Characterization of four basic models of indirect pharmacodynamic responses

Four basic models of indirect pharmacodynamic responses were characterized in terms of changing dose, I_max or S_max, and IC₅₀ or SC₅₀ to examine the effects of these fundamental drug properties on response profiles. Standard pharmacokinetic parameters were used for generating plasma concentration, and response-time profiles using computer simulations. Comparisons to theoretical expectations were made. In all four models, the maximum response (R_max) (inhibition or stimulation) and the time of its occurrence (TR_max) were dependent on the model, dose, I_max or S_max, and IC₅₀ or SC₅₀ values,. An increase in dose or a decrease in IC₅₀ or SC₅₀ by the same factor produced, as theoretically expected, identical and superimposable pharmacodynamic response patterns in each of the models. Some parameters (TR_max, ABEC) were nearly proportional to log dose, while others (R_max, CR_max) were nonlinear. Assessment of expected response signature patterns as demonstrated in this report may be helpful in experimental designs and in assigning appropriate models to pharmacodynamic data. Supported in part by Grant No. 24211 from the National Institute of General Medical Sciences, National Institutes of Health.     

Comparison of four basic models of indirect pharmacodynamic responses

Four basic models for characterizing indirect pharmacodynamic responses after drug administration have been developed and compared. The models are based on drug effects (inhibition or stimulation) on the factors controlling either the input or the dissipation of drug response. Pharmacokinetic parameters of methylprednisolone were used to generate plasma concentration and response-time profiles using computer simulations. It was found that the responses produced showed a slow onset and a slow return to baseline. The time of maximal response was dependent on the model and dose. In each case, hysteresis plots showed that drug concentrations preceded the response. When the responses were fitted with pharmacodynamic models based on distribution to a hypothetical effect compartment, the resulting parameters were dose-dependent and inferred biological implausibility. Indirect response models must be treated as distinct from conventional pharmacodynamic models which assume direct action of drugs. The assumptions, equations, and data patterns for the four basic indirect response models provide a starting point for evaluation of pharmacologie effects where the site of action precedes or follows the measured response variable.Glossary C _e Drug concentration at the hypothetical effect site - C _p Plasma concentration of drug - C _p(T_max) Plasma concentration of drug at the time of maximal response - D Dose - EC ₅₀ Drug concentration producing 50% of maximum stimulation at effect site - E _max Maximum effect attributed to drug - E _o Baseline effect prior to drug administration - IC ₅₀ Drug concentration producing 50% of maximum inhibition at effect site - K _el First-order rate constant for drug elimination - K _eo First-order rate constant for drug loss from effect site - K _in Zero-order rate constant for production of drug response - K _out First-order rate constant for loss of drug response - n Sigmoidicity factor of the sigmoid E_max equation - R Response variable - R_max Maximal (or minimal) response - R_o Initial response (time zero) prior to drug administration - t time after drug administration - T Infusion time - T_max Time to reach maximum effect following drug administration - V Volume of distribution Supported in part by Grant No. 24211 from the National Institutes of General Medical Sciences, National Institutes of Health.     

Pharmacokinetic/pharmacodynamic model for prednisolone inhibition of whole blood lymphocyte proliferation

AIMS: Mitogen-induced ex vivo whole blood lymphocyte proliferation (WBLP) is a widely used method to assess lymphocyte responsiveness to immunosuppressive therapy. A three-component complex model was developed to characterize effects of prednisolone on cell trafficking, transduction, and lymphocyte suppression. METHODS: An oral dose (0.27 mg kg-1) of prednisone was given to 32 subjects. The study consisted of baseline and prednisone phases each with 32 h of sampling. Measurements included plasma prednisolone concentrations, in vitro and ex vivo WBLP, and lymphocyte cell counts during baseline and prednisone phases. RESULTS: The final model consists of a precursor-dependent indirect response model with a first-order periodic influx rate for lymphocyte trafficking. This accounts for the rebound phenomenon and the circadian rhythm seen in all individual ex vivo WBLP effect-time profiles. Prednisolone was modelled as inhibiting lymphocyte influx from the precursor to the blood pools. The direct suppressive effect of prednisolone on WBLP was modelled with the simple Imax model. A transduction step with rate constant kt was introduced to the simple Imax model to account for the delay ( approximately 4 h) in reaching the maximum inhibition. The IC50 values obtained ex vivo were circa 10 times lower than in vitro values (3.76 vs 38.8 ng ml-1), suggesting additional in vivo factors may have enhanced lymphocyte response to the inhibitory effect of prednisolone. CONCLUSIONS: This integrated PK/PD model enables evaluation of multicomponent direct and indirect inhibition of ex vivo WBLP by steroids and other immunosuppressants in relation to sex and race.     

卡维地洛人体药动学与药效学结合模型研究

采用间接药效学模型和效应室模型两种药效学模型分别进行卡维地洛药动学与药效学关系研究, 比较两种药效学模型的拟合程度。高效液相色谱法 (荧光) 测定20名健康志愿者单次口服20 mg卡维地洛片后卡维地洛经时血药浓度, 以DAS 2.0实用药动学计算程序计算卡维地洛药动学参数。同时测定给药前后动脉收缩压和舒张压, 计算降压效果。卡维地洛片主要药动学参数t_1/2为 (4.56 ± 2.56) h, C_max为 (46.29 ± 21.07) ng·mL^-1, AUC_0-∞为 (173.76 ± 87.36) ng·mL^-1·h。间接药效模型主要参数K_in为 (0.41 ± 0.31) % h^-1, K_out为 (0.40 ± 0.26) h^-1, IC₅₀为 (24.40 ± 21.10) ng·mL^-1, AUE为 (3.82 ± 1.46) % h。效应室模型主要参数K_e0为 (0.35 ± 0.27) h^-1, EC₅₀为 (24.30 ± 24.30) ng·mL^-1, AUE为 (5.65 ± 2.54) % h。该方法可用于卡维地洛片人体药动学研究。由AIC值可知, 效应室模型可更好的应用于卡维地洛药动学-药效学结合研究。

     

Basic pharmacodynamic models for agents that alter the lifespan distribution of natural cells

A new class of basic indirect pharmacodynamic models for agents that alter the loss of natural cells based on a lifespan concept are presented. The lifespan indirect response (LIDR) models assume that cells (R) are produced at a constant rate (k(in)), survive during a certain duration T(R), and finally are lost. The rate of cell loss is equal to the production rate but is delayed by T(R). A therapeutic agent can increase or decrease the baseline cell lifespan to a new cell lifespan, T(D), by temporally changing the proportion of cells belonging to the two modes of the lifespan distribution. Therefore, the change of lifespan at time t is described according to the Hill function, H(C(t)), with capacity (E(max)) and sensitivity (EC(50)), and the pharmacokinetic function C(t). A one-compartment cell model was examined through simulations to describe the role of pharmacokinetics, pharmacodynamics and cell properties for the cases where the drug increases (T(D) > T(R)) or decreases (T(D) < T(R)) the cell lifespan. The area under the effect curve (AUCE) and explicit solutions of LIDR models for large doses were derived. The applicability of the model was further illustrated using the effects of recombinant human erythropoietin (rHuEPO) on reticulocytes. The cases of both stimulation of the proliferation of bone marrow progenitor cells and the increase of reticulocyte lifespans were used to describe mean data from healthy subjects who received single subcutaneous doses of rHuEPO ranging from 20 to 160 kIU. rHuEPO is about 4.5-fold less potent in increasing reticulocyte survival than in stimulating the precursor production. A maximum increase of 4.1 days in the mean reticulocyte lifespan was estimated and the effect duration on the lifespan distribution was dose dependent. LIDR models share similar properties with basic indirect response models describing drug stimulation or inhibition of the response loss rate with the exception of the presence of a lag time and a dose independent peak time. The current concept can be applied to describe the pharmacodynamic effects of agents affecting survival of hematopoietic cell populations yielding realistic physiological parameters.     

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 (kout) of the response variable. Partially integrated solutions for these models developed previously for i.v. 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, Imax or Smax capacity factors, IC50 or SC50 sensitivity factors, and Kout 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 i.v. bolus doses. These relationships provide both a formal and practical basis for better understanding of the time-course of basic indirect response models.     

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.     

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.     

A Dynamical Systems Analysis of the Indirect Response Model with Special Emphasis on Time to Peak Response

In this paper we present a mathematical analysis of the four classical indirect response models. We focus on characteristics such as the evolution of the response R(t) with time t, the time of maximal/minimal response T_max and the area between the response and the baseline AUC_R, and the way these quantities depend on the drug dose, the dynamic parameters such as E_max and EC₅₀ and the ratio of the fractional turnover rate k_out to the elimination rate constant k of drug in plasma. We find that depending on the model and on the drug mechanism function, T_max may increase, decrease, decrease and then increase, or stay the same, as the drug dose is increased. This has important implications for using the shift in T_max as a diagnostic tool in the selection of an appropriate model     

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.     

Multiple-pool cell lifespan model of hematologic effects of anticancer agents

The leukopenic effects of anticancer agents are described using a semi-physiologic multiple-pool cell lifespan model. The time course of myelosuppression in relation to the drug concentration vs. time profile was characterized using a three pool indirect model. The proliferation and maturation stages of myeloid cells in the bone marrow and cell removal from the circulation were quantitated with a cell life-span concept. Drug effects were assumed to take place in the bone marrow based on irreversible linear or capacity-limited cytotoxicity. Mathematical derivations and computer simulations (Adapt II) were used to examine the properties of the model. Data from the literature were also analyzed. Cell response profiles after therapy typically exhibit a lag period, reduction to a nadir, and return to baseline. The predicted values of the time periods of granulopoiesis were 10–14 days for proliferation, and 1–6 days for maturation of progenitor cells in the bone marrow. The proposed irreversible mechanism of cell killing by anticancer drugs explains previously observed relationships between leukocyte nadir counts and exposure to the drug and/or duration of drug concentrations above some threshold level. The model was applied to literature data for paclitaxel and etoposide effects on leukocyte counts. The predicted value of KC₅₀ for paclitaxel ranged from 0.004 to 0.2 g/mL and for etoposide 2 g/mL. The present model accounts for drug-induced leukopenia using a physiologic cell production and loss model and irreversible cytotoxicity in a precursor pool.     

Pharmacokinetic and pharmacodynamic studies following the intravenous and oral administration of the antiparkinsonian drug biperiden to normal subjects

Summary The pharmacokinetics and pharmacodynamics (changes in pupil size and salivary flow) of biperiden following a single oral and intravenous dose were investigated in six normal subjects.After the injection plasma concentrations declined biphasically, with half-times of 1.5 h for the rapid phase and 24 h for the terminal phase. Clearance and apparent volume of distribution were high (12 ml·min^–1·kg^–1 and 24 l·kg^–1 respectively). Absorption was rapid but the systemic availability was incomplete (33%), probably due to first-pass metabolism.Central nervous system (CNS) adverse effects and changes in pupil size were observed after both routes of administration while salivary flow was affected only by the injection.     

Indirect Pharmacodynamic Models for Responses with Multicompartmental Distribution or Polyexponential Disposition

Basic indirect response models where drug alters the production (k _in ) of the response variable (R) based on the Hill function previously assumed one-compartment distribution of the response variable and simple first-order loss (k _out ) of R. These models were extended using convolution theory to consideration of two-compartment distribution of R and/or polyexponential loss of R. Theoretical equations and methods of data analysis were developed and simulations are provided to demonstrate expected response behavior based on biexponential response dissipation. The inhibition model was applied to our previous data for inhibition of circadian cortisol secretion by prednisolone. The presence of multicompartment response variables and/or polyexponential loss complicates the response patterns and resolution of pharmacologic parameters of indirect response models and requires careful experimental and data analysis approaches in order to properly evaluate such pharmacodynamic responses. The occurrence of these alternative distribution or disposition components does not alter the area under the effect curve (AUCE) which remains identical to the basic models. Model misselection was addressed by testing fittings comparing the basic and new models. Use of the former for these more complex models does not severely perturb the calculated cardinal dynamic parameters. These models may provide improved insights into indirect responses with complexities in distribution or disposition.     

退热Ⅲ号制剂的药效试验

目的：对退热Ⅲ号口服液的药理作用进行实验研究。方法：以发热、炎症及咳嗽多种动物模型观察了退热Ⅲ号口服液的药效作用。结果：退热Ⅲ号口服液对致热家兔及大鼠体温升高均有明显的退热作用,并能抑制小鼠耳廓的炎性肿胀,并有止咳祛痰作用。结论：证实了退热Ⅲ号口服液治疗感冒、发热、止咳等疾病的作用及其机理,为临床应用提供实验依据。     

The use of pharmacokinetic and pharmacodynamic data in the assessment of drug safety in early drug development

The pharmaceutical industry continues to look for ways to reduce drug candidate attrition throughout the drug discovery and development process. A significant cause of attrition is due to safety issues arising either as a result of animal toxicity testing or in the clinical programme itself. A factor in the assessment of safety during early drug development is the pharmacokinetic profile of the compound. This allows safety data to be considered in the light of systemic drug exposure and therefore permits a quantitative assessment. This is particularly applicable when assessing the risk of a new chemical entity (NCE) in relation to safety parameters such as QT interval prolongation, where free plasma concentrations have been shown to be predictive of this property in relation to potency in preclinical testing. Prior to actual human exposure it is therefore important to be able to predict reliably the pharmacokinetic behaviour of an NCE in order to place such safety findings into a quantitative risk context. The emerging science of pharmacogenetics is likely to further our ability to assess the risk of NCEs to populations and individuals due to genetic variance. The drug metabolizing enzyme CYP2D6 has been recognized as providing the potential to result in widely differing systemic drug exposure in the patient population due to polymorphic expression. Further knowledge is likely to add to our understanding of population differences in exposure and response and aid in the identification of risk factors. One potential strategy for improving the effectiveness of the drug discovery process is to obtain clinical pharmacokinetic data more rapidly in order to assess more accurately the potential for both efficacy and safety of an NCE. Whilst procedures and technologies are available that allow this on the microdose scale, it is important that we recognize potential limitations of these approaches in order that they can be applied beneficially.     

Role of dosage regimen in controlling indirect pharmacodynamic responses

The role of drug delivery in controlling indirect pharmacodynamic responses was assessed via computer simulations and literature review. Simulations of responses related to basic indirect response mechanisms were performed for various drug input rates which allowed the importance of drug delivery rate on the overall pharmacodynamic response to be evaluated. Response versus time profiles of integrated or net responses and efficiency were examined. Rate of drug input has the greatest influence on the area under the effect curve when doses are larger and target drug concentrations are above the IC(50)/SC(50). The pharmacodynamics of drugs which elicit indirect pharmacologic responses such as corticosteroids, diuretics, growth hormone, erythropoietin and insulin indicate that sustained drug delivery enhances the therapeutic efficiency and pharmacodynamic availability.     

Role of dosage regimen in controlling indirect pharmacodynamic responses

The role of drug delivery in controlling indirect pharmacodynamic responses was assessed via computer simulations and literature review. Simulations of responses related to basic indirect response mechanisms were performed for various drug input rates which allowed the importance of drug delivery rate on the overall pharmacodynamic response to be evaluated. Response versus time profiles of integrated or net responses and efficiency were examined. Rate of drug input has the greatest influence on the area under the effect curve when doses are larger and target drug concentrations are above the IC(50)/SC(50). The pharmacodynamics of drugs which elicit indirect pharmacologic responses such as corticosteroids, diuretics, growth hormone, erythropoietin and insulin indicate that sustained drug delivery enhances the therapeutic efficiency and pharmacodynamic availability.