1
|
Mika B, Pełka M, Tkacz E. Mathematical modeling of the neutrophil production process supported by administration of glycoprotein. Biocybern Biomed Eng 2021. [DOI: 10.1016/j.bbe.2020.12.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
|
2
|
Mackey MC, Tyran-Kamińska M, Walther HO. Response of an oscillatory differential delay equation to a single stimulus. J Math Biol 2016; 74:1139-1196. [PMID: 27613016 DOI: 10.1007/s00285-016-1051-z] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2015] [Revised: 08/18/2016] [Indexed: 12/17/2022]
Abstract
Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.
Collapse
Affiliation(s)
- Michael C Mackey
- Departments of Physiology, Physics and Mathematics, McGill University, 3655 Promenade Sir William Osler, Montreal, QC, H3G 1Y6, Canada.
| | - Marta Tyran-Kamińska
- Institute of Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland
| | - Hans-Otto Walther
- Mathematisches Institut, Universität Giessen, Arndtstrasse 2, 35392, Giessen, Germany
| |
Collapse
|
3
|
Affiliation(s)
- Liza Köster
- Department of Companion Animal Clinical StudiesFaculty of Veterinary ScienceUniversity of PretoriaOnderstepoortGautengSouth Africa
- Department of Small Animal Internal MedicineRoss University School of Veterinary MedicineBasseterreSaint Kitts and Nevis
| | - Cindy Harper
- Veterinary Genetics LaboratoryFaculty of Veterinary ScienceUniversity of PretoriaOnderstepoortGautengSouth Africa
| | - Amelia Goddard
- Department of Clinical PathologyFaculty of Veterinary ScienceUniversity of PretoriaOnderstepoortGautengSouth Africa
| |
Collapse
|
4
|
Schirm S, Engel C, Loeffler M, Scholz M. Modelling chemotherapy effects on granulopoiesis. BMC SYSTEMS BIOLOGY 2014; 8:138. [PMID: 25539928 PMCID: PMC4302124 DOI: 10.1186/s12918-014-0138-7] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/02/2014] [Accepted: 12/11/2014] [Indexed: 02/05/2023]
Abstract
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.
Collapse
|
5
|
Dale DC, Mackey MC. Understanding, treating and avoiding hematological disease: better medicine through mathematics? Bull Math Biol 2014; 77:739-57. [PMID: 25213154 DOI: 10.1007/s11538-014-9995-x] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2014] [Accepted: 07/08/2014] [Indexed: 10/24/2022]
Abstract
This paper traces the experimental, clinical and mathematical modeling efforts to understand a periodic hematological disease-cyclical neutropenia. It is primarily a highly personal account by two scientists from quite different backgrounds of their interactions over almost 40 years and their attempts to understand this intriguing disease. It's also a story of their efforts to offer effective treatments for the patients who suffer from cyclic neutropenia and other conditions causing neutropenia and infections.
Collapse
Affiliation(s)
- David C Dale
- Department of Medicine, University of Washington, Seattle, WA, 98195, USA,
| | | |
Collapse
|
6
|
Schirm S, Engel C, Loeffler M, Scholz M. A combined model of human erythropoiesis and granulopoiesis under growth factor and chemotherapy treatment. Theor Biol Med Model 2014; 11:24. [PMID: 24886056 PMCID: PMC4046020 DOI: 10.1186/1742-4682-11-24] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2014] [Accepted: 05/16/2014] [Indexed: 12/03/2022] Open
Abstract
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.
Collapse
Affiliation(s)
- Sibylle Schirm
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
- LIFE Research Center of Civilization Diseases, University of Leipzig, Leipzig, Germany
| | - Christoph Engel
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
| | - Markus Loeffler
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
| | - Markus Scholz
- Institute for Medical Informatics, Statistics and Epidemiology, University of Leipzig, Leipzig, Germany
- LIFE Research Center of Civilization Diseases, University of Leipzig, Leipzig, Germany
| |
Collapse
|
7
|
Understanding and Treating Cytopenia Through Mathematical Modeling. ADVANCES IN EXPERIMENTAL MEDICINE AND BIOLOGY 2014; 844:279-302. [DOI: 10.1007/978-1-4939-2095-2_14] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
|
8
|
Krinner A, Roeder I, Loeffler M, Scholz M. Merging concepts - coupling an agent-based model of hematopoietic stem cells with an ODE model of granulopoiesis. BMC SYSTEMS BIOLOGY 2013; 7:117. [PMID: 24180697 PMCID: PMC4228322 DOI: 10.1186/1752-0509-7-117] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/19/2012] [Accepted: 10/16/2013] [Indexed: 11/11/2022]
Abstract
Background Hematopoiesis is a complex process involving different cell types and feedback mechanisms mediated by cytokines. This complexity stimulated various models with different scopes and applications. A combination of complementary models promises to provide their mutual confirmation and to explain a broader range of scenarios. Here we propose a combination of an ordinary differential equation (ODE) model of human granulopoiesis and an agent-based model (ABM) of hematopoietic stem cell (HSC) organization. The first describes the dynamics of bone marrow cell stages and circulating cells under various perturbations such as G-CSF treatment or chemotherapy. In contrast to the ODE model describing cell numbers, our ABM focuses on the organization of individual cells in the stem population. Results We combined the two models by replacing the HSC compartment of the ODE model by a difference equation formulation of the ABM. In this hybrid model, regulatory mechanisms and parameters of the original models were kept unchanged except for a few specific improvements: (i) Effect of chemotherapy was restricted to proliferating HSC and (ii) HSC regulation in the ODE model was replaced by the intrinsic regulation of the ABM. Model simulations of bleeding, chronic irradiation and stem cell transplantation revealed that the dynamics of hybrid and ODE model differ markedly in scenarios with stem cell damage. Despite these differences in response to stem cell damage, both models explain clinical data of leukocyte dynamics under four chemotherapy regimens. Conclusions ABM and ODE model proved to be compatible and were combined without altering the structure of both models. The new hybrid model introduces model improvements by considering the proliferative state of stem cells and enabling a cell cycle-dependent effect of chemotherapy. We demonstrated that it is able to explain and predict granulopoietic dynamics for a large variety of scenarios such as irradiation, bone marrow transplantation, chemotherapy and growth factor applications. Therefore, it promises to serve as a valuable tool for studies in a broader range of clinical applications, in particular where stem cell activation and proliferation are involved.
Collapse
Affiliation(s)
- Axel Krinner
- Institute for Medical Informatics and Biometry, TU Dresden, Blasewitzer str, 86, D-01307 Dresden, Germany.
| | | | | | | |
Collapse
|
9
|
Stiehl T, Ho AD, Marciniak-Czochra A. The impact of CD34+ cell dose on engraftment after SCTs: personalized estimates based on mathematical modeling. Bone Marrow Transplant 2013; 49:30-7. [DOI: 10.1038/bmt.2013.138] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2013] [Revised: 07/05/2013] [Accepted: 08/03/2013] [Indexed: 11/09/2022]
|
10
|
Brooks G, Provencher G, Lei J, Mackey MC. Neutrophil dynamics after chemotherapy and G-CSF: the role of pharmacokinetics in shaping the response. J Theor Biol 2012; 315:97-109. [PMID: 22981924 DOI: 10.1016/j.jtbi.2012.08.028] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2012] [Revised: 07/21/2012] [Accepted: 08/23/2012] [Indexed: 11/17/2022]
Abstract
Chemotherapy has profound effects on the hematopoietic system, most notably leading to neutropenia. Granulocyte colony stimulating factor (G-CSF) is often used to deal with this neutropenia, but the response is highly variable. In this paper we examine the role of pharmacokinetics and delivery protocols in shaping the neutrophil responses to chemotherapy and G-CSF. Neutrophil responses to different protocols of chemotherapy administration with varying dosages, infusion times, and schedules are studied through a mathematical model. We find that a single dose of chemotherapy produces a damped oscillation in neutrophil levels, and short-term applications of chemotherapy can induce permanent oscillations in neutrophil level if there is a bistability in the system. In addition, we confirm previous findings [Zhuge et al., J. Theor. Biol., 293(2012), 111-120] that when periodic chemotherapy is given, there is a significant period of delivery that induces resonance in the system and exacerbates the corresponding neutropenia. The width of this resonant period peak increases with the recovery rate after a single chemotherapy, which is given by the real part of the dominant eigenvalue pair at the steady state, and both are determined by a single cooperativity coefficient in the feedback function for the neutrophils. Our numerical studies show that the neutropenia caused by chemotherapy can be overcome if G-CSF is given early after chemotherapy but can actually be worsened if G-CSF is given later, consistent with results reported in Zhuge et al. (2012). The nadir in neutrophil level is found to be more sensitive to the dosage of chemotherapy than that of the G-CSF. Furthermore, dependence of our results with changes in key pharmacokinetic parameters as well as initial functions are studied. Thus, this study illuminates the potential for destructive resonance leading to neutropenia in response to periodic chemotherapy, and explores and explains why the timing of G-CSF is so crucial for successful reversal of chemotherapy induced neutropenia.
Collapse
Affiliation(s)
- Grace Brooks
- Department of Physiology, Centre for Applied Mathematics in Bioscience and Medicine, McGill University, Montreal, QC, Canada H4X 2C1
| | | | | | | |
Collapse
|
11
|
Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia. J Theor Biol 2011; 270:143-53. [DOI: 10.1016/j.jtbi.2010.11.024] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2010] [Revised: 09/21/2010] [Accepted: 11/13/2010] [Indexed: 11/19/2022]
|
12
|
Abstract
Scientists have traditionally studied complex biologic systems by reducing them to simple building blocks. Genome sequencing, high-throughput screening, and proteomics have, however, generated large datasets, revealing a high level of complexity in components and interactions. Systems biology embraces this complexity with a combination of mathematical, engineering, and computational tools for constructing and validating models of biologic phenomena. The validity of mathematical modeling in hematopoiesis was established early by the pioneering work of Till and McCulloch. In reviewing more recent papers, we highlight deterministic, stochastic, statistical, and network-based models that have been used to better understand a range of topics in hematopoiesis, including blood cell production, the periodicity of cyclical neutropenia, stem cell production in response to cytokine administration, and the emergence of imatinib resistance in chronic myeloid leukemia. Future advances require technologic improvements in computing power, imaging, and proteomics as well as greater collaboration between experimentalists and modelers. Altogether, systems biology will improve our understanding of normal and abnormal hematopoiesis, better define stem cells and their daughter cells, and potentially lead to more effective therapies.
Collapse
|
13
|
Foley C, Mackey MC. Mathematical model for G-CSF administration after chemotherapy. J Theor Biol 2008; 257:27-44. [PMID: 19007795 DOI: 10.1016/j.jtbi.2008.09.043] [Citation(s) in RCA: 46] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2008] [Revised: 09/12/2008] [Accepted: 09/23/2008] [Indexed: 11/24/2022]
Abstract
Granulocyte-colony stimulating factor (G-CSF) is used clinically for treating chemotherapy-induced neutropenia (low neutrophil levels). Here we present a delay differential equation model for the regulation of neutrophil production that accounts for the effects of G-CSF. Using a combination of analysis and numerical simulations, we use this model to study the effects of delaying G-CSF treatment following chemotherapy for two recombinant forms of G-CSF (filgrastim and pegfilgrastim). We also examine the consequences of varying the duration of filgrastim treatment. We found that varying the starting day or the duration of G-CSF treatment can lead to different qualitative responses in the neutrophil count. These changes can be explained by the coexistence of two stable solutions in the mathematical model.
Collapse
Affiliation(s)
- Catherine Foley
- Department of Mathematics and Centre for Nonlinear Dynamics, Mcgill University, 3655 Promenade Sir William Osler, Montreal, Quebec, Canada H3G 1Y6.
| | | |
Collapse
|
14
|
Dynamic hematological disease: a review. J Math Biol 2008; 58:285-322. [DOI: 10.1007/s00285-008-0165-3] [Citation(s) in RCA: 69] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2007] [Revised: 01/22/2008] [Indexed: 10/22/2022]
|