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Al-Badri G, Phillips JB, Shipley RJ, Ovenden NC. Formation of vascular-like structures using a chemotaxis-driven multiphase model. Math Biosci 2024; 372:109183. [PMID: 38554855 DOI: 10.1016/j.mbs.2024.109183] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2023] [Revised: 03/18/2024] [Accepted: 03/21/2024] [Indexed: 04/02/2024]
Abstract
We propose a continuum model for pattern formation, based on the multiphase model framework, to explore in vitro cell patterning within an extracellular matrix (ECM). We demonstrate that, within this framework, chemotaxis-driven cell migration can lead to the formation of cell clusters and vascular-like structures in 1D and 2D respectively. The influence on pattern formation of additional mechanisms commonly included in multiphase tissue models, including cell-matrix traction, contact inhibition, and cell-cell aggregation, are also investigated. Using sensitivity analysis, the relative impact of each model parameter on the simulation outcomes is assessed to identify the key parameters involved. Chemoattractant-matrix binding is further included, motivated by previous experimental studies, and found to reduce the spatial scale of patterning to within a biologically plausible range for capillary structures. Key findings from the in-depth parameter analysis of the 1D models, both with and without chemoattractant-matrix binding, are demonstrated to translate well to the 2D model, obtaining vascular-like cell patterning for multiple parameter regimes. Overall, we demonstrate a biologically-motivated multiphase model capable of generating long-term pattern formation on a biologically plausible spatial scale both in 1D and 2D, with applications for modelling in vitro vascular network formation.
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Affiliation(s)
- Georgina Al-Badri
- Department of Mathematics, University College London, London, UK; Centre for Nerve Engineering, University College London, London, UK.
| | - James B Phillips
- Centre for Nerve Engineering, University College London, London, UK; Department of Pharmacology, University College London, London, UK
| | - Rebecca J Shipley
- Centre for Nerve Engineering, University College London, London, UK; Department of Mechanical Engineering, University College London, London, UK
| | - Nicholas C Ovenden
- Department of Mathematics, University College London, London, UK; Centre for Nerve Engineering, University College London, London, UK
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2
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Yakovlev EV, Simkin IV, Shirokova AA, Kolotieva NA, Novikova SV, Nasyrov AD, Denisenko IR, Gursky KD, Shishkov IN, Narzaeva DE, Salmina AB, Yurchenko SO, Kryuchkov NP. Machine learning approach for recognition and morphological analysis of isolated astrocytes in phase contrast microscopy. Sci Rep 2024; 14:9846. [PMID: 38684715 PMCID: PMC11059356 DOI: 10.1038/s41598-024-59773-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2023] [Accepted: 04/15/2024] [Indexed: 05/02/2024] Open
Abstract
Astrocytes are glycolytically active cells in the central nervous system playing a crucial role in various brain processes from homeostasis to neurotransmission. Astrocytes possess a complex branched morphology, frequently examined by fluorescent microscopy. However, staining and fixation may impact the properties of astrocytes, thereby affecting the accuracy of the experimental data of astrocytes dynamics and morphology. On the other hand, phase contrast microscopy can be used to study astrocytes morphology without affecting them, but the post-processing of the resulting low-contrast images is challenging. The main result of this work is a novel approach for recognition and morphological analysis of unstained astrocytes based on machine-learning recognition of microscopic images. We conducted a series of experiments involving the cultivation of isolated astrocytes from the rat brain cortex followed by microscopy. Using the proposed approach, we tracked the temporal evolution of the average total length of branches, branching, and area per astrocyte in our experiments. We believe that the proposed approach and the obtained experimental data will be of interest and benefit to the scientific communities in cell biology, biophysics, and machine learning.
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Affiliation(s)
- Egor V Yakovlev
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia.
| | - Ivan V Simkin
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Anastasiya A Shirokova
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Nataliya A Kolotieva
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
- Research Center of Neurology, 80 Volokolamskoye Shosse, Moscow, 125367, Russia
| | - Svetlana V Novikova
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
- Research Center of Neurology, 80 Volokolamskoye Shosse, Moscow, 125367, Russia
| | - Artur D Nasyrov
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Ilya R Denisenko
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Konstantin D Gursky
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Ivan N Shishkov
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Diana E Narzaeva
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
- Research Center of Neurology, 80 Volokolamskoye Shosse, Moscow, 125367, Russia
| | - Alla B Salmina
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
- Research Center of Neurology, 80 Volokolamskoye Shosse, Moscow, 125367, Russia
| | - Stanislav O Yurchenko
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia
| | - Nikita P Kryuchkov
- Scientific-Educational Centre "Soft matter and physics of fluids", Bauman Moscow State Technical University, 2nd Baumanskaya Street 5, Moscow, 105005, Russia.
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3
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Crossley RM, Johnson S, Tsingos E, Bell Z, Berardi M, Botticelli M, Braat QJS, Metzcar J, Ruscone M, Yin Y, Shuttleworth R. Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist. Front Cell Dev Biol 2024; 12:1354132. [PMID: 38495620 PMCID: PMC10940354 DOI: 10.3389/fcell.2024.1354132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2023] [Accepted: 02/12/2024] [Indexed: 03/19/2024] Open
Abstract
The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.
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Affiliation(s)
- Rebecca M. Crossley
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Samuel Johnson
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Erika Tsingos
- Computational Developmental Biology Group, Institute of Biodynamics and Biocomplexity, Utrecht University, Utrecht, Netherlands
| | - Zoe Bell
- Northern Institute for Cancer Research, Newcastle University, Newcastle upon Tyne, United Kingdom
| | - Massimiliano Berardi
- LaserLab, Department of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam, Netherlands
- Optics11 life, Amsterdam, Netherlands
| | | | - Quirine J. S. Braat
- Department of Applied Physics and Science Education, Eindhoven University of Technology, Eindhoven, Netherlands
| | - John Metzcar
- Department of Intelligent Systems Engineering, Indiana University, Bloomington, IN, United States
- Department of Informatics, Indiana University, Bloomington, IN, United States
| | | | - Yuan Yin
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
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Huo X, Jüngel A. Global Existence and Weak-Strong Uniqueness for Chemotaxis Compressible Navier-Stokes Equations Modeling Vascular Network Formation. JOURNAL OF MATHEMATICAL FLUID MECHANICS : JMFM 2024; 26:11. [PMID: 38261880 PMCID: PMC10796441 DOI: 10.1007/s00021-023-00840-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 10/19/2023] [Indexed: 01/25/2024]
Abstract
A model of vascular network formation is analyzed in a bounded domain, consisting of the compressible Navier-Stokes equations for the density of the endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, which triggers the migration of the endothelial cells and the blood vessel formation. The coupling of the equations is realized by the chemotaxis force in the momentum balance equation. The global existence of finite energy weak solutions is shown for adiabatic pressure coefficients γ > 8 / 5 . The solutions satisfy a relative energy inequality, which allows for the proof of the weak-strong uniqueness property.
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Affiliation(s)
- Xiaokai Huo
- Department of Mathematics, Iowa State University, 411 Morrill Road, Ames, IA 50011-2104 USA
| | - Ansgar Jüngel
- Institute of Analysis and Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria
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Sakai K, Hayashi T, Sakai Y, Mada J, Tonami K, Uchijima Y, Kurihara H, Tokihiro T. A three-dimensional model with two-body interactions for endothelial cells in angiogenesis. Sci Rep 2023; 13:20549. [PMID: 37996513 PMCID: PMC10667370 DOI: 10.1038/s41598-023-47911-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2023] [Accepted: 11/20/2023] [Indexed: 11/25/2023] Open
Abstract
We introduce a three-dimensional mathematical model for the dynamics of vascular endothelial cells during sprouting angiogenesis. Angiogenesis is the biological process by which new blood vessels form from existing ones. It has been the subject of numerous theoretical models. These models have successfully replicated various aspects of angiogenesis. Recent studies using particle-based models have highlighted the significant influence of cell shape on network formation, with elongated cells contributing to the formation of branching structures. While most mathematical models are two-dimensional, we aim to investigate whether ellipsoids also form branch-like structures and how their shape affects the pattern. In our model, the shape of a vascular endothelial cell is represented as a spheroid, and a discrete dynamical system is constructed based on the simple assumption of two-body interactions. Numerical simulations demonstrate that our model reproduces the patterns of elongation and branching observed in the early stages of angiogenesis. We show that the pattern formation of the cell population is strongly dependent on the cell shape. Finally, we demonstrate that our current mathematical model reproduces the cell behaviours, specifically cell-mixing, observed in sprouts.
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Affiliation(s)
- Kazuma Sakai
- Graduate School of Mathematical Science, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan
| | - Tatsuya Hayashi
- Faculty of Science and Engineering, Yamato University, 2-5-1, Katayama-cho, Suita, Osaka, 564-0082, Japan.
- Research and Development Initiative, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan.
| | - Yusuke Sakai
- Graduate School of Medicine, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
| | - Jun Mada
- College of Industrial Technology, Nihon University, 1-2-1, Izumi-cho, Narashino, Chiba, 275-8575, Japan
| | - Kazuo Tonami
- Graduate School of Medicine, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
| | - Yasunobu Uchijima
- Graduate School of Medicine, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
| | - Hiroki Kurihara
- Graduate School of Medicine, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
| | - Tetsuji Tokihiro
- Graduate School of Mathematical Science, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan.
- Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo, 135-8181, Japan.
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6
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Villa C, Gerisch A, Chaplain MAJ. A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis. J Theor Biol 2022; 534:110963. [PMID: 34838584 DOI: 10.1016/j.jtbi.2021.110963] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2021] [Revised: 11/03/2021] [Accepted: 11/12/2021] [Indexed: 11/18/2022]
Abstract
The formation of new vascular networks is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, known as cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system and solve the equations numerically. We then conduct a parametric analysis of the numerical solutions of the one-dimensional problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, measured via appropriate metrics for the cluster width and compactness. We verify the key results of the parametric analysis with simulations of the two-dimensional problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, suggesting chemotaxis is responsible for the cluster topology while matrix degradation is responsible for the speed of cluster formation. Our results also indicate that the nonlocal cell-to-cell adhesion term in our model, even though it initially causes cells to aggregate, is not sufficient to ensure clusters are stable over long time periods. Consequently, new modelling strategies for cell-to-cell adhesion are required to stabilise in silico clusters. We end the paper with a thorough discussion of promising, fruitful future modelling and experimental research perspectives.
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Affiliation(s)
- Chiara Villa
- School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK.
| | - Alf Gerisch
- Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
| | - Mark A J Chaplain
- School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
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7
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Villa C, Chaplain MAJ, Gerisch A, Lorenzi T. Mechanical Models of Pattern and Form in Biological Tissues: The Role of Stress-Strain Constitutive Equations. Bull Math Biol 2021; 83:80. [PMID: 34037880 PMCID: PMC8154836 DOI: 10.1007/s11538-021-00912-5] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2020] [Accepted: 05/11/2021] [Indexed: 11/25/2022]
Abstract
Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin-Voigt model of linear viscoelasticity to represent the stress-strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress-strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin-Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress-strain relation of the ECM, while no patterns are observed when the Kelvin-Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress-strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.
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Affiliation(s)
- Chiara Villa
- School of Mathematics and Statistics, University of St Andrews, St Andrews, 16 9SS UK
| | - Mark A. J. Chaplain
- School of Mathematics and Statistics, University of St Andrews, St Andrews, 16 9SS UK
| | - Alf Gerisch
- Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany
| | - Tommaso Lorenzi
- Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, 10129 Torino, Italy
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8
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Shan G, Chuan-shan H, Ming-zhu S, Xin Z. Meshwork pattern transformed from branching pattern in spherical shell domain. J Theor Biol 2018; 455:293-302. [DOI: 10.1016/j.jtbi.2018.07.037] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2017] [Revised: 07/25/2018] [Accepted: 07/27/2018] [Indexed: 10/28/2022]
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9
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3D hybrid modelling of vascular network formation. J Theor Biol 2016; 414:254-268. [PMID: 27890575 DOI: 10.1016/j.jtbi.2016.11.013] [Citation(s) in RCA: 47] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2016] [Revised: 09/06/2016] [Accepted: 11/16/2016] [Indexed: 12/13/2022]
Abstract
We develop an off-lattice, agent-based model to describe vasculogenesis, the de novo formation of blood vessels from endothelial progenitor cells during development. The endothelial cells that comprise our vessel network are viewed as linearly elastic spheres that move in response to the forces they experience. We distinguish two types of endothelial cells: vessel elements are contained within the network and tip cells are located at the ends of vessels. Tip cells move in response to mechanical forces caused by interactions with neighbouring vessel elements and the local tissue environment, chemotactic forces and a persistence force which accounts for their tendency to continue moving in the same direction. Vessel elements are subject to similar mechanical forces but are insensitive to chemotaxis. An angular persistence force representing interactions with the local tissue is introduced to stabilise buckling instabilities caused by cell proliferation. Only vessel elements proliferate, at rates which depend on their degree of stretch: elongated elements have increased rates of proliferation, and compressed elements have reduced rates. Following division, the fate of the new cell depends on the local mechanical environment: the probability of forming a new sprout is increased if the parent vessel is highly compressed and the probability of being incorporated into the parent vessel increased if the parent is stretched. Simulation results reveal that our hybrid model can reproduce the key qualitative features of vasculogenesis. Extensive parameter sensitivity analyses show that significant changes in network size and morphology are induced by varying the chemotactic sensitivity of tip cells, and the sensitivities of the proliferation rate and the sprouting probability to mechanical stretch. Varying the chemotactic sensitivity directly influences the directionality of the networks. The degree of branching, and thereby the density of the networks, is influenced by the sprouting probability. Glyphs that simultaneously depict several network properties are introduced to show how these and other network quantities change over time and also as model parameters vary. We also show how equivalent glyphs constructed from in vivo data could be used to discriminate between normal and tumour vasculature and, in the longer term, for model validation. We conclude that our biomechanical hybrid model can generate vascular networks that are qualitatively similar to those generated from in vitro and in vivo experiments.
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Arai S. Primary Phenomenon in the Network Formation of Endothelial Cells: Effect of Charge. Int J Mol Sci 2015; 16:29148-60. [PMID: 26690133 PMCID: PMC4691096 DOI: 10.3390/ijms161226149] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2015] [Revised: 11/11/2015] [Accepted: 11/26/2015] [Indexed: 11/16/2022] Open
Abstract
Blood vessels are essential organs that are involved in the supply of nutrients and oxygen and play an important role in regulating the body's internal environment, including pH, body temperature, and water homeostasis. Many studies have examined the formation of networks of endothelial cells. The results of these studies have revealed that vascular endothelial growth factor (VEGF) affects the interactions of these cells and modulates the network structure. Though almost all previous simulation studies have assumed that the chemoattractant VEGF is present before network formation, vascular endothelial cells secrete VEGF only after the cells bind to the substrate. This suggests VEGF is not essential for vasculogenesis especially at the early stage. Using a simple experiment, we find chain-like structures which last quite longer than it is expected, unless the energetically stable cluster should be compact. Using a purely physical model and simulation, we find that the hydrodynamic interaction retard the compaction of clusters and that the chains are stabilized through the effects of charge. The charge at the surface of the cells affect the interparticle potential, and the resulting repulsive forces prevent the chains from folding. The ions surrounding the cells may also be involved in this process.
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Affiliation(s)
- Shunto Arai
- Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan.
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11
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Flegg JA, Menon SN, Maini PK, McElwain DLS. On the mathematical modeling of wound healing angiogenesis in skin as a reaction-transport process. Front Physiol 2015; 6:262. [PMID: 26483695 PMCID: PMC4588694 DOI: 10.3389/fphys.2015.00262] [Citation(s) in RCA: 44] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2015] [Accepted: 09/04/2015] [Indexed: 11/13/2022] Open
Abstract
Over the last 30 years, numerous research groups have attempted to provide mathematical descriptions of the skin wound healing process. The development of theoretical models of the interlinked processes that underlie the healing mechanism has yielded considerable insight into aspects of this critical phenomenon that remain difficult to investigate empirically. In particular, the mathematical modeling of angiogenesis, i.e., capillary sprout growth, has offered new paradigms for the understanding of this highly complex and crucial step in the healing pathway. With the recent advances in imaging and cell tracking, the time is now ripe for an appraisal of the utility and importance of mathematical modeling in wound healing angiogenesis research. The purpose of this review is to pedagogically elucidate the conceptual principles that have underpinned the development of mathematical descriptions of wound healing angiogenesis, specifically those that have utilized a continuum reaction-transport framework, and highlight the contribution that such models have made toward the advancement of research in this field. We aim to draw attention to the common assumptions made when developing models of this nature, thereby bringing into focus the advantages and limitations of this approach. A deeper integration of mathematical modeling techniques into the practice of wound healing angiogenesis research promises new perspectives for advancing our knowledge in this area. To this end we detail several open problems related to the understanding of wound healing angiogenesis, and outline how these issues could be addressed through closer cross-disciplinary collaboration.
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Affiliation(s)
- Jennifer A Flegg
- School of Mathematical Sciences, Monash University Melbourne, VIC, Australia
| | | | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford Oxford, UK
| | - D L Sean McElwain
- Institute of Health and Biomedical Innovation and School of Mathematical Sciences, Queensland University of Technology Brisbane, QLD, Australia
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12
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Dyson RJ, Green JEF, Whiteley JP, Byrne HM. An investigation of the influence of extracellular matrix anisotropy and cell–matrix interactions on tissue architecture. J Math Biol 2015; 72:1775-809. [DOI: 10.1007/s00285-015-0927-7] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2014] [Revised: 05/21/2015] [Indexed: 12/25/2022]
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13
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Spill F, Guerrero P, Alarcon T, Maini PK, Byrne HM. Mesoscopic and continuum modelling of angiogenesis. J Math Biol 2015; 70:485-532. [PMID: 24615007 PMCID: PMC5320864 DOI: 10.1007/s00285-014-0771-1] [Citation(s) in RCA: 48] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2013] [Revised: 01/14/2014] [Indexed: 10/25/2022]
Abstract
Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.
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Affiliation(s)
- F Spill
- OCCAM, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK,
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14
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Köhn-Luque A, de Back W, Yamaguchi Y, Yoshimura K, Herrero MA, Miura T. Dynamics of VEGF matrix-retention in vascular network patterning. Phys Biol 2013; 10:066007. [DOI: 10.1088/1478-3975/10/6/066007] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
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15
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Scianna M, Bell C, Preziosi L. A review of mathematical models for the formation of vascular networks. J Theor Biol 2013; 333:174-209. [DOI: 10.1016/j.jtbi.2013.04.037] [Citation(s) in RCA: 108] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2012] [Revised: 02/15/2013] [Accepted: 04/30/2013] [Indexed: 02/08/2023]
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Criaco D, Dolfin M, Restuccia L. Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2013; 10:59-73. [PMID: 23311362 DOI: 10.3934/mbe.2013.10.59] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
In a previous paper a mathematical model was developed for the dynamics of activation and clonal expansion of T cells during the immune response to a single type of antigen challenge, constructed phenomenologically in the macroscopic framework of a thermodynamic theory of continuum mechanics for reacting and proliferating fluid mixtures. The present contribution deals with approximate smooth solutions, called asymptotic waves, of the system of PDEs describing the introduced model, obtained using a suitable perturbative method. In particular, in the one-dimensional case, after deriving the expression of the velocity along the characteristic rays and the equation of the wave front, the transport equation for the first perturbation term of the asymptotic solution is obtained. Finally, it is shown that this transport equation can be reduced to an equation similar to Burgers equation.
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Affiliation(s)
- D Criaco
- Department of Mathematics and Informatics, University of Messina, Viale F. Stagno d'Alcontres n.31, 98166 Messina, Italy.
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17
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Computational Modeling of Angiogenesis: Towards a Multi-Scale Understanding of Cell–Cell and Cell–Matrix Interactions. MECHANICAL AND CHEMICAL SIGNALING IN ANGIOGENESIS 2013. [DOI: 10.1007/978-3-642-30856-7_8] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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18
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Wyczalkowski MA, Chen Z, Filas BA, Varner VD, Taber LA. Computational models for mechanics of morphogenesis. ACTA ACUST UNITED AC 2012; 96:132-52. [PMID: 22692887 DOI: 10.1002/bdrc.21013] [Citation(s) in RCA: 53] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Abstract
In the developing embryo, tissues differentiate, deform, and move in an orchestrated manner to generate various biological shapes driven by the complex interplay between genetic, epigenetic, and environmental factors. Mechanics plays a key role in regulating and controlling morphogenesis, and quantitative models help us understand how various mechanical forces combine to shape the embryo. Models allow for the quantitative, unbiased testing of physical mechanisms, and when used appropriately, can motivate new experimentaldirections. This knowledge benefits biomedical researchers who aim to prevent and treat congenital malformations, as well as engineers working to create replacement tissues in the laboratory. In this review, we first give an overview of fundamental mechanical theories for morphogenesis, and then focus on models for specific processes, including pattern formation, gastrulation, neurulation, organogenesis, and wound healing. The role of mechanical feedback in development is also discussed. Finally, some perspectives aregiven on the emerging challenges in morphomechanics and mechanobiology.
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19
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Tosin A. Initial/boundary-value problems of tumor growth within a host tissue. J Math Biol 2012; 66:163-202. [PMID: 22290313 DOI: 10.1007/s00285-012-0505-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2011] [Revised: 12/20/2011] [Indexed: 01/16/2023]
Abstract
This paper concerns multiphase models of tumor growth in interaction with a surrounding tissue, taking into account also the interplay with diffusible nutrients feeding the cells. Models specialize in nonlinear systems of possibly degenerate parabolic equations, which include phenomenological terms related to specific cell functions. The paper discusses general modeling guidelines for such terms, as well as for initial and boundary conditions, aiming at both biological consistency and mathematical robustness of the resulting problems. Particularly, it addresses some qualitative properties such as a priori non-negativity, boundedness, and uniqueness of the solutions. Existence of the solutions is studied in the one-dimensional time-independent case.
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Affiliation(s)
- Andrea Tosin
- Istituto per le Applicazioni del Calcolo M. Picone, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Rome, Italy.
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20
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Scianna M. A Multiscale Hybrid Model for Pro-angiogenic Calcium Signals in a Vascular Endothelial Cell. Bull Math Biol 2011; 74:1253-91. [DOI: 10.1007/s11538-011-9695-8] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2010] [Accepted: 09/06/2011] [Indexed: 01/07/2023]
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21
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Early embryonic vascular patterning by matrix-mediated paracrine signalling: a mathematical model study. PLoS One 2011; 6:e24175. [PMID: 21949696 PMCID: PMC3176223 DOI: 10.1371/journal.pone.0024175] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2011] [Accepted: 08/01/2011] [Indexed: 11/19/2022] Open
Abstract
During embryonic vasculogenesis, endothelial precursor cells of mesodermal origin known as angioblasts assemble into a characteristic network pattern. Although a considerable amount of markers and signals involved in this process have been identified, the mechanisms underlying the coalescence of angioblasts into this reticular pattern remain unclear. Various recent studies hypothesize that autocrine regulation of the chemoattractant vascular endothelial growth factor (VEGF) is responsible for the formation of vascular networks in vitro. However, the autocrine regulation hypothesis does not fit well with reported data on in vivo early vascular development. In this study, we propose a mathematical model based on the alternative assumption that endodermal VEGF signalling activity, having a paracrine effect on adjacent angioblasts, is mediated by its binding to the extracellular matrix (ECM). Detailed morphometric analysis of simulated networks and images obtained from in vivo quail embryos reveals the model mimics the vascular patterns with high accuracy. These results show that paracrine signalling can result in the formation of fine-grained cellular networks when mediated by angioblast-produced ECM. This lends additional support to the theory that patterning during early vascular development in the vertebrate embryo is regulated by paracrine signalling.
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22
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Spatial organization of mesenchymal stem cells in vitro--results from a new individual cell-based model with podia. PLoS One 2011; 6:e21960. [PMID: 21760935 PMCID: PMC3132757 DOI: 10.1371/journal.pone.0021960] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2011] [Accepted: 06/15/2011] [Indexed: 11/19/2022] Open
Abstract
Therapeutic application of mesenchymal stem cells (MSC) requires their extensive in vitro expansion. MSC in culture typically grow to confluence within a few weeks. They show spindle-shaped fibroblastoid morphology and align to each other in characteristic spatial patterns at high cell density. We present an individual cell-based model (IBM) that is able to quantitatively describe the spatio-temporal organization of MSC in culture. Our model substantially improves on previous models by explicitly representing cell podia and their dynamics. It employs podia-generated forces for cell movement and adjusts cell behavior in response to cell density. At the same time, it is simple enough to simulate thousands of cells with reasonable computational effort. Experimental sheep MSC cultures were monitored under standard conditions. Automated image analysis was used to determine the location and orientation of individual cells. Our simulations quantitatively reproduced the observed growth dynamics and cell-cell alignment assuming cell density-dependent proliferation, migration, and morphology. In addition to cell growth on plain substrates our model captured cell alignment on micro-structured surfaces. We propose a specific surface micro-structure that according to our simulations can substantially enlarge cell culture harvest. The 'tool box' of cell migratory behavior newly introduced in this study significantly enhances the bandwidth of IBM. Our approach is capable of accommodating individual cell behavior and collective cell dynamics of a variety of cell types and tissues in computational systems biology.
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Danino T, Volfson D, Bhatia SN, Tsimring L, Hasty J. In-silico patterning of vascular mesenchymal cells in three dimensions. PLoS One 2011; 6:e20182. [PMID: 21633504 PMCID: PMC3102094 DOI: 10.1371/journal.pone.0020182] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2010] [Accepted: 04/27/2011] [Indexed: 01/24/2023] Open
Abstract
Cells organize in complex three-dimensional patterns by interacting with proteins along with the surrounding extracellular matrix. This organization provides the mechanical and chemical cues that ultimately influence a cell's differentiation and function. Here, we computationally investigate the pattern formation process of vascular mesenchymal cells arising from their interaction with Bone Morphogenic Protein-2 (BMP-2) and its inhibitor, Matrix Gla Protein (MGP). Using a first-principles approach, we derive a reaction-diffusion model based on the biochemical interactions of BMP-2, MGP and cells. Simulations of the model exhibit a wide variety of three-dimensional patterns not observed in a two-dimensional analysis. We demonstrate the emergence of three types of patterns: spheres, tubes, and sheets, and show that the patterns can be tuned by modifying parameters in the model such as the degradation rates of proteins and chemotactic coefficient of cells. Our model may be useful for improved engineering of three-dimensional tissue structures as well as for understanding three dimensional microenvironments in developmental processes.
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Affiliation(s)
- Tal Danino
- Department of Bioengineering, University of California San Diego, La Jolla, California, United States of America
| | - Dmitri Volfson
- Department of Bioengineering, University of California San Diego, La Jolla, California, United States of America
| | - Sangeeta N. Bhatia
- Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States of America
| | - Lev Tsimring
- Biocircuits Institute, University of California San Diego, La Jolla, California, United States of America
| | - Jeff Hasty
- Department of Bioengineering, University of California San Diego, La Jolla, California, United States of America
- Biocircuits Institute, University of California San Diego, La Jolla, California, United States of America
- Molecular Biology Section, Division of Biological Science, University of California San Diego, La Jolla, California, United States of America
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24
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A multiscale hybrid approach for vasculogenesis and related potential blocking therapies. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2011; 106:450-62. [PMID: 21300081 DOI: 10.1016/j.pbiomolbio.2011.01.004] [Citation(s) in RCA: 46] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2010] [Revised: 01/25/2011] [Accepted: 01/27/2011] [Indexed: 12/16/2022]
Abstract
Solid tumors must recruit and form new blood vessels for maintenance, growth and detachments of metastases. Discovering drugs that block malignant angiogenesis is thus an important approach in cancer treatment and has given rise to multiple in vitro and in silico models. The present hybrid individual cell-based model incorporates some underlying biochemical events relating more closely the classical Cellular Potts Model (CPM) parameters to subcellular mechanisms and to the activation of specific signaling pathways. The model spans the three fundamental biological levels: at the extracellular level a continuous model describes secretion, diffusion, uptake and decay of the autocrine VEGF; at the cellular level, an extended lattice CPM, based on a system energy reduction, reproduces cell dynamics such as migration, adhesion and chemotaxis; at the subcellular level, a set of reaction-diffusion equations describes a simplified VEGF-induced calcium-dependent intracellular pathway. The results agree with the known interplay between calcium signals and VEGF dynamics and with their role in malignant vasculogenesis. Moreover, the analysis of the link between the microscopic subcellular dynamics and the macroscopic cell behaviors confirms the efficiency of some pharmacological interventions that are currently in use and, more interestingly, proposes some new therapeutic approaches, that are counter-intuitive but potentially effective.
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25
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Scianna M, Munaron L. Multiscale model of tumor-derived capillary-like network formation. ACTA ACUST UNITED AC 2011. [DOI: 10.3934/nhm.2011.6.597] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
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26
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Fractal analysis of vascular networks: Insights from morphogenesis. J Theor Biol 2010; 262:614-33. [DOI: 10.1016/j.jtbi.2009.10.037] [Citation(s) in RCA: 93] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2009] [Revised: 10/20/2009] [Accepted: 10/29/2009] [Indexed: 11/17/2022]
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27
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Jones J. Characteristics of pattern formation and evolution in approximations of Physarum transport networks. ARTIFICIAL LIFE 2010; 16:127-153. [PMID: 20067403 DOI: 10.1162/artl.2010.16.2.16202] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
Most studies of pattern formation place particular emphasis on its role in the development of complex multicellular body plans. In simpler organisms, however, pattern formation is intrinsic to growth and behavior. Inspired by one such organism, the true slime mold Physarum polycephalum, we present examples of complex emergent pattern formation and evolution formed by a population of simple particle-like agents. Using simple local behaviors based on chemotaxis, the mobile agent population spontaneously forms complex and dynamic transport networks. By adjusting simple model parameters, maps of characteristic patterning are obtained. Certain areas of the parameter mapping yield particularly complex long term behaviors, including the circular contraction of network lacunae and bifurcation of network paths to maintain network connectivity. We demonstrate the formation of irregular spots and labyrinthine and reticulated patterns by chemoattraction. Other Turing-like patterning schemes were obtained by using chemorepulsion behaviors, including the self-organization of regular periodic arrays of spots, and striped patterns. We show that complex pattern types can be produced without resorting to the hierarchical coupling of reaction-diffusion mechanisms. We also present network behaviors arising from simple pre-patterning cues, giving simple examples of how the emergent pattern formation processes evolve into networks with functional and quasi-physical properties including tensionlike effects, network minimization behavior, and repair to network damage. The results are interpreted in relation to classical theories of biological pattern formation in natural systems, and we suggest mechanisms by which emergent pattern formation processes may be used as a method for spatially represented unconventional computation.
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Affiliation(s)
- Jeff Jones
- Centre for Unconventional Computing, University of the West of England, Bristol, BS16 1QY, UK.
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28
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Lowengrub JS, Frieboes HB, Jin F, Chuang YL, Li X, Macklin P, Wise SM, Cristini V. Nonlinear modelling of cancer: bridging the gap between cells and tumours. NONLINEARITY 2010; 23:R1-R9. [PMID: 20808719 PMCID: PMC2929802 DOI: 10.1088/0951-7715/23/1/r01] [Citation(s) in RCA: 227] [Impact Index Per Article: 15.1] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.
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Affiliation(s)
- J S Lowengrub
- Department of Biomedical Engineering, Center for Mathematical and Computational Biology, University of California at Irvine, Irvine, CA 92697, USA
| | - H B Frieboes
- School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA
- Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
| | - F Jin
- School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA
- Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
| | - Y-L Chuang
- School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA
| | - X Li
- Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
| | - P Macklin
- School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA
| | - S M Wise
- Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
| | - V Cristini
- School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA
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29
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Billy F, Ribba B, Saut O, Morre-Trouilhet H, Colin T, Bresch D, Boissel JP, Grenier E, Flandrois JP. A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J Theor Biol 2009; 260:545-62. [DOI: 10.1016/j.jtbi.2009.06.026] [Citation(s) in RCA: 51] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2009] [Revised: 06/16/2009] [Accepted: 06/27/2009] [Indexed: 10/20/2022]
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30
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Scianna M, Merks RM, Preziosi L, Medico E. Individual cell-based models of cell scatter of ARO and MLP-29 cells in response to hepatocyte growth factor. J Theor Biol 2009; 260:151-60. [DOI: 10.1016/j.jtbi.2009.05.017] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2008] [Revised: 05/13/2009] [Accepted: 05/13/2009] [Indexed: 11/30/2022]
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31
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Matzavinos A, Kao CY, Green JEF, Sutradhar A, Miller M, Friedman A. Modeling oxygen transport in surgical tissue transfer. Proc Natl Acad Sci U S A 2009; 106:12091-6. [PMID: 19597143 PMCID: PMC2715514 DOI: 10.1073/pnas.0905037106] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2008] [Indexed: 11/18/2022] Open
Abstract
Reconstructive microsurgery is a clinical technique used to transfer large amounts of a patient's tissue from one location used to another in order to restore physical deformities caused by trauma, tumors, or congenital abnormalities. The trend in this field is to transfer tissue using increasingly smaller blood vessels, which decreases problems associated with tissue harvest but increases the possibility that blood supply to the transferred tissue may not be adequate for healing. It would thus be helpful to surgeons to understand the relationship between the tissue volume and blood vessel diameter to ensure success in these operations. As a first step towards addressing this question, we present a simple mathematical model that might be used to predict successful tissue transfer based on blood vessel diameter, tissue volume, and oxygen delivery.
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Affiliation(s)
| | - Chiu-Yen Kao
- Department of Mathematics, Ohio State University, Columbus, OH 43210
- Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210; and
| | - J. Edward F. Green
- Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210; and
| | - Alok Sutradhar
- Division of Plastic Surgery, Ohio State University Medical Center, Columbus, OH 43210
| | - Michael Miller
- Division of Plastic Surgery, Ohio State University Medical Center, Columbus, OH 43210
| | - Avner Friedman
- Department of Mathematics, Ohio State University, Columbus, OH 43210
- Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210; and
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32
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Merks RMH, Perryn ED, Shirinifard A, Glazier JA. Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth. PLoS Comput Biol 2008; 4:e1000163. [PMID: 18802455 PMCID: PMC2528254 DOI: 10.1371/journal.pcbi.1000163] [Citation(s) in RCA: 129] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2006] [Accepted: 07/18/2008] [Indexed: 12/16/2022] Open
Abstract
Blood vessels form either when dispersed endothelial cells (the cells lining the inner walls of fully formed blood vessels) organize into a vessel network (vasculogenesis), or by sprouting or splitting of existing blood vessels (angiogenesis). Although they are closely related biologically, no current model explains both phenomena with a single biophysical mechanism. Most computational models describe sprouting at the level of the blood vessel, ignoring how cell behavior drives branch splitting during sprouting. We present a cell-based, Glazier-Graner-Hogeweg model (also called Cellular Potts Model) simulation of the initial patterning before the vascular cords form lumens, based on plausible behaviors of endothelial cells. The endothelial cells secrete a chemoattractant, which attracts other endothelial cells. As in the classic Keller-Segel model, chemotaxis by itself causes cells to aggregate into isolated clusters. However, including experimentally observed VE-cadherin-mediated contact inhibition of chemotaxis in the simulation causes randomly distributed cells to organize into networks and cell aggregates to sprout, reproducing aspects of both de novo and sprouting blood-vessel growth. We discuss two branching instabilities responsible for our results. Cells at the surfaces of cell clusters attempting to migrate to the centers of the clusters produce a buckling instability. In a model variant that eliminates the surface-normal force, a dissipative mechanism drives sprouting, with the secreted chemical acting both as a chemoattractant and as an inhibitor of pseudopod extension. Both mechanisms would also apply if force transmission through the extracellular matrix rather than chemical signaling mediated cell-cell interactions. The branching instabilities responsible for our results, which result from contact inhibition of chemotaxis, are both generic developmental mechanisms and interesting examples of unusual patterning instabilities.
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