Pharmacodynamic analysis of time-variant cellular disposition: reticulocyte disposition changes in phlebotomized sheep

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.

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.

Distributed transit compartments for arbitrary lifespan distributions in aging populations

Transit compartment models (TCM) are often used to describe aging populations where every individual has its own lifespan. However, in the TCM approach these lifespans are gamma-distributed which is a serious limitation because often the Weibull or more complex distributions are realistic. Therefore, we extend the TCM concept to approximately describe any lifespan distribution and call this generalized concept distributed transit compartment models (DTCMs). The validity of DTCMs is obtained by convergence investigations. From the mechanistic perspective the transit rates are directly controlled by the lifespan distribution. Further, DTCMs could be used to approximate the convolution of a signal with a probability density function. As example a stimulatory effect of a drug in an aging population with a Weibull-distributed lifespan is presented where distribution and model parameters are estimated based on simulated data.

A semi-mechanistic red blood cell survival model provides some insight into red blood cell destruction mechanisms

Most mathematical models developed for the survival of haematological cell populations, in particular red blood cells (RBCs), follow the principle of parsimony. They focus on the predominant destruction mechanism of age-related cell death (senescence) and do not account for within subject variability in the RBC lifespan. However, assessment of the underlying physiological destruction mechanisms can be of interest in pathological conditions that affect RBC survival, for example sickle cell anaemia or anaemia of chronic kidney disease. We have previously proposed a semi-mechanistic RBC survival model which accounts for four different types of RBC destruction mechanisms. In this work, it is shown that the proposed model in combination with informative RBC survival data is able to provide a deeper insight into RBC destruction mechanisms. The proposed model was applied in a non-linear mixed effect modelling framework to biotin derived RBC survival data available from literature. Three mechanisms were estimable based on the available data of twelve subjects, including random destruction, senescence and destruction due to delayed failure. It was possible to identify three subjects with a decreased RBC survival in the study population. These three subjects all showed differences in the contribution of the estimated destruction mechanisms: an increased random destruction, versus an accelerated senescence, versus a combination of both.

Lifespan based indirect response models

In the field of hematology, several mechanism-based pharmacokinetic-pharmacodynamic models have been developed to understand the dynamics of several blood cell populations under different clinical conditions while accounting for the essential underlying principles of pharmacology, physiology and pathology. In general, a population of blood cells is basically controlled by two processes: the cell production and cell loss. The assumption that each cell exits the population when its lifespan expires implies that the cell loss rate is equal to the cell production rate delayed by the lifespan and justifies the use of delayed differential equations for compartmental modeling. This review is focused on lifespan models based on delayed differential equations and presents the structure and properties of the basic lifespan indirect response (LIDR) models for drugs affecting cell production or cell lifespan distribution. The LIDR models for drugs affecting the precursor cell production or decreasing the precursor cell population are also presented and their properties are discussed. The interpretation of transit compartment models as LIDR models is reviewed as the basis for introducing a new LIDR for drugs affecting the cell lifespan distribution. Finally, the applications and limitations of the LIDR models are discussed.

Erythropoietin as a novel brain and kidney protective agent

Erythropoietin is a 30.4 kDa glycoprotein produced by the kidney, which is mostly known for its physiological function in regulating red blood cell production in the bone marrow Accumulating evidence, however suggests that erythropoietin has additional organ protective effects, which may specifically be useful in protecting the brain and kidneys from injury. Experimental evidence suggests that these protective mechanisms are multi-factorial in nature and may include inhibition of apoptotic cell death, stimulation of cellular regeneration, inhibition of deleterious pathways and promotion of recovery. In this article we review the physiology of erythropoietin, assess previous work that supports the role of erythropoietin as a general tissue protective agent and explain the mechanisms by which it may achieve this tissue protective effect. We then focus on specific laboratory and clinical data that suggest that erythropoietin has a strong brain protective and kidney protective effect. In addition, we comment on the implications of these studies for clinicians at the bedside and for researchers designing controlled trials to further elucidate the true clinical utility of erythropoietin as a neuroprotective and nephroprotective agent. Finally, we describe EPO-TBI, a double-blinded multi-centre randomised controlled trial involving the authors that is being conducted to investigate the organ protective effects of erythropoietin on the brain, and also assesses its effect on the kidneys.

Receptor-based dosing optimization of erythropoietin in juvenile sheep after phlebotomy

The primary objective of this work was to determine the optimal time for administration of an erythropoietin (Epo) dose to maximize the erythropoietic effect using a simulation study based on a young sheep pharmacodynamic model. The dosing optimization was accomplished by extending a Hb production pharmacodynamic model, which evaluates the complex dynamic changes in the Epo receptor (EpoR) pool from the changes in Epo clearance. Fourteen healthy 2-month-old sheep were phlebotomized to Hb levels of 3 to 4 g/dl. Epo clearance was evaluated longitudinally in each animal by administering tracer doses of (125)I-recombinant human Epo multiple times during the experiment. Kinetic parameters were estimated by simultaneously fitting to Hb data and Epo clearance data. The phlebotomy caused a rapid temporary increase in the endogenous Epo plasma level. The Hb began to increase after the increased in the Epo level with a lag time of 1.13 ± 0.79 days. The average correlation coefficients for the fit of the model to the Hb and clearance data were 0.953 ± 0.018 and 0.876 ± 0.077, respectively. A simulation study was done in each sheep with fixed individual estimated model parameters to determine the optimal time to administer a 100 U/kg intravenous bolus Epo dose. The optimal dose administration time was 11.4 ± 6.2 days after phlebotomy. This study suggests that the Hb produced from Epo administration can be optimized by considering the dynamic changes in the EpoR pool.

Pharmacodynamic analysis of stress erythropoiesis: change in erythropoietin receptor pool size following double phlebotomies in sheep

A feedback receptor regulation model was incorporated into a pharmacodynamic model to describe the stimulation of hemoglobin (Hb) production by endogenous erythropoietin (EPO). The model considers the dynamic changes that take place in the EPO receptor (EPOR) pool under phlebotomy-induced anemia. Using a (125)I-rhEPO tracer the EPO clearance changes are evaluated longitudinally prior to and following phlebotomy-induced anemia indirectly to evaluate changes in the EPOR pool size, which has been shown to be linearly related to the clearance. The proposed model simultaneously captures the general behavior of temporal changes in Hb relative to EPO plasma clearance in five lambs (r = 0.95), while accounting for the confounding variables of phlebotomy and changes in the blood volume in the growing animals. The results indicate that under anemia the EPOR pool size is up-regulated by a factor of nearly two over baseline and that the lowest and highest EPOR pool sizes differ by a factor of approximately four. The kinetic model developed and the data-driven mechanism proposed serves as a starting point for developing an optimal EPO dosing algorithm for the treatment of neonatal anemia.

Erythropoietic response to endogenous erythropoietin in premature very low birth weight infants

Despite the common occurrence of anemia in very low birth weight (VLBW) infants, the erythropoiesis and Hb production rates and their relationship to plasma erythropoietin (EPO) concentrations remain unknown in these subjects. To determine these quantities, all blood removed by phlebotomy and administered by red blood cell (RBC) transfusion over the first 30 days of life was recorded in 14 ventilated VLBW infants born at 24 to 28 weeks of gestation. Discarded blood from frequent clinically ordered laboratory blood samples was used to construct plasma EPO, Hb, and RBC concentration-time profiles for each infant. A pharmacodynamic Hb mass balance model that accounted for the dynamic hematological conditions experienced by these infants was simultaneously fitted to the plasma EPO, Hb, and RBC concentrations from each individual subject, while accounting for subject growth. Based on the model estimates, an average of 4.69 g of Hb was produced over the first 30 days of life, compared with 5.97 g removed by phlebotomies and 12.3 g administered by transfusions. These high transfusion amounts were consistent with a relatively short RBC life span and rapidly expanding blood volume with infant growth. The estimated mean body weight-scaled Hb production rate dropped nearly 3-fold after birth to 0.144 g/day x (kg)(3/4). Although only estimated in a subset of the subjects, the mean plasma EPO EC(50) of 28.5 mU/ml and maximal Hb production rate (E(max)) indicated that a severalfold increase in Hb production rate could be achieved with only a modest increase in plasma EPO concentrations.

Flow cytometric analysis of reticulocyte maturation after erythropoietin administration in rats

We have proposed a technique for transforming reticulocyte (RET) flow cytometry count into the age distribution allowing for a quantitative characterization of RETs dynamics. This technique was evaluated in homeostatic and erythropoietically stimulated rats. We administered a single intravenous dose (4,050 IU/kg) of recombinant human erythropoietin to Wistar rats, then stained their RETs with thiazole orange, and measured the distribution of the fluorescent signal using flow cytometry. The RET age distribution was determined assuming an exponential decline of RET RNA. The RET distribution over time was characterized by median age, production rate of mature red blood cell in the blood, bone marrow and blood maturation times, and RET loss from the blood. Erythropoietin was found to cause complex changes not only in RET count but also in age distribution. Generally, an increase in RET count was accompanied by the presence of young RETs and a decrease with the presence of old RETs. The rate at which RETs were removed from the blood was affected considerably by erythropoietin administration, and their death was attributed to it. Flow cytometric analysis of RET age distribution expands our understanding of erythropoiesis and augments the information we can obtain from RET measurements.

Quantitative analysis of mechanisms that govern red blood cell age structure and dynamics during anaemia

Mathematical modelling has proven an important tool in elucidating and quantifying mechanisms that govern the age structure and population dynamics of red blood cells (RBCs). Here we synthesise ideas from previous experimental data and the mathematical modelling literature with new data in order to test hypotheses and generate new predictions about these mechanisms. The result is a set of competing hypotheses about three intrinsic mechanisms: the feedback from circulating RBC concentration to production rate of immature RBCs (reticulocytes) in bone marrow, the release of reticulocytes from bone marrow into the circulation, and their subsequent ageing and clearance. In addition we examine two mechanisms specific to our experimental system: the effect of phenylhydrazine (PHZ) and blood sampling on RBC dynamics. We performed a set of experiments to quantify the dynamics of reticulocyte proportion, RBC concentration, and erythropoietin concentration in PHZ-induced anaemic mice. By quantifying experimental error we are able to fit and assess each hypothesis against our data and recover parameter estimates using Markov chain Monte Carlo based Bayesian inference. We find that, under normal conditions, about 3% of reticulocytes are released early from bone marrow and upon maturation all cells are released immediately. In the circulation, RBCs undergo random clearance but have a maximum lifespan of about 50 days. Under anaemic conditions reticulocyte production rate is linearly correlated with the difference between normal and anaemic RBC concentrations, and their release rate is exponentially correlated with the same. PHZ appears to age rather than kill RBCs, and younger RBCs are affected more than older RBCs. Blood sampling caused short aperiodic spikes in the proportion of reticulocytes which appear to have a different developmental pathway than normal reticulocytes. We also provide evidence of large diurnal oscillations in serum erythropoietin levels during anaemia.

Pharmacodynamic modeling of the effect of changes in the environment on cellular lifespan and cellular response

Lifespan-based pharmacodynamic (PD) models of cellular response assume that the lifespan of cells is predetermined at the time of cellular production, despite recognized changes in the cellular environment following production that may alter the survival of the cells. This work extends previously proposed cellular lifespan PD models to incorporate environmental effects on the cell lifespan by considering two basic classes of models from survival analysis: accelerated life and relative risk models. Cellular responses using both model classes were simulated using a steady-state cellular production rate with changes in the environmental effects resulting from three different basic profiles. The environmental effect models were also fitted to the red blood cell (RBC) and hemoglobin concentration data from six sheep following hematopoietic ablation by busulfan administration. The simulations indicated that the basic shapes of the cellular responses were different between the accelerated life and relative risk models. Due to the more direct physical interpretation, relatively simple steady-state relationship between the cellular response and environmental effects, and the ability to reduce the model to a "point" baseline lifespan distribution, the accelerated life model appears to be a more realistic and flexible model. The analysis of the sheep RBC and hemoglobin data indicated that the environmental effect began to decrease the survival of cells 1-2 weeks following initiation of ablation and that the average "severity" of the environmental effect increased 3.49 (29.5%) (mean (C.V.)) fold under the accelerated life model. Alternative models without an environmental effect did not describe the observed data as well. The proposed environmental effect cellular lifespan PD models allow for the incorporation of arbitrary changes in the conditions of the cellular environment and modeling of environmentally dependent cellular survival. These PD models have potential applications in hematological management of end-stage renal disease, transfusion medicine, and patients undergoing chemotherapy, among other diseases and therapies.

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.

Modeling time variant distributions of cellular lifespans: increases in circulating reticulocyte lifespans following double phlebotomies in sheep

Many pharmacodynamic (PD) models of cellular response assume a single and time invariant lifespan of all cells, despite the existence of a true underlying distribution of cellular lifespans and known changes in the lifespan distributions with time. To account for these features of cellular populations, a time variant cellular lifespan distribution PD model was formulated and theoretical aspects of modeling cellular populations presented. The model extends prior work assuming time variant "point distributions" of cellular lifespans (Freise et al. J Pharmacokinet Pharmacodyn 34:519-547, 2007) and models assuming a time invariant lifespan distribution (Krzyzanski et al. J Pharmacokinet Pharmacodyn 33:125-166, 2006). The formulated time variant lifespan distribution model was fitted to endogenous plasma erythropoietin (EPO), reticulocyte, and red blood cell (RBC) concentrations in sheep phlebotomized on two occasions, 8 days apart. The time variant circulating reticulocyte lifespan was modeled as a truncated and scaled Weibull distribution, with the location parameter of the distribution non-parametrically represented by an end constrained quadratic spline function. The formulated time variant lifespan distribution model was compared to the identical time invariant distribution, time variant "point distribution", and time invariant "point distribution" cellular lifespan models. Parameters of the time variant lifespan distribution model were well estimated with low standard errors. The mean circulating reticulocyte lifespan was estimated at 0.304 days, which rapidly increased over 3-fold following the first phlebotomy to a maximum of 1.03 days (P = 0.009). On average, the percentage of erythrocytes being released as reticulocytes maximally increased an estimated two-fold following the phlebotomies. The primary features of immature RBC physiology were captured by the model and gave results consistent with other estimates in sheep and humans. The comparison of the four lifespan models gave similar parameter estimates of the stimulation function and fits to the RBC data. However, the time invariant models fit the reticulocyte data poorly, while the time variant "point distribution" cellular lifespan model gave physiologically unrealistic estimates of the changes in the circulating reticulocyte lifespan under stress erythropoiesis. Thus the underlying physiology must be considered when selecting the most appropriate cellular lifespan model and not just the goodness-of-fit criteria. The proposed PD model and the numerical implementation allows for a flexible framework to incorporate time variant lifespan distributions when modeling populations of cells whose production or stimulation depends on endogenous growth factors and/or exogenous drugs.