Modeling Interoccasion Pharmacodynamic Variability

Purpose To develop a population pharmacokinetic/pharmacodynamic (PK/PD) model that characterizes the effects of major systemic corticosteroids on lymphocyte trafficking and responsiveness. Materials and Methods Single, presumably equivalent, doses of intravenous hydrocortisone (HC), dexamethasone (DEX), methylprednisolone (MPL), and oral prednisolone (PNL) were administered to five healthy male subjects in a five - way crossover, placebo - controlled study. Measurements included plasma drug and cortisol concentrations, total lymphocyte counts, and whole blood lymphocyte proliferation (WBLP). Population data analysis was performed using a Monte Carlo-Parametric Expectation Maximization algorithm. Results The final indirect, multi-component, mechanism-based model well captured the circadian rhythm exhibited in cortisol production and suppression, lymphocyte trafficking, and WBLP temporal profiles. In contrast to PK parameters, variability of drug concentrations producing 50% maximal immunosuppression (IC₅₀) were larger between subjects (73–118%). The individual log-transformed reciprocal posterior Bayesian estimates of IC₅₀ for ex vivo WBLP were highly correlated with those determined in vitro for the four drugs (r ²  = 0.928). Conclusions The immunosuppressive dynamics of the four corticosteroids was well described by the population PK/PD model with the incorporation of inter-occasion variability for several model components. This study provides improvements in modeling systemic corticosteroid effects and demonstrates greater variability of system and dynamic parameters compared to pharmacokinetics. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

老年重症耐甲氧西林金黄色葡萄球菌感染患者替考拉宁的药动学研究和药效学评价

目的 探讨耐甲氧西林金黄色葡萄球菌（methicillin-resistant Staphylococcus aureus,MRSA）肺部感染老年患者替考拉宁的药动学差异与药效学评价。方法 研究纳入2018年9月—2019年8月诊断为MRSA肺部感染患者50例,治疗前下呼吸道痰液标本培养提示为MRSA感染并对替考拉宁敏感。所有患者予静脉输注替考拉宁,前3剂400 mg,每12 h给药1次,维持剂量400 mg每24 h给药1次。将达到稳态后第5次给药前测得C_min≤10 μg·mL^-1的患者维持剂量方案更改为600 mg qd。注射用替考拉宁400 mg或600 mg溶于100 mL 0.9%氯化钠注射液,给药时间为30 min,疗程为14~21 d。按规定时间采集静脉血2 mL,采用HPLC检测血药浓度,使用DAS 3.0软件处理,求出各例患者的药动学参数。将达稳态后第5次给药前测得C_min>10 μg·mL^-1的药动学参数设为A;对于C_min≤10 μg·mL^-1的患者,将维持剂量调整为600 mg qd,并将调整剂量后第5次给药前测得C_min>10 μg·mL^-1的药动学参数设为B。按照APACHE II评分和SOFA评分将患者分为重症和非重症2组。结合PK/PD原理,比较2组患者在C_min处于10~20 μg·mL^-1和20~30 μg·mL^-1以及AUC/MIC≥345和<345的临床治疗有效率、细菌清除率及不良反应发生情况。结果 替考拉宁药动学以二室模型描述最佳。负荷剂量相同时,维持剂量为400 mg的A组与维持剂量为600 mg的B组药动学参数比较如下：C_max[（32.28±15.16） mg·L^-1vs（65.73±28.96） mg·L^-1],t_1/2[（86.24±10.61） h vs（70.51±11.78） h],V_d[（2.73±1.32） L·kg^-1vs（2.58±1.02） L·kg^-1],CL[（0.11±0.05） L·h^-1·kg^-1vs（0.13±0.06） L·h^-1·kg^-1],AUC_（0~t）[（2 698.16±1 603.25） mg·h·L^-1vs（4 076.85±1 873.09） mg·h·L^-1],AUC_（0~∞）[（4 509.33±2 786.54） mg·h·L^-1vs（7 193.58±4 109.81） mg·h·L^-1],差异有统计学意义（P<0.05）。患者SOFA评分≤5和>5的临床有效率为65.71%和53.33%,细菌清除率分别为65.79%和47.37%。APACHEII评分≤15和>15的临床有效率分别为63.64%和58.82%,细菌清除率分别为63.16%和52.63%。结论 MRSA肺部感染老年患者替考拉宁的药动学存在较大差异,结合PK/PD原理,能够为个体化治疗提供科学的给药方案。     

Effects of naringin on the pharmacokinetics of verapamil and one of its metabolites, norverapamil, in rabbits

It was reported that verapamil is metabolized via hepatic microsomal cytochrome P450 (CYP) 3A4 and that naringin (a component of grapefruit juice) inhibits CYP3A4 in humans. Hence, after oral administration of verapamil, the total area under the plasma concentration-time curve from time zero to time infinity (AUC) of verapamil and the AUC(verapamil)/AUC(D-617 (a metabolite of verapamil)) ratio were significantly greater after oral grapefruit juice in humans. The aim of this study was to determine whether similar results could be obtained from rabbits. The pharmacokinetics of verapamil and one of its metabolites, norverapamil, were investigated after oral administration of verapamil at a dose of 9 mg/kg without or with oral naringin at a dose of 7.5 mg/kg in rabbits. With naringin, the AUC of verapamil was significantly greater (28.4 versus 18.4 microg min/ml). Although, the AUC values of norverapamil were not significantly different between groups without and with naringin, the AUC(verapamil)/AUC(norverapamil) ratio was considerably greater (1.49 versus 1.11) with naringin. The above data suggested that the metabolism of verapamil and the formation of norverapamil was inhibited by naringin possibly by inhibition of CYP3A in rabbits.     

Pharmacokinetic and Pharmacodynamic Evaluation for Tissue-Selective Inhibition of Cholesterol Synthesis by Pravastatin

The tissue-selective inhibition of cholesterol synthesis by pravastatin was evaluated pharmacokinetically and pharmacodynamically. Plasma, tissue, urine, and bile concentrations were measured after iv bolus injection of pravastatin to rats at various doses. The total body clearance and steady state volume of distribution decreased with increasing dose. A saturable biliary excretion was also observed. The time course of plasma and liver concentrations was described by a three-compartment model, consisting of a central compartment, a deep compartment with an nonsaturable uptake process, and a shallow compartment with saturable uptake and nonsaturable elimination processes. It suggests that a mechanism for the decrease in the total body clearance and distribution volume might be explained by a saturation of pravastatin uptake into the liver. Plasma concentration data after oral administration was also fitted to the same model by connecting an absorption compartment to the shallow compartment. The inhibitory activity of pravastatin against cholesterol synthesis in liver could be related to the concentration in the shallow compartment via a sigmoidal Emax model and the obtained pharmacodynamic parameters were comparable to those in vitro. Results suggest that the carrier-mediated hepatic uptake of pravastatin is actually responsible for the hepatoselective inhibition of cholesterol synthesis under physiological conditions.     

Pharmacodynamic Models for Agents that Alter Production of Natural Cells with Various Distributions of Lifespans

Indirect pharmacodynamic response (IDR) models were developed for agents which alter the generation of cell populations with arbitrary lifespan distributions. These models extend lifespan based IDR models introduced previously [J. Pharmacokinet. Biopharm. 27: 467, 1999] for cell populations with the same lifespan (“delta” distribution). Considered are cell populations exhibiting time-invariant lifespan distributions described by the probability density function ℓ(τ). It is assumed that cell response (R) is produced at a zero-order rate (k _in(t)) and is eliminated from the population when the cell lifespan expires. The cell loss rate is calculated as k _in*ℓ(t), where ‘*’ denotes the convolution operator. Therapeutic agents can stimulate or inhibit production rates according to the Hill function: 1 ± H(C(t)) where H(C(t)) contains the capacity (S _max) and potency (SC ₅₀) parameters and C(t) is a pharmacokinetic function. The production rate is k _in(t)=k _in· [ 1±H(C(t))]. The operational model is dR/dt = k _in(t)−k _in*ℓ(t) with the baseline condition R ₀ = k _in· T _R , where T _R is the mean lifespan. Single populations as well as populations with precursors were examined by simulation to establish the role of lifespan distribution parameters (mean and standard deviation) in controlling the response vs. time profile. Estimability of parameters was assessed. Numerical techniques of solving differential equations with the convolution integral were proposed. In addition, the models were applied to literature data to describe the stimulatory effects of single doses of recombinant human erythropoietin on reticulocytes in blood. The estimates of S _max and SC ₅₀ for these agents were obtained along with means and standard deviations for reticulocyte lifespan distributions. The proposed models can be used to analyze the pharmacodynamics of agents which alter natural cell production yielding parameters describing their efficacy and potency as well as means and standard deviations for cell lifespan distributions. This work was supported in part by Grant No. GM 57980 from the National Institute of General Medical Sciences, National Institutes of Health.     

Pharmacodynamic and pharmacokinetic comparison of two formulations of lorazepam and placebo

Nine healthy male volunteers took part in a comparison of the pharmacokinetic profiles and effects on psychomotor performance and memory following single dosing with an experimental sustained release (SR) and a conventional release (CR) formulation of lorazepam. There was evidence of a sustained release profile for the SR formulation which had a later t_max (median 8 h) and significantly lower peak serum concentration (mean 12·1 ng ml^?1) compared with CR lorazepam (2 h and 21.8 ng ml^?1 respectively). The effect of SR lorazepam on pharmacodynamic variables occurred later and was less intense than with CR lorazepam. Good correlations were obtained between most of the pharmacodynamic parameters measured and the logarithm of the serum concentration for both formulations, with the exception of the choice reaction time test for SR lorazepam. However, both formulations produced a similar degree of impairment of 24h recall.     

Pharmacodynamic Analysis of Analgesic Clinical Trials Using Empirical Methods

Data from analgesic clinical trials have characteristics such as ordered categorical longitudinal responses with repeated measures, delay of effect with respect to analgesic plasma concentration, and right-hand censoring of response due to remedication. In order to determine the concentration-effect relationship of such data, we propose convolving an empirical function for plasma concentration, in the form of broken lines which connect each pair of neighboring observations, with a monoexponential function, to generate effect site concentration Effect site concentration and time are used, simultaneously, as independent variables in the fit of the model for the logit of the probability of having a specific pain relief (PR) score at each time point pre-remedication, via maximum likelihood. Using corresponding effect site concentration, the probabilities of having specific PR scores post-remedication are predicted via the concentration-response relationship established. The overall (pre- and post-remedication) predictions and corresponding standard errors for the responses are then estimated. Inference of the PR scoring, using a posterior method, is proposed. An illustration using real data is used to demonstrate these methods.     

Pharmacodynamic Assessment of the Benztropine Analogues AHN-1055 and AHN-2005 Using Intracerebral Microdialysis to Evaluate Brain Dopamine Levels and Pharmacokinetic/Pharmacodynamic Modeling

No HeadingPurpose. The benztropine (BZT) analogues bind with high affinity to the dopamine transporter (DAT) and demonstrate a behavioral and pharmacokinetic profile unlike that of cocaine. The development of a predictive pharmacokinetic/pharmacodynamic (PK/PD) model to characterize the concentration-effect relationship between the BZT analogues and brain dopamine (DA) levels is an important step in the evaluation of these compounds as potential cocaine abuse pharmacotherapies. Hence, the objective of this study was to mathematically characterize the PD of BZT analogues and cocaine, using appropriate PK/PD models.Methods. Dialysis probes were stereotaxically implanted into the nucleus accumbens of Sprague-Dawley rats (275–300 g). Extracellular fluid (ECF) DA levels were measured after intravenous administration of the BZT analogues AHN-1055 and AHN-2005, as well as cocaine using high performance liquid chromatography-electrochemical detection (HPLC-ECD). PD models were used to describe the relationship between the BZT analogues or cocaine and brain microdialysate DA, and suitability was based on standard goodness-of-fit criteria.Results. The BZT analogues produced a sustained increase in brain microdialysate DA levels in comparison to cocaine. The time of maximum concentration (T_max) for brain microdialysate DA was 2 h for AHN-1055 and 1 h for AHN-2005 compared to a T_max of 10 min for cocaine. The duration of brain microdialysate DA elevation was 12–24 h for the BZTs in comparison to 1 h for cocaine. An indirect model with inhibition of loss of response and a sigmoid E_max model best described the PK/PD for the BZT analogues and cocaine, respectively. The 50% of maximum inhibition (IC₅₀) of the loss of DA was lower for AHN-2005 (226 ± 27.5 ng/ml) compared to AHN-1055 (321 ± 19.7 ng/ml). In addition, the EC₅₀ for cocaine was 215 ± 11.2 ng/ml.Conclusions. The slow onset and long duration of BZT analogue–induced DA elevation may avoid the reinforcing effects and craving of cocaine. Further, the developed models will be useful in characterizing the PK/PD of other analogues and aid in the assessment of the therapeutic efficacy of the BZT analogues as substitute medications for cocaine abuse.     

Inter- and intra-subject variation in the first-pass elimination of highly cleared drugs during chronic dosing

Summary The pharmacokinetics of verapamil in five healthy volunteers were investigated on 4 occasions during chronic administration of deuterated verapamil. There was no statistically significant difference in oral clearance, terminal half-life, bioavailability, morning trough level and peak concentration or in the time of their occurrence on the four occasions. The plasma clearance, however, exhibited considerable inter- and intra-individual variation, ranging between 26.3% and 85.4% and 12.0% and 48.0%, respectively. Comparison of these pharmacokinetic parameters with data from previous single dose studies in the same subjects revealed a significant (p<0.05) decrease in the clearance and an increase in the apparent bioavailability of verapamil during chronic administration, although no difference in the half-life was found. Due to the considerable variation in the oral clearance of verapamil during chronic dosing, steady-state conditions in a strict pharmacokinetic sense may never be attained, and pharmacokinetic data obtained in single dose studies will be of limited value in predicting steady-state plasma concentrations.