Wojciech Krzyzanski

Basic pharmacodynamic models for agents that alter production of natural cells.

Basic indirect pharmacodynamic models for agents which alter the generation of natural cells based on a life-span concept are introduced. It is assumed that cells (R) are produced at a constant rate (kin), survive for a specific duration TR, and then are lost. The rate of cell loss must equal the production rate but is delayed by TR, A therapeutic agent can stimulate or inhibit the production rate according to the Hill function: 1 ±H(C(t)) where H(C(t)) contains capacity (Smax) and sensitivity (SC50) constants and C(t) is a pharmacokinetic function. Thus an operative model is dR/dt=kin · [1± H(C(t))]-kin ·[1 ± H(C(t-TR))] with the baseline condition R0= kin· TR. One- and two-compartment catenary cell models were examined by simulation to describe the role of pharmacokinetics and cell properties. The area under the effect curve (AUCE) was derived. The models were applied to literature data to describe the stimulatory effects of single doses of hematopoietic growth factors such as granulocyte colony-stimulating factor (G-CSF) on neutrophils, thrombopoietin (TPO) on platelets, and erythropoietin (EPO) on reticulocytes in blood. The models described experimental data adequately and provided cell life-spans and SC50values. The proposed cell production/loss models can be readily used to analyze the pharmacodynamics of agents which alter cell production yielding realistic physiological parameters.

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 IC50values. 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 L2-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 L2-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.

Pharmacokinetic and Pharmacodynamic Modeling of Recombinant Human Erythropoietin after Intravenous and Subcutaneous Administration in Rats

The pharmacokinetics (PK) and pharmacodynamics (PD) of recombinant human erythropoietin (rHuEPO) were studied in rats after single i.v. and s.c. administration at three dose levels (450, 1350, and 4050 IU/kg). The plasma concentrations of rHuEPO and its erythropoietic effects including reticulocyte (RET), red blood cell (RBC), and hemoglobin (Hb) levels were determined. A two-compartment model with dual input rate and nonlinear disposition was used to characterize the PK of rHuEPO. The catenary indirect response model with several compartments reflecting the bone marrow and circulating erythropoietic cells was applied. The s.c. doses exhibited slow absorption (Tmax = 12 h) and incomplete bioavailability (F = 0.59). In placebo groups, RBC and Hb values gradually increased over time with growth of the rats, and the changes in the baselines monitored from 8 to 32 weeks of age were described by a nonlinear growth function. All doses resulted in dose-dependent increases in RET counts followed by an immediate decline below the baseline at around 6 days and returned to the predose level in 21–24 days after dosing. A subsequent steady increase of RBC and Hb levels followed and reached peaks at 6 days. A tolerance phenomenon observed at all dose levels was modeled by a negative feedback inhibition with the relative change in Hb level. The PK/PD model well described the erythropoietic effects of rHuEPO as well as tolerance, thereby yielding important PD parameters (Smax = 1.87 and SC50 = 65.37 mIU/ml) and mean lifespans of major erythropoietic cell populations in rats.

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, Imax or Smax, and kout 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.

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 (kin(t)) and is eliminated from the population when the cell lifespan expires. The cell loss rate is calculated as kin*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 (Smax) and potency (SC50) parameters and C(t) is a pharmacokinetic function. The production rate is kin(t)=kin· [ 1±H(C(t))]. The operational model is dR/dt = kin(t)−kin*ℓ(t) with the baseline condition R0 = kin· TR, where TR 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 Smax and SC50 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.

Pharmacodynamic Analysis of Recombinant Human Erythropoietin Effect on Reticulocyte Production Rate and Age Distribution in Healthy Subjects

AbstractObjective: To evaluate the effect of recombinant human erythropoietin (rHuEPO) on the reticulocyte production rate and age distribution in healthy subjects. Methods: Extensive pharmacokinetic and pharmacodynamic data collected from 88 subjects who received a single subcutaneous dose of rHuEPO (dose range 20–160 kIU) were analysed. Four nonlinear mixed-effects models were evaluated to describe the time course of the percentage of reticulocytes and their age distribution in relation to rHuEPO pharmacokinetics. Model A accounted for stimulation of the production of progenitor cells in bone marrow, and model B implemented shortening of differentiation and maturation times of early progenitors in bone marrow. Model C was the combination of models A and B, and model D was the combination of model A with an increase in the maturation times of the circulating reticulocytes. Model evaluation was performed using goodness-of-fit plots, a nonparametric bootstrap and a posterior predictive check. Results: Model D was selected as the best model, and evidenced accurate and precise estimation of model parameters and prediction of the time course of the percentage of reticulocytes. At baseline, the estimated circulating reticulocyte maturation time was 2.6 days, whereas the lifespan of the precursors in the bone marrow was about 5 days. The rHuEPO potency for the stimulatory effect (7.61 IU/L) was higher than that for the increase in reticulocyte maturation times (56.3 IU/L). There was a significant 1- to 2-day lag time in the reticulocyte response. The effect of rHuEPO on the reticulocyte age distribution consisted of a transient increase in the reticulocyte maturation time from baseline up to 6–7 days, occurring 1 day after administration. The dose-dependent amplitude of the changes in the age distribution lasted for 12–14 days. The model-predicted peak increase in the reticulocyte release rate ranged from 140% to 160% of the baseline value and was maximal on days 7–8 following rHuEPO administration. Conclusions: A semiphysiological model quantifying the effect of rHuEPO on the reticulocyte production rate and age distribution was developed. The validated model predicts that rHuEPO increases the reticulocyte production rate and modifies the reticulocyte age distribution in a dose-dependent manner.

Population cell life span models for effects of drugs following indirect mechanisms of action.

Pharmacokinetic/pharmacodynamic (PK/PD) models for hematological drug effects exist that assume that cells are produced by a zero- or first-order process, survive for a specific duration (cell lifespan), and then are lost. Due to the fact that delay differential equations (DDE) are needed for cell lifespan models, their software implementation is not straightforward. Our objective is to demonstrate methods to implement three different cell lifespan models for dealing with hematological drug effects and to evaluate the performance of NONMEM to estimate the model parameters. For the basic lifespan indirect response (LIDR) model, cells are produced by a zero-order process and removed due to senescence. The modified LIDR model adds a precursor pool. The LIDR model of cytotoxicity assumes a three-pool indirect model to account for the cell proliferation with capacity-limited cytotoxicity followed by maturation, and removal from the circulation. A numerical method (method of steps) implementing DDE in NONMEM was introduced. Simulation followed by estimation was used to evaluate NONMEM performance and the impact of the minimization algorithm (first-order method vs. first-order conditional estimation method) and the model for residual variability on the estimates of the population parameters. The FOCE method combined with log-transformation of data was found to be superior. This report provides methodology that will assist in application of population methods for assessing hematological responses to various types of drugs

Pharmacokinetic model of target-mediated disposition of thrombopoietin.

Thrombopoietin, TPO, a 353 amino acid cytokine, is a primary regulator of platelet production that was cloned recently. A target-mediated (platelet receptors) pharmacokinetic model was developed to characterize the disposition of TPO. Receptor-mediated endocytosis was assigned as the major elimination pathway in the model. A nonspecific binding compartment was also incorporated into the model. TPO concentration vs time profiles from a published phase 1 and 2 clinical trial were used to apply this model. Noncompartmental analysis demonstrated that TPO exhibits nonlinear kinetics. The proposed model captured the concentration-time profiles relatively well. The first-order internalization rate constant was estimated as 0.1 h−1. The endogenous binding capacity was estimated as 164.0 pM. The second-order binding association constant (kon) was 0.055 h−1·pM−1 and the first-order dissociation constant (koff) was estimated as 2.5 h−1, rendering the equilibrium dissociation constant Kd as 45.5 pM. This model may be relevant to other therapeutic agents with receptor-mediated endocytotic disposition.

Interpretation of transit compartments pharmacodynamic models as lifespan based indirect response models

Transit compartments (TC) models are used to describe pharmacodynamic responses that involve drug action on cells undergoing differentiation and maturation. Such pharmacodynamic systems can also be described by lifespan based indirect response (LIDR) models. The purpose of this report is to investigate conditions under which the transit compartments models can be considered a special case of LIDR models. An integral representation of a solution to TC model has been used to determine the lifespan distribution for cell population described by this model. The distribution served as a basis for definition of new LIDRE (lifespan based indirect response with an effect on the lifespan distribution) models. Time courses of responses described by both types of models were simulated for a monoexponential pharmacokinetic function. The limit response was calculated as the number of transit compartments approached infinity. The difference between the limit response and TC responses were evaluated by computer simulations using MATLAB 7.7. TC models are a special case of LIDR models with the lifespan distribution described by the gamma function. If drug affects only the production of cells, then the cell lifespan distribution is time invariant. In this case an increase in the number of compartments results in a basic LIDR model with a point lifespan distribution. When the drug inhibits or stimulates cell aging, the cell lifespan distribution becomes time dependent revealing a new mechanism for drug effect on the gamma probability density function. The TC model with a large number of transit compartments converges to an LIDRE model. The limit LIDR models are approximated by the TC models when the number of compartments is at least 5. A moderate improvement in the approximation is observed if this number exceeds 20. The lifespan distribution for a cell population described by a TC model is described by the gamma probability density function. A drug affects this distribution only if it stimulates or inhibits the rate of cell maturation. If the number of transit compartments increases, then the TC model converges to a new type of LIDR model.

Assessment of Drug Interactions Relevant to Pharmacodynamic Indirect Response Models

The assessment of drug interactions for a simple turnover system when the basic pharmacodynamic response is governed by indirect mechanisms was explored. This report describes a diverse array of possible in vivo pharmacodynamic effects from a combination of two drugs acting by similar or different indirect mechanisms. Various conditions of pharmacodynamic drug combinations were explored mathematically and by simulation: (a) interactions of two drugs acting simultaneously either on the production (kin) or on the dissipation (kout) processes controlling the in vivo response by competitive (four cases) or non-competitive interaction (six cases); and (b) combinations of two drugs acting on separate kin and kout processes simultaneously (four cases). A range of different combinations of drug doses was used. Plasma concentration time profiles were generated according to monoexponential disposition. Pharmacodynamic response profiles were simulated using the above conditions and characterized by descriptors such as Area Between Effect (and Baseline) Curve (ABEC) values. The interaction of agents by competitive mechanisms produced net responses that were additive in nature. Response profiles for non-competitive interactions on the same process were both antagonistic (for two drugs with effects that oppose each other) and synergistic (for two drugs that produce the same response). On the other hand, response signatures for agents acting non-competitively on the production and dissipation factors by opposing mechanisms (e.g. inhibiting kin plus stimulating kout) showed dramatic changes in net effects and produced apparent drug synergy. Net indirect response profiles for joint use of two or more drugs measured by ABEC values may look “additive”, “antagonistic”, or “synergistic” depending on doses, their intrinsic potencies, the nature of interaction (competitive or non-competitive) as well as their mechanisms of action. These models may help explain changes in pharmacologic responses to two agents in a more rational and mechanistic fashion than older empirical methods.