A Systems Pharmacology Model of Erythropoiesis in Mice Induced by Small Molecule Inhibitor of Prolyl Hydroxylase Enzymes.

Mammalian erythropoiesis is a conserved process tightly controlled by the hypoxia-inducible factor (HIF1) pathway. In this study, a small molecule inhibitor (PHI-1) of prolyl-hydroxylase-2 (PHD2) enzyme involved in regulating HIF1α levels was orally administered to male BALB/c mice at 10 and 30 mg/kg. A systems pharmacology model was developed based on the measured PHI-1 plasma exposures, kidney HIF1α, kidney erythropoietin (EPO) mRNA, plasma EPO, reticulocyte counts, red blood cells, and hemoglobin levels. The model fit resulted in the estimation of drug potency (IC50: 1.7µM), and systems parameters such as EPO mRNA turnover (kdeg_EPOmRNA: 0.43 hr(-1)) and mean lifespan of reticulocytes (Tr : 81 hours). The model correctly described the observed 30-40-fold increase in kidney HIF1α protein, ∼1,000 fold increase in EPO mRNA and 2-3-fold increase in the reticulocytes at 30 mg/kg. This study presents the first parsimonious systems model of erythropoiesis to quantitatively describe the in vivo effects of PHD2 inhibition.

vRNA structured population model for Hepatitis C Virus dynamics.

Improvements in the understanding of the Hepatitis C Virus (HCV) life-cycle have led to the identification of targets and the development of drugs affecting the intracellular reproduction of the virus. These advancements have presented new modeling challenges as the classic models have focused on describing the macroscopic viral kinetics only. Our primary objective is to apply the existing theory of Physiologically Structured Population (PSP) modeling to describe dynamics of viral RNA (vRNA) in infected hepatocytes of patients receiving treatment with Direct-acting Antiviral Agents (DAA). Using vRNA as a physiological structure this work expands on previous structured population models allowing exploration of micro- and macroscopic implications of such treatments. The PSP model provides a description of vRNA distribution in the infected cells at steady state and its time evolution following treatment. The long term behavior of the model predicts viral load time courses in plasma and permits to quantify conditions for the virus eradication. Finally, we demonstrate that PSP models can account for additional structures, which are essential for the viral replication process with potentially far reaching implications in our understanding of HCV infections and treatment options.

Quantitative assessment of minimal effective concentration of erythropoiesis-stimulating agents.

Minimal effective concentration (MEC) was proposed to explain why subcutaneous (SC) administration of erythropoietin (EPO) induces a higher hemoglobin (HGB) increase than intravenous (IV) administration. It has been further used to explain the paradox that erythropoiesis-stimulating agent (ESA) with lower receptor binding affinity may have higher in vivo activity. We have developed a pharmacokinetic and pharmacodynamic (PK/PD) model with incorporation of the operational model of agonism to characterize the data from two clinical trials. By using model-based simulations, we demonstrate that SC route is more efficacious than IV route and explain the paradoxical behavior of ESAs. We determined that MEC can be quantified by C50, which represents the concentration of an ESA producing its half-maximal effect of stimulating the proliferation of erythroid precursor cells. The model used may allow joint PK/PD modeling of data from different ESAs, and provide a platform for dosing regimen optimizations and future clinical study designs.CPT: Pharmacometrics & Systems Pharmacology (2013) 2, e62; doi:10.1038/psp.2013.39; published online 7 August 2013.

Evaluation of epoetin alpha (rHuEPO) and darbepoetin alpha (DARB) on human burst-colony formation (BFU-E) in culture.

The erythropoietic effect of recombinant human erythropoietin, epoetin alpha (rHuEPO), in promoting the growth of erythroid burst-forming units (BFU-E) was compared with darbepoetin alpha (DARB), a rHuEPO analogue obtained by site-directed mutagenesis. Human bone marrow cells derived from healthy donors were cultured with different concentrations of rHuEPO or DARB for 12 - 21 days and BFU-E were counted using an inverted microscope. The EC50 of rHuEPO was about 10-fold lower than DARB and the size of the colonies was significantly larger in rHuEPO-containing cultures using comparable concentrations. The maximum number of colonies obtained in some rHuEPO-containing cultures was also higher than for DARB. The number of colonies in DARB-containing cultures was increased, in part, by the addition of low concentrations of rHuEPO, but not by DARB, even at high concentrations. We conclude that DARB is not as effective as rHuEPO in supporting the in vitro growth of human BFU-E.

Pharmacodynamic modeling of lansoprazole using an indirect irreversible response model.

A mechanism-based pharmacokinetic/pharmacodynamic model was used to assess lansoprazole effects on gastric pH. The irreversible inactivation of the H+/K+-ATPase enzyme by lansoprazole controls the secretion rate of H+ ions and gastric pH values. The basal circadian rhythm of gastric acid production was taken into account as well as the effects of food intake. A model was applied to multiple-dose data from a crossover study of four dosage regimens of lansoprazole in two groups of normal male subjects. Model parameters were estimated by nonlinear regression and were compared to historical values reported in the literature. The predicted mean gastric ion concentration was 23.2 mM (pH 1.6) with the peak time at 12.6 hours (8:30 p.m.), and the half-time for H+ removal from the stomach averaged 1.7 hours. The estimated half-life of gastric food removal was 0.8 hours. The rate constant for normal H+/K+-ATPase degradation was 0.045 h(-1). The pharmacodynamic parameter describing lansoprazole action on gastric acid secretion was the second-order enzyme inactivation constant, which averaged 0.16 microg(-1) x L x h(-1). The parameters obtained for both the baseline and drug treatment data were consistent with the literature and physiologically relevant with the exception of effective food volume, which was large presumably due to buffer effects. The model successfully incorporated the physiological regulation of gastric acid production, the effects of food on gastric acid, and the effects of multiple-dosing regimens of lansoprazole on gastric acid production to give reasonable profiles of gastric pH.

Indirect pharmacodynamic models for responses with multicompartmental distribution or polyexponential disposition.

Basic indirect response models where drug alters the production (kin) of the response variable (R) based on the Hill function previously assumed one-compartment distribution of the response variable and simple first-order loss (kout) 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.

An application of a weak solution of the cable equation to the Rall model of a nerve cell.

The Rall model of a nerve cell with the lumped soma and natural termination boundary conditions at the nerve ends is considered as a particular case of the cable equation with oblique linear boundary conditions. A variational approach is taken to derive a weak solution of the problem. This extends classical results and removes a major obstacle of nonorthogonal boundary conditions in standard analysis of the Rall model. Additionally, new estimates of nerve cell time constants are obtained. The weak solution is applied to describe the nerve membrane potential following current injection at the soma, synaptic input, and instant unit current charge.

Algorithm for application of Fourier analysis for biorhythmic baselines of pharmacodynamic indirect response models.

The change of an indirect pharmacological response R(t) can be described by a periodic time-dependent production rate kin(t) and a first-order loss constant kout. If kin(t) follows some biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. A new approach for describing the input function in indirect response models with biorhythmic baselines of physiologic substances is introduced. The present approach uses the baseline (placebo) response Rb(t) to recover the equation for kin(t). Fourier analysis provides an approximate equation for Rb(t) that consists of terms (usually two or three) of the Fourier series (harmonics) that contribute most to the overall sum. The model differential equation is solved backward for kin(t), yielding the equation involving Rb(t). A computer program was developed to perform the square L2-norm approximation technique. Fourier analysis was also performed based on nonlinear regression. Cortisol suppression after inhalation of fluticasone propionate (FP) was modeled based on the inhibition of the secretion rate kin(t) using ADAPT II. The pharmacodynamic parameters kout and IC50 were estimated from the model equation with kin(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of kin(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method.

Asymptotics of the total net direct pharmacological effect for large drug doses.

The direct pharmacological effect E is described by the Emax model relating E to the drug plasma concentration Cp. The area under the effect vs. time curve (AUCE) is used as the measurement of the total net pharmacological effect. The drug plasma concentrations are solutions of compartmental systems of ordinary differential equations with the input terminated after a finite time and controlled in a proportional manner by a single dose-like parameter. The asymptotics of the time derivative of Cp for large doses are derived and used as conditions which have to be satisfied by functions for which the asymptotics of the integral defining AUCE are derived. The AUCE is proportional to the time TC > EC50 for which the drug concentration stays above the threshold level EC50. The threshold EC50 denotes the drug plasma concentration which elicits 50% of the maximum effect. The parameter TC > EC50 is proportional to the logarithm of drug dose for large doses and its asymptotics is calculated up to the order o(1) as dose increases to infinity. The results are applied to basic pharmacokinetic systems.

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 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.

Analysis of a model of membrane potential for a skin receptor.

A linear model for the membrane potential of a nerve ending associated with a sense-of-touch skin receptor is formulated and analyzed. Because the nerve has cytoplasmic extensions and is stimulated through a membrane strain mechanism, the model is a generalized cable equation with linear oblique boundary conditions (i.e. boundary conditions involving both time and space derivatives). The model is stimulated through a certain conductance parameter. We establish solutions for the problem and we consider a particular physiological question, both analytically and computationally, of determining conditions on parameters and stimulus which give threshold level potential values at a particular boundary point.

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 [equation: see text] 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 SC50 values. The proposed cell production/loss models can be readily used to analyze the pharmacodynamics of agents which alter cell production yielding realistic physiological parameters.

Integrated functions for four basic models of indirect pharmacodynamic response.

The integrated solutions (ABEC, area between baseline and effect curve) of four basic models of indirect pharmacodynamic responses are developed. These models assume that drug can inhibit or stimulate the production or loss of the response variable. For two models (I and III) with monoexponential drug disposition, explicit formulas for the ABEC were obtained, where ABEC is a function of ln (1 + (D/V)/IC50) or ln (1 + (D/V)/SC50) where D = dose, V = volume, and IC50 or SC50 = 50% effective concentration. Two other models (II and IV) were treated asymptotically with respect to small and large doses. Approximate formulas [e.g., ABEC = constant(1) x ln (1 + (D/V)/IC50) + constant (2)] were derived and the asymptotic behavior of the ABEC was established. In addition, simulations were performed to assess the effects of drug absorption rates and polyexponential disposition on ABEC values. These models show how pharmacokinetic and pharmacodynamic factors jointly determine the net response to a single dose of drug.

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.

Characterization of pharmacodynamic recession slopes for direct and indirect response models.

Direct pharmacologic effects are known to recede over time with largely linear slopes (Levy's k.m product, J. Pharm. Sci. 53:342, 1964) and indirect responses have similar behavior. Pharmacodynamic slope properties were examined mathematically for the Hill function with monoexponential drug disposition and simulations were carried out for other pharmacokinetic functions. Both types of pharmacodynamic profiles exhibit a single terminal inflection point (fp) when drug concentrations exceed the EC50 (that concentration causing one-half maximum effect, Emax). For direct effects it was found that Cfp (the drug concentration at fp) = EC50, the determinants of inflection time were identified, and Slopefp = -lambda z gamma Emax/4 where lambda z is the terminal disposition slope and gamma is the Hill coefficient. These characteristics were explored for the four basic indirect response models which also exhibit recession profiles with slight sigmoidity and a single terminal inflection point at higher doses. The drug concentration at inflection Cfp is < or = IC50 or SC50 (drug concentrations causing half-maximal inhibition or stimulation), while the inflection response (Rfp) attains constant values at larger doses. Indirect Response Models I, III, and IV have nearly linear return slopes for a wide range of doses which are governed by the disposition slope lambda z of the drug, loss constant kout of the response, maximum inhibition (Imax) or stimulation (Smax) factors, and a unique fractional constant (0 < G < or = 1). Model II exhibits more complex behavior with recession slopes which are less likely to be parallel for various doses. Most indirect responses are expected to show nearly linear recession slopes which are parallel for moderate to large doses and mainly governed by an identical combination of pharmacokinetic (lambda z), system (kout), and dynamic capacity factors (Imax or Smax).

Note: caution in use of empirical equations for pharmacodynamic indirect response models.

An empirical equation was recently proposed and used to characterize inhibitory drug effects on production of an endogenous substance. The limitations of this empirical equation are described utilizing mathematical equations and simulations based on the exact differential equation for Indirect Response Model I. The latter is preferable for fitting data using indirect response models.

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.