A mathematical model of thrombopoiesis in rats

Flow-accelerated platelet biogenesis is due to an elasto-hydrodynamic instability

Blood platelets are formed by fragmentation of long membrane extensions from bone marrow megakaryocytes in the blood flow. Using lattice-Boltzmann/immersed boundary simulations we propose a biological Rayleigh-Plateau instability as the biophysical mechanism behind this fragmentation process. This instability is akin to the surface tension-induced breakup of a liquid jet but is driven by active cortical processes including actomyosin contractility and microtubule sliding. Our fully three-dimensional simulations highlight the crucial role of actomyosin contractility, which is required to trigger the instability, and illustrate how the wavelength of the instability determines the size of the final platelets. The elasto-hydrodynamic origin of the fragmentation explains the strong acceleration of platelet biogenesis in the presence of an external flow, which we observe in agreement with experiments. Our simulations then allow us to disentangle the influence of specific flow conditions: While a homogeneous flow with uniform velocity leads to the strongest acceleration, a shear flow with a linear velocity gradient can cause fusion events of two developing platelet-sized swellings during fragmentation. A fusion event may lead to the release of larger structures which are observable as preplatelets in experiments. Together, our findings strongly indicate a mainly physical origin of fragmentation and regulation of platelet size in flow-accelerated platelet biogenesis.

Evaluating the Role of Janus Kinase Pathways in Platelet Homeostasis Using a Systems Modeling Approach

Maintaining platelet homeostasis is important to avoid spontaneous bleeding and organ damage. Thrombopoietin, the primary regulator of platelet production, is affected by and acts in part via Janus kinase (JAK)‐signal transducer and activator of transcription (STAT)–mediated mechanisms. Interleukin‐6 is also partly responsible for inducing thrombopoietin production via the JAK‐STAT pathway. Although current understanding suggests that JAK2 is a primary mediator of platelet regulation, the emerging data show that a JAK1‐specific inhibitor resulted in the modulation of platelet numbers following dosing. To gain a mechanistic understanding, a model describing platelet regulation based on known physiology and JAK‐STAT pathways was built. The model provides a tool to coalesce biological understanding of platelet physiology and an in silico experimental platform to explore drug effects on platelet homeostasis. In this article, we explain the model construction and demonstrate the use of JAK‐inhibitor programs as informing probes of the physiology, gaining insights on dosing paradigms that avoid platelet‐related safety concerns.

Modeling individual time courses of thrombopoiesis during multi-cyclic chemotherapy

Background

Thrombocytopenia is a major side-effect of cytotoxic cancer therapies. The aim of precision medicine is to develop therapy modifications accounting for the individual’s risk.

Methodology/Principle findings

To solve this task, we develop an individualized bio-mechanistic model of the dynamics of bone marrow thrombopoiesis, circulating platelets and therapy effects thereon. Comprehensive biological knowledge regarding cell differentiation, amplification, apoptosis rates, transition times and corresponding regulations are translated into ordinary differential equations. A model of osteoblast/osteoclast interactions was incorporated to mechanistically describe bone marrow support of quiescent cell stages. Thrombopoietin (TPO) as a major regulator is explicitly modelled including pharmacokinetics and–dynamics of TPO injections. Effects of cytotoxic drugs are modelled by transient depletions of proliferating cells.

To calibrate the model, we used population data from the literature and close-meshed individual data of N = 135 high-grade non-Hodgkin’s lymphoma patients treated with CHOP-like chemotherapies. To limit the number of free parameters, several parsimony assumptions were derived from biological data and tested via Likelihood methods. Heterogeneity of patients was explained by a few model parameters. The over-fitting issue of individual parameter estimation was successfully dealt with a virtual participation of each patient in population-based experiments.

The model qualitatively and quantitatively explains a number of biological observations such as the role of osteoblasts in explaining long-term toxic effects, megakaryocyte-mediated feedback on stem cells, bi-phasic stimulation of thrombopoiesis by TPO, dynamics of megakaryocyte ploidies and non-exponential platelet degradation. Almost all individual time series could be described with high precision. We demonstrated how the model can be used to provide predictions regarding individual therapy adaptations.

Conclusions

We propose a mechanistic thrombopoiesis model of unprecedented comprehensiveness in both, biological mechanisms considered and experimental data sets explained. Our innovative method of parameter estimation allows robust determinations of individual parameter settings facilitating the development of individual treatment adaptations during chemotherapy.

Chemotherapy is ubiquitously used to treat cancer diseases. Due to general toxicity of the drugs, chemotherapy results in a number of side effects especially with respect to blood formation. Here we study the loss of platelets during chemotherapy which is dose limiting in many situations. However, this side-effect greatly varies between patients with respect to both, severity and necessity of clinical countermeasures.We therefore developed a mathematical model to predict the time course of platelets of patients under chemotherapy and to propose possible treatment adaptations in cases of intolerable toxicity. The model is based on available biological knowledge and data of platelet formation and therapeutic effects thereon. As a major result, we could describe individual time series data of 135 patients under chemotherapy. Conversely, the model can be used to make predictions regarding alternative therapy schedules such as postponement of therapy or chemotherapy dose reductions. Our model is intended to support clinical decision making on an individual patient level.

Cyclic thrombocytopenia with statistically significant neutrophil oscillations

Cyclic thrombocytopenia is often misdiagnosed as immune thrombocytopenia due to similar clinical features, a fact of significance because cyclic thrombocytopenia generally responds poorly to treatments used successfully in immune thrombocytopenia. A precise diagnosis must establish the statistical significance of periodicity of the platelet counts using statistical methods (eg, Lomb-Scargle periodogram).

Towards Quantitative Systems Pharmacology Models of Chemotherapy-Induced Neutropenia

Neutropenia is a serious toxic complication of chemotherapeutic treatment. For years, mathematical models have been developed to better predict hematological outcomes during chemotherapy in both the traditional pharmaceutical sciences and mathematical biology disciplines. An increasing number of quantitative systems pharmacology (QSP) models that combine systems approaches, physiology, and pharmacokinetics/pharmacodynamics have been successfully developed. Here, I detail the shift towards QSP efforts, emphasizing the importance of incorporating systems-level physiological considerations in pharmacometrics.

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 cell life span models for effects of drugs following indirect mechanisms of action

Pharmacokinetic/pharmacodynamic (PK/PD) models for hematological drug effects exist that assume that cells are produced by a zero- or first-order process, survive for a specific duration (cell lifespan), and then are lost. Due to the fact that delay differential equations (DDE) are needed for cell lifespan models, their software implementation is not straightforward. Our objective is to demonstrate methods to implement three different cell lifespan models for dealing with hematological drug effects and to evaluate the performance of NONMEM to estimate the model parameters. For the basic lifespan indirect response (LIDR) model, cells are produced by a zero-order process and removed due to senescence. The modified LIDR model adds a precursor pool. The LIDR model of cytotoxicity assumes a three-pool indirect model to account for the cell proliferation with capacity-limited cytotoxicity followed by maturation, and removal from the circulation. A numerical method (method of steps) implementing DDE in NONMEM was introduced. Simulation followed by estimation was used to evaluate NONMEM performance and the impact of the minimization algorithm (first-order method vs. first-order conditional estimation method) and the model for residual variability on the estimates of the population parameters. The FOCE method combined with log-transformation of data was found to be superior. This report provides methodology that will assist in application of population methods for assessing hematological responses to various types of drugs.

Basic pharmacodynamic models for agents that alter production of natural cells

Basic indirect pharmacodynamic models for agents which alter the generation of natural cells based on a life-span concept are introduced. It is assumed that cells (R) are produced at a constant rate (kin), survive for a specific duration TR, and then are lost. The rate of cell loss must equal the production rate but is delayed by TR. A therapeutic agent can stimulate or inhibit the production rate according to the Hill function: 1 +/- H(C(t)) where H(C(t)) contains capacity (Smax) and sensitivity (SC50) constants and C(t) is a pharmacokinetic function. Thus an operative model is [equation: see text] with the baseline condition R0 = kin.TR. One- and two-compartment catenary cell models were examined by simulation to describe the role of pharmacokinetics and cell properties. The area under the effect curve (AUCE) was derived. The models were applied to literature data to describe the stimulatory effects of single doses of hematopoietic growth factors such as granulocyte colony-stimulating factor (G-CSF) on neutrophils, thrombopoietin (TPO) on platelets, and erythropoietin (EPO) on reticulocytes in blood. The models described experimental data adequately and provided cell life-spans and SC50 values. The proposed cell production/loss models can be readily used to analyze the pharmacodynamics of agents which alter cell production yielding realistic physiological parameters.

[Thrombopoiesis, thrombocyte count and thrombocyte function before and following cell separation]

About 50% of the circulating platelets can be taken from healthy donors by means of the cell separation technique. In order to investigate the effect of cell separation on thrombopoesis bone marrow samples were taken from 12 voluntary donors up to 5 times. The megacaryocytes were analyzed before and after cell separation by morphometric measures and determinations of platelet count and platelet function (adhesion, aggregation, spreading, retraction) were carried out. Removing of large amounts of platelets form the circulation by stimulation of thrombopoesis resulted in an increase of the peripherical platelet count already 48 h after cell separation. Of the platelet functions tested only aggregation induced by collagen was found to be increased after cell separation as compared to that before the procedure.

[Mathematical models in hematology]

The use of mathematical models in haematology is shown by some examples concerning stem cell kinetics, erythropoiesis and thrombopoiesis. At first, model assumptions are formulated which include the biological knowledge and some regulatory hypotheses. Then, the reaction of the model on stimulation and suppression is calculated. Finally, by comparison with experimental or clinical data one can evaluate how far the model assumptions are sufficient to understand the measurements. Thus one can exclude wrong hypotheses and identify the important regulatory influences.