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

Evaluation of performance of distributed delay model for chemotherapy-induced myelosuppression

The distributed delay model has been introduced that replaces the transit compartments in the classic model of chemotherapy-induced myelosuppression with a convolution integral. The maturation of granulocyte precursors in the bone marrow is described by the gamma probability density function with the shape parameter (ν). If ν is a positive integer, the distributed delay model coincides with the classic model with ν transit compartments. The purpose of this work was to evaluate performance of the distributed delay model with particular focus on model deterministic identifiability in the presence of the shape parameter. The classic model served as a reference for comparison. Previously published white blood cell (WBC) count data in rats receiving bolus doses of 5-fluorouracil were fitted by both models. The negative two log-likelihood objective function (-2LL) and running times were used as major markers of performance. Local sensitivity analysis was done to evaluate the impact of ν on the pharmacodynamics response WBC. The ν estimate was 1.46 with 16.1% CV% compared to ν = 3 for the classic model. The difference of 6.78 in - 2LL between classic model and the distributed delay model implied that the latter performed significantly better than former according to the log-likelihood ratio test (P = 0.009), although the overall performance was modestly better. The running times were 1 s and 66.2 min, respectively. The long running time of the distributed delay model was attributed to computationally intensive evaluation of the convolution integral. The sensitivity analysis revealed that ν strongly influences the WBC response by controlling cell proliferation and elimination of WBCs from the circulation. In conclusion, the distributed delay model was deterministically identifiable from typical cytotoxic data. Its performance was modestly better than the classic model with significantly longer running time.

Estimation of adult and neonatal RBC lifespans in anemic neonates using RBCs labeled at several discrete biotin densities

BACKGROUND

Prior conclusions that autologous neonatal red blood cells (RBC) have substantially shorter lifespans than allogeneic adult RBCs were not based on direct comparison of autologous neonatal vs. allogeneic adult RBCs performed concurrently in the same infant. Biotin labeling of autologous neonatal RBCs and allogeneic adult donor RBCs permits concurrent direct comparison of autologous vs. allogeneic RBC lifespan.

METHODS

RBCs from 15 allogeneic adult donors and from 15 very-low-birth-weight (VLBW) neonates were labeled at separate biotin densities and transfused simultaneously into the 15 neonates. Two mathematical models that account for the RBC differences were employed to estimate lifespans for the two RBC populations.

RESULTS

Mean ± SD lifespan for adult allogeneic RBC was 70.1 ± 19.1 d, which is substantially shorter than the 120 d lifespan of both autologous and adult allogeneic RBC in healthy adults. Mean ± SD lifespan for neonatal RBC was 54.2 ± 11.3 d, which is only about 30% shorter than that of the adult allogeneic RBCs.

CONCLUSION

This study provides evidence that extrinsic environmental factors primarily determine RBC survival (e.g., small bore of the capillaries of neonates, rate of oxygenation/deoxygenation cycles) rather than factors intrinsic to RBC.

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.

Pharmacodynamic models of age-structured cell populations

The purpose of this work is to review basic pharmacodynamic (PD) models describing drug effects on cell populations and expand them to age-structured models using the theory of physiologically structured populations. The plasma drug concentrations are interpreted as the environment affecting the cell production and mortality rates. An explicit solution to model equations provides the age density distribution that serves to establish a relationship between the cell lifespan distribution and the hazard of cell removal. Given the lifespan distributions, the age distributions for most commonly applied PD models of cell responses including basic cell turnover, transit compartments, and basic lifespan models have been derived both for the baseline conditions and drug treatment. The steady-state age distribution for basic indirect response models is exponential, and it is uniform for the basic lifespan model. As an example of more complex cell population, the age distribution of human red blood cells has been simulated based on a recent model of red blood cell survival. The age distribution for cells in the transit compartment model is the sum of the gamma functions. Means and variances of age distributions for all discussed models were calculated. A brief discussion of numerical challenges and possible future model developments is presented.

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 Method to Evaluate Fetal Erythropoiesis from Postnatal Survival of Fetal RBCs

Fetal RBCs are produced during a period of very rapid growth and stimulated erythropoiesis under hypoxic intrauterine conditions. Fetal RBC life span varies with gestational age (GA) and is shorter than that in healthy adults. Due to the special kinetic properties of life span-based survival of human RBCs, a mathematical model-based kinetic analysis of the survival of fetal RBCs shortly after birth provides a unique opportunity to "look backward in time" to evaluate fetal erythropoiesis. This work introduces a novel method that utilizes postnatal in vivo RBC survival data collected within 2 days after birth to study both nonsteady-state (non-SS) in utero RBC production and changing fetal RBC life span over time. The effect of changes in erythropoiesis rate and RBC life span and the effect of multiple postnatal phlebotomies on the RBC survival curves were investigated using model-based simulations. This mathematical model, which considers both changes in the rate of erythropoiesis and RBC life span and which accurately accounts for the confounding effect of multiple phlebotomies, was applied to survival curves for biotin-labeled RBCs from ten anemic very low birth weight preterm infants. The estimated mean fetal RBC production rate scaled by body weight was 1.07 × 10(7) RBCs/day g, and the mean RBC life span at birth was 52.1 days; these values are consistent with reported values. The in utero RBC life span increased at a rate of 0.51 days per day of gestation. We conclude that the proposed mathematical model and its implementation provide a flexible framework to study in utero non-SS fetal erythropoiesis in newborn infants.

Solution and implementation of distributed lifespan models

We consider a population where every individual has a unique lifespan. After expiring of its lifespan the individual has to leave the population. A realistic approach to describe these lifespans is by a continuous distribution. Such distributed lifespan models (DLSMs) were introduced earlier in the indirect response context and consist of the mathematical convolution operator to describe the rate of change. Therefore, DLSMs could not directly be implemented in standard PKPD software. In this work we present the solution representation of DLSMs with and without a precursor population and an implementation strategy for DLSMs in ADAPT , NONMEM and MATLAB . We fit hemoglobin measurements from literature and investigate computational properties.

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.

Population pharmacokinetic and pharmacodynamic model-based comparability assessment of a recombinant human Epoetin Alfa and the Biosimilar HX575

The aim of this study was to develop an integrated pharmacokinetic and pharmacodynamic (PK/PD) model and assess the comparability between epoetin alfa HEXAL/Binocrit (HX575) and a comparator epoetin alfa by a model-based approach. PK/PD data-including serum drug concentrations, reticulocyte counts, red blood cells, and hemoglobin levels-were obtained from 2 clinical studies. In sum, 149 healthy men received multiple intravenous or subcutaneous doses of HX575 (100 IU/kg) and the comparator 3 times a week for 4 weeks. A population model based on pharmacodynamics-mediated drug disposition and cell maturation processes was used to characterize the PK/PD data for the 2 drugs. Simulations showed that due to target amount changes, total clearance may increase up to 2.4-fold as compared with the baseline. Further simulations suggested that once-weekly and thrice-weekly subcutaneous dosing regimens would result in similar efficacy. The findings from the model-based analysis were consistent with previous results using the standard noncompartmental approach demonstrating PK/PD comparability between HX575 and comparator. However, due to complexity of the PK/PD model, control of random effects was not straightforward. Whereas population PK/PD model-based analyses are suited for studying complex biological systems, such models have their limitations (statistical), and their comparability results should be interpreted carefully.

Modeling red blood cell survival data

Anaemia of chronic kidney disease (CKD) is a common complication in patients with renal impairment, especially in end-stage renal failure. As well as erythropoietin deficiency, decreased red blood cell survival is a contributing factor. However, it remains unclear which mechanism underlies the altered survival of red blood cells (RBCs). In this work a previously developed statistical model for RBC survival was applied to clinical data obtained from 14 patients with CKD undergoing hemodialysis as well as 14 healthy controls using radioactive chromium (⁵¹Cr) as random labelling method. A classical two-stage approach and a full population analysis were applied to estimate the key parameters controlling random destruction and senescence in the model. Estimating random destruction was preferred over estimating an accelerated senescence in both approaches and both groups as it provided the better fit to the data. Due to significant nonspecific random loss of the label from the cells that cannot be quantified directly only a relative RBC survival can be obtained from data using ⁵¹Cr as labelling method. Nevertheless, RBC survival was found to be significantly reduced in CKD patients compared to the controls with a relative reduction of 20-30% depending on the analysis method used.

Evaluation of red blood cell labelling methods based on a statistical model for red blood cell survival

The aim of this work is to compare different labelling methods that are commonly used to estimate the lifespan of red blood cells (RBCs), e.g. in anaemia of renal failure, where the effect of treatment with erythropoietin depends on the lifespan of RBCs. A previously developed model for the survival time of RBCs that accounts for plausible physiological processes of RBC destruction was used to simulate ideal random and cohort labelling methods for RBCs, as well as the flaws associated with these methods (e.g. reuse of label and loss of the label from the surviving RBCs). Random labelling with radioactive chromium and cohort labelling using heavy nitrogen were considered. Blood sampling times were determined for RBC survival studies using both labelling methods by applying the theory of optimal design. It was assessed whether the underlying parameter values of the model are estimable from these studies, and the precision of the parameter estimates were calculated. In theory, parameter estimation would be possible for both types of ideal labelling methods without flaws. However, flaws associated with random labelling are significant and not all parameters controlling RBC survival in the model can be estimated with good precision. In contrast, cohort labelling shows good precision in the parameter estimates even in the presence of reuse and prolonged incorporation of the label. A model based analysis of RBC survival studies is recommended in future to account for limitations in methodology as well as likely causes of RBC destruction.

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.

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.

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.