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.     

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.     

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.     

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.     

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     

A Method of Obtaining Starting Values of k in and k out for the Indirect Response Models

A method based on the multivariate technique known as principal component analysis is proposed to obtain starting values for the rate constants of indirect response models. The method is not iterative and only requires standard deviation calculations for two quantities, which are simple functions of the measured pharmacodynamic response. An algorithm is provided which can be implemented in a spreadsheet. The methodology is justified theoretically herein, nevertheless, two examples are provided to illustrate the method and demonstrate its viability.     

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.     

Mathematical Modeling of Circadian Cortisol Concentrations Using Indirect Response Models: Comparison of Several Methods

Six mathematical functions to describe the chronobiology of cortisol concentrations were assessed. Mean data from a dose-proportionality study of inhaled fluticasone propionate were fitted with an indirect response model using various biorhythmic functions (single cosine, dual ramps, dual zero-order, dual cosines, and Fourier series with 2 and n-harmonics) for production rate. Data with known parameters and random variation were also generated and fitted using the ADAPT II program. Fitted parameters, model estimation criteria, and runs tests were compared. Models with preassigned functions: the dual ramps, the dual zero-order and the dual cosines provide maximum and minimum times for cortisol release rate, were suitable for describing asymmetric circadian patterns and yielding IC₅₀ values. Fourier analysis differs from the other methods in that it uses the placebo data to recover equations for cortisol secretion rate rather than by postulation. Nonlinear regression for Fourier analysis, instead of the L ² -norm method, was useful to characterize the baseline cortisol data but was restricted to a maximum of two harmonics. Apart from the single cosine function, which predicts symmetrical cortisol concentrations, all methods were satisfactory in describing the baseline and suppressed cortisol concentrations. On the other hand, Fourier series with L ² -norm produced the best unbiased estimate for baseline patterns. The Fourier method is flexible, accurate, and can be extended to other drug-induced changes in normal periodic rhythms.     

Population Pharmacokinetic-Pharmacodynamic Model of Craving in an Enforced Smoking Cessation Population: Indirect Response and Probabilistic Modeling

Purpose. A population pharmacokinetic-pharmacodynamic model accounting for placebo effect was used to relate nicotine concentration and enforced smoking cessation craving score measured by the Tiffany rating scale short form. Methods. Twenty-four smokers were enrolled in a placebo-controlled, randomized, double-blind, three periods, crossover trial. The study objective was to describe the nicotine-induced changes on craving scores. Two modeling strategies based on a mechanistic (indirect response models with drug-related inhibition on the k_in synthesis rate and with a drug-related stimulation of the k_out removal rate were evaluated) and a probabilistic (logistic regression) approach were used. Results. Placebo response model properly fitted the circadian changes on craving scores. The analysis revealed that the indirect response model with inhibition on k_in was the preferred model for the smoking data whereas the preferred model for the Nicotine Replacement Therapy data was the one with stimulation on k_out. The logistic analysis showed that the nicotine concentration was a significant predictor of reduction in craving during the free-smoking period. Conclusions. Nicotine dosage regimen can influence the nicotine mechanism of action: an instantaneous delivery at an individually selected time seems to inhibit the onset of craving while constant delivery at a pre-defined time seems to attenuate the craving.     

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.     

Collapsing Mechanistic Models: An Application to Dose Selection for Proof of Concept of a Selective Irreversible Antagonist

When data fail to support fully mechanistic models, alternative modeling strategies must be pursued. Simpler, more empirical models or the fixing of various rate constants are necessary to avoid over-parameterization. Fitting empirical models can dilute information, limit interpretation, and cloud inference. Fixing rate constants requires external, relevant, and reliable information on the mechanism and can introduce subjectivity as well as complicate determining the validity of model extrapolation. Furthermore, both these methods ignore the possibility that failure of the data to support the mechanistic model could contain information about the pharmacodynamic process. If the pathway has processes with “fast” dynamics, these steps could collapse yielding parametrically simpler classes of models. The collapsed models would retain the mechanistic interpretation of the full model, which is crucial for performing substantive inference, while reducing the number of parameters to be estimated. These concepts are illustrated through their manifestations on the dose–effect relationship and ensuing dose selection for a proof of concept study. Specifically, a mechanistic model for a selective irreversible antagonist was posited and candidate classes of models were derived utilizing “fast dynamics” assumptions. Model assessment determined the rate-limiting step facilitating pertinent inference with respect to the mechanism. For comparison, inference using a more empirical modeling strategy is also presented. A general solution for the collapse of the typical PK–PD model differential equations is provided in Appendix A     

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.     

Further Developments of the Fisher Information Matrix in Nonlinear Mixed Effects Models with Evaluation in Population Pharmacokinetics

We extend the development of the expression of the Fisher information matrix in nonlinear mixed effects models for designs evaluation. We consider the dependence of the marginal variance of the observations with the mean parameters and assume an heteroscedastic variance error model. Complex models with interoccasions variability and parameters quantifying the influence of covariates are introduced. Two methods using a Taylor expansion of the model around the expectation of the random effects or a simulated value, using then Monte Carlo integration, are proposed and compared. Relevance of the resulting standard errors is investigated in a simulation study with NONMEM.     

In-vitro Cell Culture Models of the Nasal Epithelium: A Comparative Histochemical Investigation of Their Suitability for Drug Transport Studies

Purpose. To evaluate different in-vitro cell culture models for their suitability to study drug transport through cell monolayers. Methods. Bovine turbinate cells (BT; ATCC CRL 1390), human nasal septum tumor cells (RPMI, 2650; ATCC CCL 30), and primary cell cultures of human nasal epithelium were characterized morphologically and histochemically by their lectin binding properties. The development of tight junctions in culture was monitored by actin staining and transepithelial electrical resistance measurements. Results. The binding pattern of thin-sections of excised human nasal respiratory epithelium was characterized using a pannel of fluorescently-labelled lectins. Mucus in goblet cells was stained by PNA, WGA and SBA, demonstrating the presence of terminal N-acetylglucosamine, N-acetylgalactosamine and galactose residues respectively in the mucus of human nasal cells. Ciliated cells revealed binding sites for N-acetylglucosamine, stained by WGA, whereas Con A, characteristic for mannose moieties, labelled the apical cytoplasm of epithelial cells. Binding sites for DBA were not present in this tissue. Comparing three different cell culture models: BT, RPMI 2650, and human nasal cells in primary culture using three lectins (PNA, WGA, Con A) as well as intracellular actin staining and transepithelial electrical resistance measurements we found, that only human nasal epithelial cells in primary culture showed differentiated epithelial cells, ciliated nasal cells and mucus producing goblet cells, which developed confluent cell monolayers with tight junctions. Conclusions. Of the in-vitro cell culture models studied, only human nasal cells in primary culture appears to be suitable for drug transport studies.