Lifespan based indirect response models

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.

Modelling haemoglobin incremental loss on chronic red blood cell transfusions

BACKGROUND AND OBJECTIVES

Understanding the impact of red blood cell (RBC) lifespan, initial RBC removal, and transfusion intervals on patient haemoglobin (Hb) levels and total iron exposure is not accessible for chronic transfusion scenarios. This article introduces the first model to help clinicians optimize chronic transfusion intervals to minimize transfusion frequency.

MATERIALS AND METHODS

Hb levels and iron exposure from multiple transfusions were calculated from Weibull residual lifespan distributions, the fraction effete RBC removed within 24-h (X_e ) and the nominal Hb increment. Two-unit transfusions of RBCs initiated at patient [Hb] = 7 g/dl were modelled for different RBC lifespans and transfusion intervals from 18 to 90 days, and X_e from 0.1 to 0.5.

RESULTS

Increased X_e requires shorter transfusion intervals to achieve steady-state [Hb] of 9 g/dl as follows: 30 days between transfusions at X_e  = 0.5, 36 days at X_e  = 0.4, 42 days at X_e  = 0.3, 48 days at X_e  = 0.2 and 54 days at X_e  = 0.1. The same transfusion interval/X_e pairs result in a steady-state [Hb] = 8 g/dl when the RBC lifespan was halved. By reducing transfused RBC increment loss from 30% to 10%, annual transfusions were decreased by 22% with iron addition decreased by 24%. Acute dosing of iron occurs at the higher values of X_e on the day after a transfusion event.

CONCLUSION

Systematic trends in fractional Hb incremental loss X_e have been modelled and have a significant and calculatable impact on transfusion intervals and associated introduction of iron.

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.

Unraveling Disease Pathophysiology with Mathematical Modeling

Modeling has enabled fundamental advances in our understanding of the mechanisms of health and disease for centuries, since at least the time of William Harvey almost 500 years ago. Recent technological advances in molecular methods, computation, and imaging generate optimism that mathematical modeling will enable the biomedical research community to accelerate its efforts in unraveling the molecular, cellular, tissue-, and organ-level processes that maintain health, predispose to disease, and determine response to treatment. In this review, we discuss some of the roles of mathematical modeling in the study of human physiology and pathophysiology and some challenges and opportunities in general and in two specific areas: in vivo modeling of pulmonary function and in vitro modeling of blood cell populations.

Delayed logistic indirect response models: realization of oscillating behavior

Indirect response (IDR) models are probably the most frequently applied tools relating the effect of a signal to a baseline response. A response modeled by such a classical IDR model will always return monotonously to its baseline after drug administration. We extend IDR models with a delay process, i.e. a retarded response state, that leads to oscillating response behavior. First, IDR models with a first-order production and second-order loss term based on the famous logistic equation are constructed. Second, a delay process similar to the delayed logistic equation is included. Relations of the classical IDR model with our extended IDR model concerning response and model parameters are revealed. Simulations of typical response profiles are presented and data fitting of a model for leptin and cholesterol dynamics after administration of methylprednisolone is performed. The influence of the delay parameter on the other model parameters is discussed.

Clinical Pharmacokinetics and Pharmacodynamics of Monoclonal Antibodies Approved to Treat Rheumatoid Arthritis

Monoclonal antibodies (mAbs) are increasingly used to treat rheumatoid arthritis (RA). At present, anti-tumor necrosis factor-α drugs (infliximab, adalimumab, certolizumab pegol, and golimumab), rituximab, and tocilizumab are approved for RA treatment. This review focuses on the pharmacokinetics and pharmacodynamics of mAbs approved in RA. Being large proteins, mAbs exhibit complex pharmacokinetic and pharmacodynamic properties. In particular, owing to the interactions of mAbs with their antigenic targets, the pharmacokinetics of mAbs depends on target turnover and exhibits non-specific (linear) and target-mediated (often nonlinear) clearances. Their volume of distribution is low (3-4 L) and their elimination half-life usually ranges from 2 to 3 weeks. The inter-individual pharmacokinetic variability of mAbs is usually large and is partly explained by differences in antigenic burden or by anti-drug antibodies, which accelerate mAb elimination. The inter-individual variability of clinical response is large and influenced by the pharmacokinetics. The analysis of mAbs concentration-effect relationship relies more and more often on pharmacokinetic-pharmacodynamic modeling; these models being suitable for dosing optimization. Even if adverse effects of mAbs used in RA are well known, the relationship between mAb concentration and adverse effects is poorly documented, especially for anti-tumor necrosis factor-α mAbs. Overall, RA patients treated with mAbs should benefit from individualized dosing strategies. Because of the complexity of their pharmacokinetics and mechanisms of action, the current dosing strategy of mAbs is not based on sound knowledge. New studies are needed to assess individual dosing regimen, adjusted notably to disease activity.

Models for the red blood cell lifespan

The lifespan of red blood cells (RBCs) plays an important role in the study and interpretation of various clinical conditions. Yet, confusion about the meanings of fundamental terms related to cell survival and their quantification still exists in the literature. To address these issues, we started from a compartmental model of RBC populations based on an arbitrary full lifespan distribution, carefully defined the residual lifespan, current age, and excess lifespan of the RBC population, and then derived the distributions of these parameters. For a set of residual survival data from biotin-labeled RBCs, we fit models based on Weibull, gamma, and lognormal distributions, using nonlinear mixed effects modeling and parametric bootstrapping. From the estimated Weibull, gamma, and lognormal parameters we computed the respective population mean full lifespans (95 % confidence interval): 115.60 (109.17-121.66), 116.71 (110.81-122.51), and 116.79 (111.23-122.75) days together with the standard deviations of the full lifespans: 24.77 (20.82-28.81), 24.30 (20.53-28.33), and 24.19 (20.43-27.73). We then estimated the 95th percentiles of the lifespan distributions (a surrogate for the maximum lifespan): 153.95 (150.02-158.36), 159.51 (155.09-164.00), and 160.40 (156.00-165.58) days, the mean current ages (or the mean residual lifespans): 60.45 (58.18-62.85), 60.82 (58.77-63.33), and 57.26 (54.33-60.61) days, and the residual half-lives: 57.97 (54.96-60.90), 58.36 (55.45-61.26), and 58.40 (55.62-61.37) days, for the Weibull, gamma, and lognormal models respectively. Corresponding estimates were obtained for the individual subjects. The three models provide equally excellent goodness-of-fit, reliable estimation, and physiologically plausible values of the directly interpretable RBC survival parameters.

Toxicodynetics: A new discipline in clinical toxicology

OBJECTIVES

Regarding the different disciplines that encompass the pharmacology and the toxicology, none is specifically dedicated to the description and analysis of the time-course of relevant toxic effects both in experimental and clinical studies. The lack of a discipline devoted to this major field in toxicology results in misconception and even in errors by clinicians.

MATERIAL AND METHODS

Review of the basic different disciplines that encompass pharmacology toxicology and comparing with the description of the time-course of effects in conditions in which toxicological analysis was not performed or with limited analytical evidence.

RESULTS

Review of the literature clearly shows how misleading is the current extrapolation of toxicokinetic data to the description of the time-course of toxic effects.

CONCLUSION

A new discipline entitled toxicodynetics should be developed aiming at a more systematic description of the time-course of effects in acute human and experimental poisonings. Toxicodynetics might help emergency physicians in risk assessment when facing a poisoning and contribute to a better assessment of quality control of data collected by poison control centres. Toxicodynetics would also allow a quantitative approach to the clinical effects resulting from drug-drug interaction.

A Joint Model for the Kinetics of CTC Count and PSA Concentration During Treatment in Metastatic Castration-Resistant Prostate Cancer

Assessment of treatment efficacy in metastatic castration-resistant prostate cancer (mCRPC) is limited by frequent nonmeasurable bone metastases. The count of circulating tumor cells (CTCs) is a promising surrogate marker that may replace the widely used prostate-specific antigen (PSA). The purpose of this study was to quantify the dynamic relationships between the longitudinal kinetics of these markers during treatment in patients with mCRPC. Data from 223 patients with mCRPC treated by chemotherapy and/or hormonotherapy were analyzed for up to 6 months of treatment. A semimechanistic model was built, combining the following several pharmacometric advanced features: (1) Kinetic-Pharmacodynamic (K-PD) compartments for treatments (chemotherapy and hormonotherapy); (2) a latent variable linking both marker kinetics; (3) modeling of CTC kinetics with a cell lifespan model; and (4) a negative binomial distribution for the CTC random sampling. Linked with survival, this model would potentially be useful for predicting treatment efficacy during drug development or for therapeutic adjustment in treated patients.

Lifespan based pharmacokinetic-pharmacodynamic model of tumor growth inhibition by anticancer therapeutics

Accurate prediction of tumor growth is critical in modeling the effects of anti-tumor agents. Popular models of tumor growth inhibition (TGI) generally offer empirical description of tumor growth. We propose a lifespan-based tumor growth inhibition (LS TGI) model that describes tumor growth in a xenograft mouse model, on the basis of cellular lifespan T. At the end of the lifespan, cells divide, and to account for tumor burden on growth, we introduce a cell division efficiency function that is negatively affected by tumor size. The LS TGI model capability to describe dynamic growth characteristics is similar to many empirical TGI models. Our model describes anti-cancer drug effect as a dose-dependent shift of proliferating tumor cells into a non-proliferating population that die after an altered lifespan T_A. Sensitivity analysis indicated that all model parameters are identifiable. The model was validated through case studies of xenograft mouse tumor growth. Data from paclitaxel mediated tumor inhibition was well described by the LS TGI model, and model parameters were estimated with high precision. A study involving a protein casein kinase 2 inhibitor, AZ968, contained tumor growth data that only exhibited linear growth kinetics. The LS TGI model accurately described the linear growth data and estimated the potency of AZ968 that was very similar to the estimate from an established TGI model. In the case study of AZD1208, a pan-Pim inhibitor, the doubling time was not estimable from the control data. By fixing the parameter to the reported in vitro value of the tumor cell doubling time, the model was still able to fit the data well and estimated the remaining parameters with high precision. We have developed a mechanistic model that describes tumor growth based on cell division and has the flexibility to describe tumor data with diverse growth kinetics.

Romiplostim dose-response in patients with myelodysplastic syndromes

AIM

To characterize the romiplostim dose-response in subjects with low or intermediate-1 risk myelodysplastic syndromes (MDS) receiving subcutaneous romiplostim.

METHODS

Data from 44 MDS subjects receiving subcutaneous romiplostim (dose range 300-1500 μg week(-1) ) were used to develop a pharmacodynamic model consisting of a romiplostim-sensitive progenitor cell compartment linked to the peripheral blood compartment through four transit compartments representing the maturation in the bone marrow from megakaryocytes to platelets. A kinetics of drug effect model was used to quantify the stimulatory effect of romiplostim on the proliferation of sensitive progenitor cells and pharmacodynamics-mediated disposition was modelled by assuming the kinetics of drug effect constant (kDE ) to be proportional to the change in platelet count relative to baseline.

RESULTS

The estimated values (between subject variability) for baseline platelet count, mean transit time, and kDE were 24 × 10(9) l(-1) (47%), 9.6 days (44%) and 0.28 days(-1) , respectively. MDS subjects had a shorter platelet lifespan (42 h) than healthy subjects (257 h). Romiplostim effect was described for responders (78%) and non-responders (22%). The average weekly stimulatory effect of romiplostim on the production rate of sensitive progenitor cells at baseline was 269% per 100 μg week(-1) for responders. Body weight, age, gender and race were not statistically related to romiplostim pharmacodynamic parameters. Visual predictive checks confirmed the model adequacy.

CONCLUSION

The time course of platelet counts in MDS subjects receiving subcutaneous administration of escalating doses of romiplostim was characterized and showed a linear dose-response for romiplostim responders to increase the platelet counts.

Pharmacokinetic and pharmacodynamic modeling of romiplostim in animals

PURPOSE

Romiplostim is a novel thrombopoiesis-stimulating peptibody that targets the thrombopoietin c-Mpl receptor, resulting in increased platelet production. The pharmacodynamic-mediated disposition (PDMDD) and its stimulatory effect on platelet production in Sprague-Dawley rats, rhesus monkeys, and cynomolgus monkeys following IV bolus and SC administration at various dose levels were determined.

METHODS

The pharmacokinetic (PK) profile was described by a PDMDD model that accounts for romiplostim binding to the c-Mpl receptor. The PD model contained a series of aging compartments for precursor cells in bone marrow and platelets. The stimulatory function was described by an on-and-off function operating on the fractional receptor occupancy (RO). The threshold effect, RO(thr), and K(D) parameters were determinants of drug potency, whereas S(max) reflected drug efficacy.

RESULTS

The model implicated that receptor-mediated clearance was negligible. RO(thr) estimated occupancies were 0.288, 0.385, 0.771 for rats, rhesus, and cynomolgus monkeys, respectively. The analogous estimated values of K(D) were 4.05, 2320, and 429 ng/mL, implying that romiplostim was much more potent in rats, which was confirmed by a dose-response (ratio of peak platelet count to baseline) relationship.

CONCLUSIONS

The model adequately described romiplostim serum concentrations and platelet counts in rats, rhesus monkeys, and cynomolgus monkeys, and quantified linear clearance, PDMDD, and potency of romiplostim.