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Busse JE, Cuadrado S, Marciniak-Czochra A. Local asymptotic stability of a system of integro-differential equations describing clonal evolution of a self-renewing cell population under mutation. J Math Biol 2022; 84:10. [PMID: 34988700 DOI: 10.1007/s00285-021-01708-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2020] [Revised: 11/01/2021] [Accepted: 11/19/2021] [Indexed: 11/30/2022]
Abstract
In this paper we consider a system of non-linear integro-differential equations (IDEs) describing evolution of a clonally heterogeneous population of malignant white blood cells (leukemic cells) undergoing mutation and clonal selection. We prove existence and uniqueness of non-trivial steady states and study their asymptotic stability. The results are compared to those of the system without mutation. Existence of equilibria is proved by formulating the steady state problem as an eigenvalue problem and applying a version of the Krein-Rutmann theorem for Banach lattices. The stability at equilibrium is analysed using linearisation and the Weinstein-Aronszajn determinant which allows to conclude local asymptotic stability.
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Affiliation(s)
- Jan-Erik Busse
- Institute of Applied Mathematics, Interdisciplinary Center for Scientific Computing (IWR) and BIOQUANT Center, Heidelberg, Germany
| | - Sílvia Cuadrado
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Spain
| | - Anna Marciniak-Czochra
- Institute of Applied Mathematics, Interdisciplinary Center for Scientific Computing (IWR) and BIOQUANT Center, Heidelberg, Germany.
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2
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Bonnet C, Gou P, Girel S, Bansaye V, Lacout C, Bailly K, Schlagetter MH, Lauret E, Méléard S, Giraudier S. Multistage hematopoietic stem cell regulation in the mouse: A combined biological and mathematical approach. iScience 2021; 24:103399. [PMID: 34877482 PMCID: PMC8627979 DOI: 10.1016/j.isci.2021.103399] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2021] [Revised: 07/29/2021] [Accepted: 11/02/2021] [Indexed: 11/24/2022] Open
Abstract
We have reconciled steady-state and stress hematopoiesis in a single mathematical model based on murine in vivo experiments and with a focus on hematopoietic stem and progenitor cells. A phenylhydrazine stress was first applied to mice. A reduced cell number in each progenitor compartment was evidenced during the next 7 days through a drastic level of differentiation without proliferation, followed by a huge proliferative response in all compartments including long-term hematopoietic stem cells, before a return to normal levels. Data analysis led to the addition to the 6-compartment model, of time-dependent regulation that depended indirectly on the compartment sizes. The resulting model was finely calibrated using a stochastic optimization algorithm and could reproduce biological data in silico when applied to different stress conditions (bleeding, chemotherapy, HSC depletion). In conclusion, our multi-step and time-dependent model of immature hematopoiesis provides new avenues to a better understanding of both normal and pathological hematopoiesis. We describe a new 6-compartment time-dependent regulated model of hematopoiesis Biological data under steady state and stress and cell dynamics were used Modeling is able to recapitulate effects from chemotherapy, bleeding, or HSC depletion
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Affiliation(s)
- Céline Bonnet
- CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France
| | - Panhong Gou
- Centre Hayem, Université de Paris, Hôpital Saint Louis, INSERM U1131, 1 Avenue Claude Vellefaux, 75010 Paris, France
| | - Simon Girel
- CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France
| | - Vincent Bansaye
- CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France
| | - Catherine Lacout
- Centre Hayem, Université de Paris, Hôpital Saint Louis, INSERM U1131, 1 Avenue Claude Vellefaux, 75010 Paris, France
| | - Karine Bailly
- Université de Paris, Institut Cochin, INSERM U1016, CNRS UMR8104, 75014 Paris, France
| | - Marie-Hélène Schlagetter
- Centre Hayem, Université de Paris, Hôpital Saint Louis, INSERM U1131, 1 Avenue Claude Vellefaux, 75010 Paris, France
| | - Evelyne Lauret
- Université de Paris, Institut Cochin, INSERM U1016, CNRS UMR8104, 75014 Paris, France
| | - Sylvie Méléard
- CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France.,Institut Universitaire de France, Paris, France
| | - Stéphane Giraudier
- Centre Hayem, Université de Paris, Hôpital Saint Louis, INSERM U1131, 1 Avenue Claude Vellefaux, 75010 Paris, France
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Chulián S, Martínez-Rubio Á, Marciniak-Czochra A, Stiehl T, Goñi CB, Rodríguez Gutiérrez JF, Ramírez Orellana M, Castillo Robleda A, Pérez-García VM, Rosa M. Dynamical properties of feedback signalling in B lymphopoiesis: A mathematical modelling approach. J Theor Biol 2021; 522:110685. [PMID: 33745905 DOI: 10.1016/j.jtbi.2021.110685] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2020] [Revised: 12/09/2020] [Accepted: 03/15/2021] [Indexed: 12/11/2022]
Abstract
Haematopoiesis is the process of generation of blood cells. Lymphopoiesis generates lymphocytes, the cells in charge of the adaptive immune response. Disruptions of this process are associated with diseases like leukaemia, which is especially incident in children. The characteristics of self-regulation of this process make them suitable for a mathematical study. In this paper we develop mathematical models of lymphopoiesis using currently available data. We do this by drawing inspiration from existing structured models of cell lineage development and integrating them with paediatric bone marrow data, with special focus on regulatory mechanisms. A formal analysis of the models is carried out, giving steady states and their stability conditions. We use this analysis to obtain biologically relevant regions of the parameter space and to understand the dynamical behaviour of B-cell renovation. Finally, we use numerical simulations to obtain further insight into the influence of proliferation and maturation rates on the reconstitution of the cells in the B line. We conclude that a model including feedback regulation of cell proliferation represents a biologically plausible depiction for B-cell reconstitution in bone marrow. Research into haematological disorders could benefit from a precise dynamical description of B lymphopoiesis.
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Affiliation(s)
- Salvador Chulián
- Department of Mathematics, Universidad de Cádiz, Puerto Real, Cádiz, Spain; Biomedical Research and Innovation Institute of Cádiz (INiBICA), Hospital Universitario Puerta del Mar, Cádiz, Spain.
| | - Álvaro Martínez-Rubio
- Department of Mathematics, Universidad de Cádiz, Puerto Real, Cádiz, Spain; Biomedical Research and Innovation Institute of Cádiz (INiBICA), Hospital Universitario Puerta del Mar, Cádiz, Spain
| | - Anna Marciniak-Czochra
- Institute of Applied Mathematics, BioQuant and Interdisciplinary Center of Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany
| | - Thomas Stiehl
- Institute of Applied Mathematics, BioQuant and Interdisciplinary Center of Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany
| | | | | | - Manuel Ramírez Orellana
- Department of Paediatric Haematology and Oncology, Hospital Infantil Universitario Niño Jesús, Instituto Investigación Sanitaria La Princesa, Madrid, Spain
| | - Ana Castillo Robleda
- Department of Paediatric Haematology and Oncology, Hospital Infantil Universitario Niño Jesús, Instituto Investigación Sanitaria La Princesa, Madrid, Spain
| | - Víctor M Pérez-García
- Department of Mathematics, Mathematical Oncology Laboratory (MOLAB), Universidad de Castilla-La Mancha, Ciudad Real, Spain; Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, Ciudad Real, Spain; ETSI Industriales, Universidad de Castilla-La Mancha, Ciudad Real, Spain
| | - María Rosa
- Department of Mathematics, Universidad de Cádiz, Puerto Real, Cádiz, Spain; Biomedical Research and Innovation Institute of Cádiz (INiBICA), Hospital Universitario Puerta del Mar, Cádiz, Spain
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4
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Jost F, Zierk J, Le TTT, Raupach T, Rauh M, Suttorp M, Stanulla M, Metzler M, Sager S. Model-Based Simulation of Maintenance Therapy of Childhood Acute Lymphoblastic Leukemia. Front Physiol 2020; 11:217. [PMID: 32256384 PMCID: PMC7093595 DOI: 10.3389/fphys.2020.00217] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2019] [Accepted: 02/25/2020] [Indexed: 01/19/2023] Open
Abstract
Acute lymphoblastic leukemia is the most common malignancy in childhood. Successful treatment requires initial high-intensity chemotherapy, followed by low-intensity oral maintenance therapy with oral 6-mercaptopurine (6MP) and methotrexate (MTX) until 2–3 years after disease onset. However, intra- and inter-individual variability in the pharmacokinetics (PK) and pharmacodynamics (PD) of 6MP and MTX make it challenging to balance the desired antileukemic effects with undesired excessive myelosuppression during maintenance therapy. A model to simulate the dynamics of different cell types, especially neutrophils, would be a valuable contribution to improving treatment protocols (6MP and MTX dosing regimens) and a further step to understanding the heterogeneity in treatment efficacy and toxicity. We applied and modified a recently developed semi-mechanistic PK/PD model to neutrophils and analyzed their behavior using a non-linear mixed-effects modeling approach and clinical data obtained from 116 patients. The PK model of 6MP influenced the accuracy of absolute neutrophil count (ANC) predictions, whereas the PD effect of MTX did not. Predictions based on ANC were more accurate than those based on white blood cell counts. Using the new cross-validated mathematical model, simulations of different treatment protocols showed a linear dose-effect relationship and reduced ANC variability for constant dosages. Advanced modeling allows the identification of optimized control criteria and the weighting of specific influencing factors for protocol design and individually adapted therapy to exploit the optimal effect of maintenance therapy on survival.
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Affiliation(s)
- Felix Jost
- Department of Mathematics, Institute of Mathematical Optimization, Otto-von-Guericke University Magdeburg, Magdeburg, Germany
| | - Jakob Zierk
- Department of Paediatrics and Adolescent Medicine, University Hospital Erlangen, Erlangen, Germany
| | - Thuy T T Le
- Department of Mathematics, Institute of Mathematical Optimization, Otto-von-Guericke University Magdeburg, Magdeburg, Germany
| | - Thomas Raupach
- Department of Paediatrics and Adolescent Medicine, University Hospital Erlangen, Erlangen, Germany
| | - Manfred Rauh
- Department of Paediatrics and Adolescent Medicine, University Hospital Erlangen, Erlangen, Germany
| | - Meinolf Suttorp
- Pediatric Hematology and Oncology, University Hospital "Carl Gustav Carus", Dresden, Germany
| | - Martin Stanulla
- Department of Pediatric Hemato-Oncology, Hannover Medical School, Hanover, Germany
| | - Markus Metzler
- Department of Paediatrics and Adolescent Medicine, University Hospital Erlangen, Erlangen, Germany
| | - Sebastian Sager
- Department of Mathematics, Institute of Mathematical Optimization, Otto-von-Guericke University Magdeburg, Magdeburg, Germany.,Health Campus "Immunology, Infectiology and Inflammation (GC-I3)", Otto-von-Guericke University, Magdeburg, Germany
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5
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Lorenzi T, Marciniak-Czochra A, Stiehl T. A structured population model of clonal selection in acute leukemias with multiple maturation stages. J Math Biol 2019; 79:1587-1621. [DOI: 10.1007/s00285-019-01404-w] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2018] [Revised: 07/05/2019] [Indexed: 12/19/2022]
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