Age-structured and two-delay models for erythropoiesis

Extracellular vesicles: effectors of transfusion-related acute lung injury

Respiratory transfusion reactions represent some of the most severe adverse reactions related to receiving blood products. Of those, transfusion-related acute lung injury (TRALI) is associated with elevated morbidity and mortality. TRALI is characterized by severe lung injury associated with inflammation, pulmonary neutrophil infiltration, lung barrier leak, and increased interstitial and airspace edema that cause respiratory failure. Presently, there are few means of detecting TRALI beyond clinical definitions based on physical examination and vital signs or preventing/treating TRALI beyond supportive care with oxygen and positive pressure ventilation. Mechanistically, TRALI is thought to be mediated by the culmination of two successive proinflammatory hits, which typically comprise a recipient factor (1st hit-e.g., systemic inflammatory conditions) and a donor factor (2nd hit-e.g., blood products containing pathogenic antibodies or bioactive lipids). An emerging concept in TRALI research is the contribution of extracellular vesicles (EVs) in mediating the first and/or second hit in TRALI. EVs are small, subcellular, membrane-bound vesicles that circulate in donor and recipient blood. Injurious EVs may be released by immune or vascular cells during inflammation, by infectious bacteria, or in blood products during storage, and can target the lung upon systemic dissemination. This review assesses emerging concepts such as how EVs: 1) mediate TRALI, 2) represent targets for therapeutic intervention to prevent or treat TRALI, and 3) serve as biochemical biomarkers facilitating TRALI diagnosis and detection in at-risk patients.

Modeling and forecasting age-specific drug overdose mortality in the United States

Drug overdose deaths continue to increase in the United States for all major drug categories. Over the past two decades the total number of overdose fatalities has increased more than fivefold; since 2013 the surge in overdose rates is primarily driven by fentanyl and methamphetamines. Different drug categories and factors such as age, gender, and ethnicity are associated with different overdose mortality characteristics that may also change in time. For example, the average age at death from a drug overdose has decreased from 1940 to 1990 while the overall mortality rate has steadily increased. To provide insight into the population-level dynamics of drug overdose mortality, we develop an age-structured model for drug addiction. Using an augmented ensemble Kalman filter (EnKF), we show through a simple example how our model can be combined with synthetic observation data to estimate mortality rate and an age-distribution parameter. Finally, we use an EnKF to combine our model with observation data on overdose fatalities in the United States from 1999 to 2020 to forecast the evolution of overdose trends and estimate model parameters.

Why Are Periodic Erythrocytic Diseases so Rare in Humans?

Many studies have shown that periodic erythrocytic (red blood cell linked) diseases are extremely rare in humans. To explain this observation, we develop here a simple model of erythropoiesis in mammals and investigate its stability in the parameter space. A bifurcation analysis enables us to sketch stability diagrams in the plane of key parameters. Contrary to some other mammal species such as rabbits, mice or dogs, we show that human-specific parameter values prevent periodic oscillations of red blood cells levels. In other words, human erythropoiesis seems to lie in a region of parameter space where oscillations exclusively concerning red blood cells cannot appear. Further mathematical analysis show that periodic oscillations of red blood cells levels are highly unusual and if exist, might only be due to an abnormally high erythrocytes destruction rate or to an abnormal hematopoietic stem cell commitment into the erythrocytic lineage. We also propose numerical results only for an improved version of our approach in order to give a more realistic but more complex approach of our problem.

Multistage feedback-driven compartmental dynamics of hematopoiesis

Human hematopoiesis is surprisingly resilient to disruptions, providing suitable responses to severe bleeding, long-lasting immune activation, and even bone marrow transplants. Still, many blood disorders exist which push the system past its natural plasticity, resulting in abnormalities in the circulating blood. While proper treatment of such diseases can benefit from understanding the underlying cell dynamics, these are non-trivial to predict due to the hematopoietic system's hierarchical nature and complex feedback networks. To characterize the dynamics following different types of perturbations, we investigate a model representing hematopoiesis as a sequence of compartments covering all maturation stages-from stem to mature cells-where feedback regulates cell production to ongoing necessities. We find that a stable response to perturbations requires the simultaneous adaptation of cell differentiation and self-renewal rates, and show that under conditions of continuous disruption-as found in chronic hemolytic states-compartment cell numbers evolve to novel stable states.

A generalized within-host model of dengue infection with a non-constant monocyte production rate

In this paper, we generalize a previous model of within-host dengue infection with a nonconstant monocyte production rate. We establish the existence of three equilibria and give some local stability results. We then estimate three parameters in the model from clinical data for dengue virus serotype 2. It is then shown that the model can exhibit behaviours that are not possible under the assumption of constant monocyte production. Lastly, we perform a sensitivity analysis of the model in two contexts, antiviral treatment and immunostimulatory treatment. The results predict that antiviral treatments that reduce the viral replication rate in infected monocytes are the most effective, while immunostimulatory treatments that increase the rate at which infected monocytes are removed are best.

On the Effect of Age-Dependent Mortality on the Stability of a System of Delay-Differential Equations Modeling Erythropoiesis

We present an age-structured model for erythropoiesis in which the mortality of mature cells is described empirically by a physiologically realistic probability distribution of survival times. Under some assumptions, the model can be transformed into a system of delay differential equations with both constant and distributed delays. The stability of the equilibrium of this system and possible Hopf bifurcations are described for a number of probability distributions. Physiological motivation and interpretation of our results are provided.

Modelling stem cell ageing: a multi-compartment continuum approach

Stem cells are important to generate all specialized tissues at an early life stage, and in some systems, they also have repair functions to replenish the adult tissues. Repeated cell divisions lead to the accumulation of molecular damage in stem cells, which are commonly recognized as drivers of ageing. In this paper, a novel model is proposed to integrate stem cell proliferation and differentiation with damage accumulation in the stem cell ageing process. A system of two structured PDEs is used to model the population densities of stem cells (including all multiple progenitors) and terminally differentiated (TD) cells. In this system, cell cycle progression and damage accumulation are modelled by continuous dynamics, and damage segregation between daughter cells is considered at each division. Analysis and numerical simulations are conducted to study the steady-state populations and stem cell damage distributions under different damage segregation strategies. Our simulations suggest that equal distribution of the damaging substance between stem cells in a symmetric renewal and less damage retention in stem cells in the asymmetric division are favourable strategies, which reduce the death rate of the stem cells and increase the TD cell populations. Moreover, asymmetric damage segregation in stem cells leads to less concentrated damage distribution in the stem cell population, which may be more robust to the stochastic changes in the damage. The feedback regulation from stem cells can reduce oscillations and population overshoot in the process, and improve the fitness of stem cells by increasing the percentage of cells with less damage in the stem cell population.

Recent advances in physiologically based pharmacokinetic and pharmacodynamic models for anticancer nanomedicines

Nanoparticles (NPs) have distinct pharmacokinetic (PK) properties and can potentially improve the absorption, distribution, metabolism, and elimination (ADME) of small-molecule drugs loaded therein. Owing to the unwanted toxicities of anticancer agents in healthy organs and tissues, their precise delivery to the tumor is an essential requirement. There have been numerous advancements in the development of nanomedicines for cancer therapy. Physiologically based PK (PBPK) models serve as excellent tools for describing and predicting the ADME properties and the efficacy and toxicity of drugs, in combination with pharmacodynamic (PD) models. The recent preliminary application of these modeling approaches to NPs demonstrated their potential benefits in research and development processes relevant to the ADME and pharmacodynamics of NPs and nanomedicines. Here, we comprehensively review the pharmacokinetics of NPs, the developed PBPK models for anticancer NPs, and the developed PD model for anticancer agents.

Mathematical Modeling of RBC Count Dynamics after Blood Loss

The regeneration of red blood cells (RBCs) after blood loss is an individual complex process. We present a novel simple compartment model which is able to capture the most important features and can be personalized using parameter estimation. We compare predictions of the proposed and personalized model to a more sophisticated state-of-the-art model for erythropoiesis, and to clinical data from healthy subjects. We discuss the choice of model parameters with respect to identifiability. We give an outlook on how extensions of this novel mathematical model could have an important impact for personalized clinical decision support in the case of polycythemia vera (PV). PV is a slow-growing type of blood cancer, where especially the production of RBCs is increased. The principal treatment targeting the symptoms of PV is bloodletting (phlebotomy), at regular intervals that are based on personal experiences of the physicians. Model-based decision support might help to identify optimal and individualized phlebotomy schedules.

Stem cell biomanufacturing under uncertainty: A case study in optimizing red blood cell production

As breakthrough cellular therapy discoveries are translated into reliable, commercializable applications, effective stem cell biomanufacturing requires systematically developing and optimizing bioprocess design and operation. This article proposes a rigorous computational framework for stem cell biomanufacturing under uncertainty. Our mathematical tool kit incorporates: high‐fidelity modeling, single variate and multivariate sensitivity analysis, global topological superstructure optimization, and robust optimization. The advantages of the proposed bioprocess optimization framework using, as a case study, a dual hollow fiber bioreactor producing red blood cells from progenitor cells were quantitatively demonstrated. The optimization phase reduces the cost by a factor of 4, and the price of insuring process performance against uncertainty is approximately 15% over the nominal optimal solution. Mathematical modeling and optimization can guide decision making; the possible commercial impact of this cellular therapy using the disruptive technology paradigm was quantitatively evaluated. © 2017 American Institute of Chemical Engineers AIChE J, 64: 3011–3022, 2018

Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues

OBJECTIVE

Modeling and analysis of cell population dynamics enhance our understanding of cancer. Here we introduce and explore a new model that may apply to many tissues.

ANALYSES

An age-structured model describing coexistence between mutated and ordinary stem cells is developed and explored. The model is transformed into a nonlinear time-delay system governing the dynamics of healthy cells, coupled to a nonlinear differential-difference system describing dynamics of unhealthy cells. Its main features are highlighted and an advanced stability analysis of several steady states is performed, through specific Lyapunov-like functionals for descriptor-type systems.

RESULTS

We propose a biologically based model endowed with rich dynamics. It incorporates a new parameter representing immunoediting processes, including the case where proliferation of cancer cells is locally kept under check by the immune cells. It also considers the overproliferation of cancer stem cells, modeled as a subpopulation of mutated cells that is constantly active in cell division. The analysis that we perform here reveals the conditions of existence of several steady states, including the case of cancer dormancy, in the coupled model of interest. Our study suggests that cancer dormancy may result from a plastic sensitivity of mutated cells to their shared environment, different from that - fixed - of healthy cells, and this is related to an action (or lack of action) of the immune system. Next, the stability analysis that we perform is essentially oriented towards the determination of sufficient conditions, depending on all the model parameters, that ensure either a regionally (i.e., locally) stable dormancy steady state or eradication of unhealthy cells. Finally, we discuss some biological interpretations, with regards to our findings, in light of current and emerging therapeutics. These final insights are particularly formulated in the paradigmatic case of hematopoiesis and acute leukemia, which is one of the best known malignancies for which it is always hard, in presence of a clinical and histological remission, to decide between cure and dormancy of a tumoral clone.

Prediction of hemoglobin levels in individual hemodialysis patients by means of a mathematical model of erythropoiesis

Anemia commonly occurs in people with chronic kidney disease (CKD) and is associated with poor clinical outcomes. The management of patients with anemia in CKD is challenging, due to its severity, frequent hypo-responsiveness to treatment with erythropoiesis stimulating agents (ESA) and common hemoglobin cycling. Nonlinear dose-response curves and long delays in the effect of treatment on red blood cell population size complicate predictions of hemoglobin (Hgb) levels in individual patients. A comprehensive physiology based mathematical model for erythropoiesis was adapted individually to 60 hemodialysis patients treated with ESAs by identifying physiologically meaningful key model parameters from temporal Hgb data. Crit-Line^® III monitors provided non-invasive Hgb measurements for every hemodialysis treatment. We used Hgb data during a 150-day baseline period together to estimate a patient’s individual red blood cell lifespan, effects of the ESA on proliferation of red cell progenitor cells, endogenous erythropoietin production and ESA half-life. Estimated patient specific parameters showed excellent alignment with previously conducted clinical studies in hemodialysis patients. Further, the model qualitatively and quantitatively reflected empirical hemoglobin dynamics in demographically, anthropometrically and clinically diverse patients and accurately predicted the Hgb response to ESA therapy in individual patients for up to 21 weeks. The findings suggest that estimated model parameters can be used as a proxy for parameters that are clinically very difficult to quantify. The presented method has the potential to provide new insights into the individual pathophysiology of renal anemia and its association with clinical outcomes and can potentially be used to guide personalized anemia treatment.

Whole-body iron transport and metabolism: Mechanistic, multi-scale model to improve treatment of anemia in chronic kidney disease

Iron plays vital roles in the human body including enzymatic processes, oxygen-transport via hemoglobin and immune response. Iron metabolism is characterized by ~95% recycling and minor replenishment through diet. Anemia of chronic kidney disease (CKD) is characterized by a lack of synthesis of erythropoietin leading to reduced red blood cell (RBC) formation and aberrant iron recycling. Treatment of CKD anemia aims to normalize RBC count and serum hemoglobin. Clinically, the various fluxes of iron transport and accumulation are not measured so that changes during disease (e.g., CKD) and treatment are unknown. Unwanted iron accumulation in patients is known to lead to adverse effects. Current whole-body models lack the mechanistic details of iron transport related to RBC maturation, transferrin (Tf and TfR) dynamics and assume passive iron efflux from macrophages. Hence, they are not predictive of whole-body iron dynamics and cannot be used to design individualized patient treatment. For prediction, we developed a mechanistic, multi-scale computational model of whole-body iron metabolism incorporating four compartments containing major pools of iron and RBC generation process. The model accounts for multiple forms of iron in vivo, mechanisms involved in iron uptake and release and their regulation. Furthermore, the model is interfaced with drug pharmacokinetics to allow simulation of treatment dynamics. We calibrated our model with experimental and clinical data from peer-reviewed literature to reliably simulate CKD anemia and the effects of current treatment involving combination of epoietin-alpha and iron dextran. This in silico whole-body model of iron metabolism predicts that a year of treatment can potentially lead to 90% downregulation of ferroportin (FPN) levels, 15-fold increase in iron stores with only a 20% increase in iron flux from the reticulo-endothelial system (RES). Model simulations quantified unmeasured iron fluxes, previously unknown effects of treatment on FPN-level and iron stores in the RES. This mechanistic whole-body model can be the basis for future studies that incorporate iron metabolism together with related clinical experiments. Such an approach could pave the way for development of effective personalized treatment of CKD anemia.

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with three examples arising from cancer invasion.

Universal Dichotomy for Dynamical Systems with Variable Delay

We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.

Mathematical model of susceptibility, resistance, and resilience in the within-host dynamics between a Plasmodium parasite and the immune system

We developed a coupled age-structured partial differential equation model to capture the disease dynamics during blood-stage malaria. The addition of age structure for the parasite population, with respect to previous models, allows us to better characterize the interaction between the malaria parasite and red blood cells during infection. Here we prove that the system we propose is well-posed and there exist at least two global states. We further demonstrate that the numerical simulation of the system coincides with clinically observed outcomes of primary and secondary malaria infection. The well-posedness of this system guarantees that the behavior of the model remains smooth, bounded, and continuously dependent on initial conditions; calibration with clinical data will constrain domains of parameters and variables to physiological ranges.

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.

Comparison of mathematical frameworks for modeling erythropoiesis in the context of malaria infection

Malaria is an infectious disease present all around the globe and responsible for half a million deaths per year. A within-host model of this infection requires a framework capable of properly approximating not only the blood stage of the infection but also the erythropoietic process that is in charge of overcoming the malaria induced anemia. Within this context, we compare ordinary differential equations (ODEs) with and without age classes, delayed differential equations (DDEs), and discrete recursive equations (DREs) with age classes. Results show that ODEs without age classes are fair approximations that do not provide a crisp temporal representation of the processes involved, and inclusion of age classes only mitigates the problem to some degree. DDEs perform well with respect to generating the essentially fixed delay between cell production and cell removal due to age, but the inclusion of any other processes, such as sudden blood loss, becomes cumbersome. The framework that was found to perform best in representing the dynamics of red blood cells during malaria infection is a DRE with age classes. In this model structure, the amount of time a cell remains alive is easily controlled, and the addition of age dependent or independent processes is straightforward. All events that populations of cells face during their lifespan, like growth or adaptation in differentiation or maturation rate, are properly represented in this framework.

Dynamical properties induced by state-dependent delays in photonic systems

In many dynamical systems and complex networks time delays appear naturally in feedback loops or coupling connections of individual elements. Moreover, in a whole class of systems, these delay times can depend on the state of the system. Nevertheless, so far the understanding of the impact of such state-dependent delays remains poor with a particular lack of systematic experimental studies. Here we fill this gap by introducing a conceptually simple photonic system that exhibits dynamics of self-organised switching between two loops with two different delay times, depending on the state of the system. On the basis of experiments and modelling on semiconductor lasers with frequency-selective feedback mirrors, we characterize the switching between the states defined by the individual delays. Our approach opens new perspectives for the study of this class of dynamical systems and enables applications in which the self-organized switching can be exploited.

Time delays in feedback loops and connections in dynamical systems and complex networks can depend on the state of the system, but these state-dependent delays are poorly understood. Here, the authors use a photonic system to characterize the switching between two loops with different delay times.

Modelling chemotherapy effects on granulopoiesis

Background

Although the growth-factor G-CSF is widely used to prevent granulotoxic side effects of cytotoxic chemotherapies, its optimal use is still unknown since treatment outcome depends on many parameters such as dosing and timing of chemotherapies, pharmaceutical derivative of G-CSF used and individual risk factors. We showed in the past that a pharmacokinetic and –dynamic model of G-CSF and human granulopoiesis can be used to predict the performance of yet untested G-CSF schedules. However, only a single chemotherapy was considered so far.

In the present paper, we propose a comprehensive model of chemotherapy toxicity and combine it with our cell kinetic model of granulopoiesis. Major assumptions are: proportionality of cell numbers and cell loss, delayed action of chemotherapy, drug, drug-dose and cell stage specific toxicities, no interaction of drugs and higher toxicity of drugs at the first time of application. Correspondingly, chemotherapies can be characterized by a set of toxicity parameters which can be estimated by fitting the predictions of our model to clinical time series data of patients under therapy. Data were either extracted from the literature or were received from cooperating clinical study groups.

Results

Model assumptions proved to be feasible in explaining granulotoxicity of 10 different chemotherapeutic drugs or drug-combinations applied in 33 different schedules with and without G-CSF. Risk groups of granulotoxicity were traced back to differences in toxicity parameters.

Conclusion

We established a comprehensive model of combined G-CSF and chemotherapy action in humans which allows us to predict and compare the outcome of alternative G-CSF schedules. We aim to apply the model in different clinical contexts to optimize and individualize G-CSF treatment.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-014-0138-7) contains supplementary material, which is available to authorized users.

Optimization of anemia treatment in hemodialysis patients via reinforcement learning

OBJECTIVE

Anemia is a frequent comorbidity in hemodialysis patients that can be successfully treated by administering erythropoiesis-stimulating agents (ESAs). ESAs dosing is currently based on clinical protocols that often do not account for the high inter- and intra-individual variability in the patient's response. As a result, the hemoglobin level of some patients oscillates around the target range, which is associated with multiple risks and side-effects. This work proposes a methodology based on reinforcement learning (RL) to optimize ESA therapy.

METHODS

RL is a data-driven approach for solving sequential decision-making problems that are formulated as Markov decision processes (MDPs). Computing optimal drug administration strategies for chronic diseases is a sequential decision-making problem in which the goal is to find the best sequence of drug doses. MDPs are particularly suitable for modeling these problems due to their ability to capture the uncertainty associated with the outcome of the treatment and the stochastic nature of the underlying process. The RL algorithm employed in the proposed methodology is fitted Q iteration, which stands out for its ability to make an efficient use of data.

RESULTS

The experiments reported here are based on a computational model that describes the effect of ESAs on the hemoglobin level. The performance of the proposed method is evaluated and compared with the well-known Q-learning algorithm and with a standard protocol. Simulation results show that the performance of Q-learning is substantially lower than FQI and the protocol. When comparing FQI and the protocol, FQI achieves an increment of 27.6% in the proportion of patients that are within the targeted range of hemoglobin during the period of treatment. In addition, the quantity of drug needed is reduced by 5.13%, which indicates a more efficient use of ESAs.

CONCLUSION

Although prospective validation is required, promising results demonstrate the potential of RL to become an alternative to current protocols.

Stochastic modeling of stress erythropoiesis using a two-type age-dependent branching process with immigration

The erythroid lineage is a particularly sensitive target of radiation injury. We model the dynamics of immature (BFU-E) and mature (CFU-E) erythroid progenitors, which have markedly different kinetics of recovery, following sublethal total body irradiation using a two-type reducible age-dependent branching process with immigration. Properties of the expectation and variance of the frequencies of both types of progenitors are presented. Their explicit expressions are derived when the process is Markovian, and their asymptotic behavior is identified in the age-dependent (non-Markovian) case. Analysis of experimental data on the kinetics of BFU-E and CFU-E reveals that the probability of self-renewal increases transiently for both cell types following sublethal irradiation. In addition, the probability of self-renewal increased more for CFU-E than for BFU-E. The strategy adopted by the erythroid lineage ensures replenishment of the BFU-E compartment while optimizing the rate of CFU-E recovery. Finally, our analysis also indicates that radiation exposure causes a delay in BFU-E recovery consistent with injury to the hematopoietic stem/progenitor cell compartment that give rise to BFU-E. Erythroid progenitor self-renewal is thus an integral component of the recovery of the erythron in response to stress.

A combined model of human erythropoiesis and granulopoiesis under growth factor and chemotherapy treatment

BACKGROUND

Haematotoxicity of conventional chemotherapies often results in delays of treatment or reduction of chemotherapy dose. To ameliorate these side-effects, patients are routinely treated with blood transfusions or haematopoietic growth factors such as erythropoietin (EPO) or granulocyte colony-stimulating factor (G-CSF). For the latter ones, pharmaceutical derivatives are available, which differ in absorption kinetics, pharmacokinetic and -dynamic properties. Due to the complex interaction of cytotoxic effects of chemotherapy and the stimulating effects of different growth factor derivatives, optimal treatment is a non-trivial task. In the past, we developed mathematical models of thrombopoiesis, granulopoiesis and erythropoiesis under chemotherapy and growth-factor applications which can be used to perform clinically relevant predictions regarding the feasibility of chemotherapy schedules and cytopenia prophylaxis with haematopoietic growth factors. However, interactions of lineages and growth-factors were ignored so far.

RESULTS

To close this gap, we constructed a hybrid model of human granulopoiesis and erythropoiesis under conventional chemotherapy, G-CSF and EPO applications. This was achieved by combining our single lineage models of human erythropoiesis and granulopoiesis with a common stem cell model. G-CSF effects on erythropoiesis were also implemented. Pharmacodynamic models are based on ordinary differential equations describing proliferation and maturation of haematopoietic cells. The system is regulated by feedback loops partly mediated by endogenous and exogenous EPO and G-CSF. Chemotherapy is modelled by depletion of cells. Unknown model parameters were determined by fitting the model predictions to time series data of blood counts and cytokine profiles. Data were extracted from literature or received from cooperating clinical study groups. Our model explains dynamics of mature blood cells and cytokines after growth-factor applications in healthy volunteers. Moreover, we modelled 15 different chemotherapeutic drugs by estimating their bone marrow toxicity. Taking into account different growth-factor schedules, this adds up to 33 different chemotherapy regimens explained by the model.

CONCLUSIONS

We conclude that we established a comprehensive biomathematical model to explain the dynamics of granulopoiesis and erythropoiesis under combined chemotherapy, G-CSF, and EPO applications. We demonstrate how it can be used to make predictions regarding haematotoxicity of yet untested chemotherapy and growth-factor schedules.

The role of spatial organization of cells in erythropoiesis

Erythropoiesis, the process of red blood cell production, occurs mainly in the bone marrow. The functional unit of mammalian erythropoiesis, the erythroblastic island, consists of a central macrophage surrounded by adherent erythroid progenitor cells (CFU-E/Pro-EBs) and their differentiating progeny, the erythroblasts. Central macrophages display on their surface or secrete various growth or inhibitory factors that influence the fate of the surrounding erythroid cells. CFU-E/Pro-EBs have three possible fates: (a) expansion of their numbers without differentiation, (b) differentiation into reticulocytes that are released into the blood, (c) death by apoptosis. CFU-E/Pro-EB fate is under the control of a complex molecular network, that is highly dependent upon environmental conditions in the erythroblastic island. In order to assess the functional role of space coupled with the complex network behavior in erythroblastic islands, we developed hybrid discrete-continuous models of erythropoiesis. A model was developed in which cells are considered as individual physical objects, intracellular regulatory networks are modeled with ordinary differential equations and extracellular concentrations by partial differential equations. We used the model to investigate the impact of an important difference between humans and mice in which mature late-stage erythroblasts produce the most Fas-ligand in humans, whereas early-stage erythroblasts produce the most Fas-ligand in mice. Although the global behaviors of the erythroblastic islands in both species were similar, differences were found, including a relatively slower response time to acute anemia in humans. Also, our modeling approach was very consistent with in vitro culture data, where the central macrophage in reconstituted erythroblastic islands has a strong impact on the dynamics of red blood cell production. The specific spatial organization of erythroblastic islands is key to the normal, stable functioning of mammalian erythropoiesis, both in vitro and in vivo. Our model of a simplified molecular network controlling cell decision provides a realistic functional unit of mammalian erythropoiesis that integrates multiple microenvironmental influences within the erythroblastic island with those of circulating regulators of erythropoiesis, such as EPO and glucocorticosteroids, that are produced at remote sites.

Erythropoiesis: from molecular pathways to system properties

Erythropoiesis is regulated through a long-range negative feedback loop, whereby tissue hypoxia stimulates erythropoietin (Epo) secretion, which promotes an increase in erythropoietic rate. However, this long-range feedback loop, by itself, cannot account for the observed system properties of erythropoiesis, namely, a wide dynamic range, stability in the face of random perturbations, and a rapid stress response. Here, we show that three Epo-regulated erythroblast survival pathways each give rise to distinct system properties. The induction of Bcl-xL by signal transducer and activator of transcription 5 (Stat5) is responsive to the rate of change in Epo levels, rather than to its absolute level, and is therefore maximally but transiently activated in acute stress. By contrast, Epo-mediated suppression of the pro-survival Fas and Bim pathways is proportional to the levels of stress/Epo and persists throughout chronic stress. Together, these elements operate in a manner reminiscent of a "proportional-integral-derivative (PID)" feedback controller frequently found in engineering applications. A short-range negative autoregulatory loop within the early erythroblast compartment, operated by Fas/FasL, filters out random noise and controls a reserve pool of early erythroblasts that is poised to accelerate the response to acute stress. Both these properties have previously been identified as inherent to negative regulatory motifs. Finally, we show that signal transduction by Stat5 combines binary and graded modalities, thereby increasing signaling fidelity over the wide dynamic range of Epo found in health and disease.

Effect of mature blood-stage Plasmodium parasite sequestration on pathogen biomass in mathematical and in vivo models of malaria

Parasite biomass and microvasculature obstruction are strongly associated with disease severity and death in Plasmodium falciparum-infected humans. This is related to sequestration of mature, blood-stage parasites (schizonts) in peripheral tissue. The prevailing view is that schizont sequestration leads to an increase in pathogen biomass, yet direct experimental data to support this are lacking. Here, we first studied parasite population dynamics in inbred wild-type (WT) mice infected with the rodent species of malaria, Plasmodium berghei ANKA. As is commonly reported, these mice became moribund due to large numbers of parasites in multiple tissues. We then studied infection dynamics in a genetically targeted line of mice, which displayed minimal tissue accumulation of parasites. We constructed a mathematical model of parasite biomass dynamics, incorporating schizont-specific host clearance, both with and without schizont sequestration. Combined use of mathematical and in vivo modeling indicated, first, that the slowing of parasite growth in the genetically targeted mice can be attributed to specific clearance of schizonts from the circulation and, second, that persistent parasite growth in WT mice can be explained solely as a result of schizont sequestration. Our work provides evidence that schizont sequestration could be a major biological process driving rapid, early increases in parasite biomass during blood-stage Plasmodium infection.

A biomathematical model of human erythropoiesis under erythropoietin and chemotherapy administration

Anaemia is a common haematologic side effect of dose-dense multi-cycle cytotoxic polychemotherapy requiring erythrocyte transfusions or erythropoietin (EPO) administration. To simulate the effectiveness of different EPO application schedules, we performed both modelling of erythropoiesis under chemotherapy and pharmacokinetic and dynamic modelling of EPO applications in the framework of a single comprehensive biomathematical model. For this purpose, a cell kinetic model of bone marrow erythropoiesis was developed that is based on a set of differential compartment equations describing proliferation and maturation of erythropoietic cell stages. The system is regulated by several feedback loops comprising those mediated by EPO. We added a model of EPO absorption after injection at different sites and a pharmacokinetic model of EPO derivatives to account for the effects of external EPO applications. Chemotherapy is modelled by a transient depletion of bone marrow cell stages. Unknown model parameters were determined by fitting the predictions of the model to data sets of circulating erythrocytes, haemoglobin, haematocrit, percentage of reticulocytes or EPO serum concentrations derived from the literature or cooperating clinical study groups. Parameter fittings resulted in a good agreement of model and data. Depending on site of injection and derivative (Alfa, Beta, Delta, Darbepoetin), nine groups of EPO applications were distinguished differing in either absorption kinetics or pharmacokinetics. Finally, eight different chemotherapy protocols were modelled. The model was validated on the basis of scenarios not used for parameter fitting. Simulations were performed to analyze the impact of EPO applications on the risk of anaemia during chemotherapy. We conclude that we established a model of erythropoiesis under chemotherapy that explains a large set of time series data under EPO and chemotherapy applications. It allows predictions regarding yet untested EPO schedules. Prospective clinical studies are needed to validate model predictions and to explore the feasibility and effectiveness of the proposed schedules.

Parameter identification of hematopoiesis mathematical model - periodic chronic myelogenous leukemia

Periodic chronic myelogenous leukemia (PCML) is a dynamic hematopoietic disease which causes oscillations of circulating leukocytes, platelets and reticulocytes. Mathematical modeling is an invaluable tool to help in predicting hematopoiesis behavior. In this paper we modify the existing models based on improving the parameters of the model. Also more parameters are estimated regarding the proposed model. It is our major intention to construct a physiological model which can map major identified mechanisms of leukopoiesis to provide a deeper insight into this complex biological process. In the proposed model the leukocytes line has been modeled more precisely. In fact, precursor cells have been considered as two separate groups: proliferating precursor cells and non-proliferating precursor cells. As a result, more parameters have appeared in the model and identifying the new parameters has resulted in a better fit of clinical data and the data extracted from the model for both platelets and leukocytes. That is, the new model describes the leukocytes and platelets of the system in a way that is closer to clinical data, so the proposed model can be more useful for predicting the behavior of leukocytes and platelets for PCML disease. Compared with the previous works, it is shown that the new model has a better fit of the quantitative data on leukocytes and platelets.

Preferential invasion by Plasmodium merozoites and the self-regulation of parasite burden

The preferential invasion of particular red blood cell (RBC) age classes may offer a mechanism by which certain species of Plasmodia regulate their population growth. Asexual reproduction of the parasite within RBCs exponentially increases the number of circulating parasites; limiting this explosion in parasite density may be key to providing sufficient time for the parasite to reproduce, and for the host to develop a specific immune response. It is critical that the role of preferential invasion in infection is properly understood to model the within-host dynamics of different Plasmodia species. We develop a simulation model to show that limiting the range of RBC age classes available for invasion is a credible mechanism for restricting parasite density, one which is equally as important as the maximum parasite replication rate and the duration of the erythrocytic cycle. Different species of Plasmodia that regularly infect humans exhibit different preferences for RBC invasion, with all species except P. falciparum appearing to exhibit a combination of characteristics which are able to self-regulate parasite density.

Understanding immunology via engineering design: the role of mathematical prototyping

A major challenge in immunology is how to translate data into knowledge given the inherent complexity and dynamics of human physiology. Both the physiology and engineering communities have rich histories in applying computational approaches to translate data obtained from complex systems into knowledge of system behavior. However, there are some differences in how disciplines approach problems. By referring to mathematical models as mathematical prototypes, we aim to highlight aspects related to the process (i.e., prototyping) rather than the product (i.e., the model). The objective of this paper is to review how two related engineering concepts, specifically prototyping and "fitness for use," can be applied to overcome the pressing challenge in translating data into improved knowledge of basic immunology that can be used to improve therapies for disease. These concepts are illustrated using two immunology-related examples. The prototypes presented focus on the beta cell mass at the onset of type 1 diabetes and the dynamics of dendritic cells in the lung. This paper is intended to illustrate some of the nuances associated with applying mathematical modeling to improve understanding of the dynamics of disease progression in humans.

A model of erythropoiesis in adults with sufficient iron availability

In this paper we present a model for erythropoiesis under the basic assumption that sufficient iron availability is guaranteed. An extension of the model including a sub-model for the iron dynamics in the body is topic of present research efforts. The model gives excellent results for a number of important situations: recovery of the red blood cell mass after blood donation, adaptation of the number of red blood cells to changes in the altitude of residence and, most important, the reaction of the body to different administration regimens of erythropoiesis stimulating agents, as for instance in the case of pre-surgical administration of Epoetin-α. The simulation results concerning the last item show that choosing an appropriate administration regimen can reduce the total amount of the administered drug considerably. The core of the model consists of structured population equations for the different cell populations which are considered. A key feature of the model is the incorporation of neocytolysis.

Comparative performance of cell life span and cell transit models for describing erythropoietic drug effects

Prolonged time delay in response to drug action is a common feature of hematological responses to pharmacotherapy such as erythropoiesis. The objective of this study was to compare the performance of two competing modeling approaches for delayed drug effects, mechanistic cell life span models, and semi-mechanistic cell transit models. The comparison was performed with an experimental dataset from multiple dose administrations of an erythropoietin mimetic to Cynomolgus monkeys. Comparative performance measures include visual predictive checks, goodness-of-fit plots, model estimation time, estimation status, and estimation error. The analysis revealed that both models resulted in a similarly good description of the erythropoietic drug effect, with precision and bias of the model-based predictions of red blood cell counts of less than 11%. The cell transit model needed slightly longer time to converge compared to the cell life span model. The system and drug effect parameters were similar in both models indicating that the models can be interchangeably used to describe the current data. Thus, model selection would be dependent on the purpose of the modeling exercise, the available data, and the time allocated for model development.

Time delay in physiological systems: analyzing and modeling its impact

This article examines the functional and clinical impact of time delays that arise in human physiological systems, especially control systems. An overview of the mathematical and physiological contexts for considering time delays will be illustrated, from the system level to cell level, by examining models that incorporate time delays. This examination will highlight how such delays in combination with other system structures and parameters influence system dynamics. Model analysis that reveals the influence of delays can also reveal related physiological effects which may have medical consequences and clinical applications.

Toward integration of biological and physiological functions at multiple levels

An aim of systems physiology today can be stated as to establish logical and quantitative bridges between phenomenological attributes of physiological entities such as cells and organs and physical attributes of biological entities, i.e., biological molecules, allowing us to describe and better understand physiological functions in terms of underlying biological functions. This article illustrates possible schema that can be used for promoting systems physiology by integrating quantitative knowledge of biological and physiological functions at multiple levels of time and space with the use of information technology infrastructure. Emphasis will be made for systematic, modular, hierarchical, and standardized descriptions of mathematical models of the functions and advantages for the use of them.