A distributed delay approach for modeling delayed outcomes in pharmacokinetics and pharmacodynamics studies

Pharmacokinetic-Pharmacodynamic Analysis of pH-Responsive Doxorubicin-Releasing Micelles with Anticancer Activity

This study aimed to investigate the effect of in vivo pH-responsive doxorubicin (DOX) release and the targetability of pilot molecules in folic acid (FA)-modified micelles using a pharmacokinetic-pharmacodynamic (PK-PD) model. The time profiles of intratumoral DOX concentrations in Walker256 tumor-bearing rats were monitored using a microdialysis probe, followed by compartmental analysis, to evaluate intratumoral tissue pharmacokinetics. Maximal DOX was released from micelles 350 min after the administration of pH-responsive DOX-releasing micelles. However, FA modification of the micelles shortened the time to peak drug concentration to 150 min. Additionally, FA modification resulted in a 27-fold increase in the tumor inflow rate constant. Walker256 tumor-bearing rats were subsequently treated with DOX, pH-responsive DOX-releasing micelles, and pH-responsive DOX-releasing FA-modified micelles to monitor the tumor growth-time profiles. An intratumoral threshold concentration of DOX (55-64 ng/g tumor) was introduced into the drug efficacy compartment to construct a PD model, followed by PK-PD analysis of the tumor growth-time profiles. Similar results of threshold concentration and drug potency of DOX were obtained across all three formulations. Cell proliferation was delayed as the drug delivery ability of DOX was improved. The PK model, which was developed using the microdialysis method, revealed the intratumoral pH-responsive DOX distribution profiles. This facilitated the estimation of intratumoral PK parameters. The PD model with threshold concentrations contributed to the estimation of PD parameters in the three formulations, with consistent mechanisms observed. We believe that our PK-PD model can objectively assess the contributions of pH-responsive release ability and pilot molecule targetability to pharmacological effects.

Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

Pharmacodynamic Age Structured Population Model For Cell Trafficking

OBJECTIVES

Cell trafficking encompasses movement of the immune system cells (e.g., granulocytes, lymphocytes) between the blood and the extravascular tissues (e.g., lymph nodes). Corticosteroids are known to suppress cell trafficking. The age-structured cell population models introduce the transit time as a structure that allows one to quantify the distribution of times the immune cells spend in the blood and the extravascular tissues. The objective of this work is to develop an age-structured cell population model describing drug effects on cell trafficking and to implement the model in pharmacometric software to enable parameter estimation and simulations.

METHODS

We adopted the well-known McKendrick age-structured population model to describe the age distributions in two cell populations: blood cells and cells in the extravascular space. The hazard of cell recirculation from the extravascular tissues was age dependent and described by the Weibull function with the shape ν and scale β parameters. The drug effect on cell trafficking was modeled as the product of the Emax function of the drug plasma concentration and the Weibull hazard. The model was implemented in NONMEM 7.5.1. The model was applied to the basophil data in 34 healthy subjects who received a single intramuscular or oral dose of 6 mg dexamethasone (DEX). A recently published pharmacokinetic model was applied to describe DEX plasma concentration. Typical values of parameter estimates were further used to simulate the DEX effect of the basophil mean transit time in the extravascular tissues.

RESULTS

Simulations of basophil time courses for varying ν demonstrated that the rebound in the blood count data following drug administration is only possible for ν >1. The estimates of model parameters were ν = 3.02, β = 0.00863 1/h, and IC50 = 7.47 ng/mL. The calculated baseline mean transit times of basophils in the blood 7.2 h and extravascular tissues 104.9 h agree with the values reported in the literature.

CONCLUSIONS

We introduced an age-structured population model to describe cell trafficking between the blood and extravascular tissues. The model was adopted to account for the inhibitory drug effect on the cell recirculation. We showed that the age structure is essential to explain the rebound observed in the blood count response to a single dose drug administration. The model was validated using the basophil responses to DEX treatment in healthy subjects.

Numerical methods and hypoexponential approximations for gamma distributed delay differential equations

Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.

R-praziquantel integrated population pharmacokinetics in preschool- and school-aged African children infected with Schistosoma mansoni and S. haematobium and Lao adults infected with Opisthorchis viverrini

Racemic praziquantel (PZQ) is the standard treatment for schistosomiasis and liver fluke infections (opisthorchiasis and clonorchiasis). The development of an optimal pediatric formulation and dose selection would benefit from a population pharmacokinetic (popPK) model. A popPK model was developed for R-PZQ, the active enantiomer of PZQ, in 664 subjects, 493 African children (2-15 years) infected with Schistosoma mansoni and S. haematobium, and 171 Lao adults (15-78 years) infected with Opisthorchis viverrini. Racemate tablets were administered as single doses of 20, 40 and 60 mg/kg in children and 30, 40 and 50 mg/kg in 129 adults, and as 3 × 25 mg/kg apart in 42 adults. Samples collected by the dried-blood-spot technique were assayed by LC-MS/MS. A two-compartment disposition model, with allometric scaling and dual first-order and transit absorption, was developed using Phoenix™ software. Inversely parallel functions of age described the apparent oral bioavailability (BA) and clearance maturation in children and ageing in adults. BA decreased slightly in children with dose increase, and by 35% in adults with multiple dosing. Crushing tablets for preschool-aged children increased the first-order absorption rate by 64%. The mean transit absorption time was 70% higher in children. A popPK model for R-PZQ integrated African children over 2 years of age with schistosomiasis and Lao adults with opisthorchiasis, and should be useful to support dose optimization in children. In vitro hepatic and intestinal metabolism data would help refining and validating the model in younger children as well as in target ethnic pediatric and adult groups.

Delay differential equations based models in NONMEM

Delay differential equations (DDEs) are commonly used in pharmacometric models to describe delays present in pharmacokinetic and pharmacodynamic data analysis. Several DDE solvers have been implemented in NONMEM 7.5 for the first time. Two of them are based on algorithms already applied elsewhere, while others are extensions of existing ordinary differential equations (ODEs) solvers. The purpose of this tutorial is to introduce basic concepts underlying DDE based models and to show how they can be developed using NONMEM. The examples include previously published DDE models such as logistic growth, tumor growth inhibition, indirect response with precursor pool, rheumatoid arthritis, and erythropoiesis-stimulating agents. We evaluated the accuracy of NONMEM DDE solvers, their ability to handle stiff problems, and their performance in parameter estimation using both first-order conditional estimation (FOCE) and the expectation-maximization (EM) method. NONMEM control streams and excerpts from datasets are provided for all discussed examples. All DDE solvers provide accurate and precise solutions with the number of significant digits controlled by the error tolerance parameters. For estimation of population parameters, the EM method is more stable than FOCE regardless of the DDE solver.

A mathematical model of viral oncology as an immuno-oncology instigator

We develop and analyse a mathematical model of tumour-immune interaction that explicitly incorporates heterogeneity in tumour cell cycle duration by using a distributed delay differential equation. We derive a necessary and sufficient condition for local stability of the cancer-free equilibrium in which the amount of tumour-immune interaction completely characterizes disease progression. Consistent with the immunoediting hypothesis, we show that decreasing tumour-immune interaction leads to tumour expansion. Finally, by simulating the mathematical model, we show that the strength of tumour-immune interaction determines the long-term success or failure of viral therapy.

Pharmacokinetic Model Analysis of Supralingual, Oral and Intravenous Deliveries of Mycophenolic Acid

Mycophenolic acid (MPA) is commonly used for organ rejection prophylaxis via oral administration in the clinic. Recent studies have shown that MPA also has anticancer activities. To explore new therapeutic options for oral precancerous/cancerous lesions, MPA was designed to release topically on the dorsal tongue surface via a mucoadhesive patch. The objective of this study was to establish the pharmacokinetic (PK) and tongue tissue distribution of mucoadhesive MPA patch formulation after supralingual administration in rats and also compare the PK differences between oral, intravenous, and supralingual administration of MPA. Blood samples were collected from Sprague Dawley rats before and after a single intravenous bolus injection, a single oral dose, or a mucoadhesive patch administration on the dorsal tongue surface for 4 h, all with a dose of 0.5 mg/kg of MPA. Plots of MPA plasma concentration versus time were obtained. As multiple peaks were found in all three curves, the enterohepatic recycling (EHR) model in the Phoenix software was adapted to describe their PK parameters with an individual PK analysis method. The mean half-lives of intravenous and oral administrations were 10.5 h and 7.4 h, respectively. The estimated bioavailability after oral and supralingual administration was 72.4% and 7.6%, respectively. There was a 0.5 h lag-time presented after supralingual administration. The results suggest that the systemic plasma MPA concentrations were much lower in rats receiving supralingual administration compared to those receiving doses from the other two routes, and the amount of MPA accumulated in the tongue after patch application showed a sustained drug release pattern. Studies on the dynamic of drug retention in the tongue after supralingual administration showed that ~3.8% of the dose was accumulated inside of tongue right after the patch removal, ~0.11% of the dose remained after 20 h, and ~20.6% of MPA was not released from the patches 4 h after application. The data demonstrate that supralingual application of an MPA patch can deliver a high amount of drug at the site of administration with little systemic circulation exposure, hence lowering the potential gastrointestinal side effects associated with oral administration. Thus, supralingual administration is a potential alternative route for treating oral lesions.

Population Pharmacokinetic Analysis of Yimitasvir in Chinese Healthy Volunteers and Patients With Chronic Hepatitis C Virus Infection

Yimitasvir is a novel, oral hepatitis C virus (HCV) non-structural protein 5A inhibitor for the treatment of chronic HCV genotype 1 infection. The objective of this analysis was to develop a population pharmacokinetic model of yimitasvir in Chinese healthy volunteers and HCV infection patients. The model was performed using data from 219 subjects across six studies. Nonlinear mixed effects models were developed using Phoenix NLME software. The covariates were evaluated using a stepwise forward inclusion (p < 0.01) and then a backward exclusion procedure (p < 0.001). A two-compartment model with sequential zero-first order absorption and first-order elimination reasonably described yimitasvir pharmacokinetics (PK). The apparent oral clearance and central volume of distribution were 13.8 l·h⁻¹ and 188 l, respectively. The bioavailability (F) of yimitasvir decreased 12.9% for each 100 mg dose increase. Food was found to affect absorption rate (Ka) and F. High-fat meal decreased Ka and F by 90.9% and 38.5%, respectively. Gender and alanine aminotransferase were identified as significant covariates on apparent oral clearance. Female subjects had lower clearance than male subjects. Zero-order absorption duration was longer in healthy volunteers (2.17 h) than that in patients (1.43 h). The population pharmacokinetic model described yimitasvir PK profile well. Food decreased Ka and F significantly, so it was recommended to take yimitasvir at least 2 h before or after a meal. Other significant covariates were not clinically important.

Exact solutions and equi-dosing regimen regions for multi-dose pharmacokinetics models with transit compartments

Compartmental models which yield linear ordinary differential equations (ODEs) provide common tools for pharmacokinetics (PK) analysis, with exact solutions for drug levels or concentrations readily obtainable for low-dimensional compartment models. Exact solutions enable valuable insights and further analysis of these systems. Transit compartment models are a popular semi-mechanistic approach for generalising simple PK models to allow for delayed kinetics, but computing exact solutions for multi-dosing inputs to transit compartment systems leading to different final compartments is nontrivial. Here, we find exact solutions for drug levels as functions of time throughout a linear transit compartment cascade followed by an absorption compartment and a central blood compartment, for the general case of n transit compartments and M equi-bolus doses to the first compartment. We further show the utility of exact solutions to PK ODE models in finding constraints on equi-dosing regimen parameters imposed by a prescribed therapeutic range. This leads to the construction of equi-dosing regimen regions (EDRRs), providing new, novel visualisations which summarise the safe and effective dosing parameter space. EDRRs are computed for classical and transit compartment models with two- and three-dimensional parameter spaces, and are proposed as useful graphical tools for informing drug dosing regimen design.