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Ben-Ami Y, Pitt-Francis JM, Maini PK, Byrne HM. Using a probabilistic approach to derive a two-phase model of flow-induced cell migration. Biophys J 2024; 123:799-813. [PMID: 38414238 PMCID: PMC10995429 DOI: 10.1016/j.bpj.2024.02.017] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2023] [Revised: 01/22/2024] [Accepted: 02/21/2024] [Indexed: 02/29/2024] Open
Abstract
Interstitial fluid flow is a feature of many solid tumors. In vitro experiments have shown that such fluid flow can direct tumor cell movement upstream or downstream depending on the balance between the competing mechanisms of tensotaxis (cell migration up stress gradients) and autologous chemotaxis (downstream cell movement in response to flow-induced gradients of self-secreted chemoattractants). In this work we develop a probabilistic-continuum, two-phase model for cell migration in response to interstitial flow. We use a kinetic description for the cell velocity probability density function, and model the flow-dependent mechanical and chemical stimuli as forcing terms that bias cell migration upstream and downstream. Using velocity-space averaging, we reformulate the model as a system of continuum equations for the spatiotemporal evolution of the cell volume fraction and flux in response to forcing terms that depend on the local direction and magnitude of the mechanochemical cues. We specialize our model to describe a one-dimensional cell layer subject to fluid flow. Using a combination of numerical simulations and asymptotic analysis, we delineate the parameter regime where transitions from downstream to upstream cell migration occur. As has been observed experimentally, the model predicts downstream-oriented chemotactic migration at low cell volume fractions, and upstream-oriented tensotactic migration at larger volume fractions. We show that the locus of the critical volume fraction, at which the system transitions from downstream to upstream migration, is dominated by the ratio of the rate of chemokine secretion and advection. Our model also predicts that, because the tensotactic stimulus depends strongly on the cell volume fraction, upstream, tensotaxis-dominated migration occurs only transiently when the cells are initially seeded, and transitions to downstream, chemotaxis-dominated migration occur at later times due to the dispersive effect of cell diffusion.
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Affiliation(s)
- Yaron Ben-Ami
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK.
| | | | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK; Ludwig Institute for Cancer Research, University of Oxford, Oxford, UK
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2
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Köry J, Narain V, Stolz BJ, Kaeppler J, Markelc B, Muschel RJ, Maini PK, Pitt-Francis JM, Byrne HM. Enhanced perfusion following exposure to radiotherapy: A theoretical investigation. PLoS Comput Biol 2024; 20:e1011252. [PMID: 38363799 PMCID: PMC10903964 DOI: 10.1371/journal.pcbi.1011252] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2023] [Revised: 02/29/2024] [Accepted: 01/23/2024] [Indexed: 02/18/2024] Open
Abstract
Tumour angiogenesis leads to the formation of blood vessels that are structurally and spatially heterogeneous. Poor blood perfusion, in conjunction with increased hypoxia and oxygen heterogeneity, impairs a tumour's response to radiotherapy. The optimal strategy for enhancing tumour perfusion remains unclear, preventing its regular deployment in combination therapies. In this work, we first identify vascular architectural features that correlate with enhanced perfusion following radiotherapy, using in vivo imaging data from vascular tumours. Then, we present a novel computational model to determine the relationship between these architectural features and blood perfusion in silico. If perfusion is defined to be the proportion of vessels that support blood flow, we find that vascular networks with small mean diameters and large numbers of angiogenic sprouts show the largest increases in perfusion post-irradiation for both biological and synthetic tumours. We also identify cases where perfusion increases due to the pruning of hypoperfused vessels, rather than blood being rerouted. These results indicate the importance of considering network composition when determining the optimal irradiation strategy. In the future, we aim to use our findings to identify tumours that are good candidates for perfusion enhancement and to improve the efficacy of combination therapies.
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Affiliation(s)
- Jakub Köry
- School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Vedang Narain
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Bernadette J. Stolz
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
- Laboratory for Topology and Neuroscience, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
| | - Jakob Kaeppler
- Cancer Research UK and MRC Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, United Kingdom
| | - Bostjan Markelc
- Cancer Research UK and MRC Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, United Kingdom
- Department of Experimental Oncology, Institute of Oncology Ljubljana, Ljubljana, Slovenia
| | - Ruth J. Muschel
- Cancer Research UK and MRC Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, United Kingdom
| | - Philip K. Maini
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Joe M. Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
| | - Helen M. Byrne
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
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3
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Celora GL, Bader SB, Hammond EM, Maini PK, Pitt-Francis JM, Byrne HM. Corrigendum to "A DNA-structured mathematical model of cell-cycle progression in cyclic hypoxia" [J. Theor. Biol. 545 (2022) 111104]. J Theor Biol 2023:111507. [PMID: 37286445 DOI: 10.1016/j.jtbi.2023.111507] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Affiliation(s)
| | - Samuel B Bader
- Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, UK
| | - Ester M Hammond
- Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, UK
| | | | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford, UK
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Ben-Ami Y, Atkinson GW, Pitt-Francis JM, Maini PK, Byrne HM. Structural Features of Microvascular Networks Trigger Blood Flow Oscillations. Bull Math Biol 2022; 84:85. [PMID: 35802265 PMCID: PMC9270315 DOI: 10.1007/s11538-022-01046-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2022] [Accepted: 06/22/2022] [Indexed: 11/29/2022]
Abstract
We analyse mathematical models in order to understand how microstructural features of vascular networks may affect blood flow dynamics, and to identify particular characteristics that promote the onset of self-sustained oscillations. By focusing on a simple three-node motif, we predict that network “redundancy”, in the form of a redundant vessel connecting two main flow-branches, together with differences in haemodynamic resistance in the branches, can promote the emergence of oscillatory dynamics. We use existing mathematical descriptions for blood rheology and haematocrit splitting at vessel branch-points to construct our flow model; we combine numerical simulations and stability analysis to study the dynamics of the three-node network and its relation to the system’s multiple steady-state solutions. While, for the case of equal inlet-pressure conditions, a “trivial” equilibrium solution with no flow in the redundant vessel always exists, we find that it is not stable when other, stable, steady-state attractors exist. In turn, these “nontrivial” steady-state solutions may undergo a Hopf bifurcation into an oscillatory state. We use the branch diameter ratio, together with the inlet haematocrit rate, to construct a two-parameter stability diagram that delineates regimes in which such oscillatory dynamics exist. We show that flow oscillations in this network geometry are only possible when the branch diameters are sufficiently different to allow for a sufficiently large flow in the redundant vessel, which acts as the driving force of the oscillations. These microstructural properties, which were found to promote oscillatory dynamics, could be used to explore sources of flow instability in biological microvascular networks.
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Affiliation(s)
- Y Ben-Ami
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK.
| | - G W Atkinson
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - J M Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford, UK
| | - P K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - H M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
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Celora GL, Bader SB, Hammond EM, Maini PK, Pitt-Francis JM, Byrne HM. DNA-structured mathematical model of cell-cycle progression in cyclic hypoxia. J Theor Biol 2022; 545:111104. [PMID: 35337794 DOI: 10.1016/j.jtbi.2022.111104] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2021] [Revised: 03/15/2022] [Accepted: 03/18/2022] [Indexed: 01/22/2023]
Abstract
New experimental data have shown how the periodic exposure of cells to low oxygen levels (i.e., cyclic hypoxia) impacts their progress through the cell-cycle. Cyclic hypoxia has been detected in tumours and linked to poor prognosis and treatment failure. While fluctuating oxygen environments can be reproduced in vitro, the range of oxygen cycles that can be tested is limited. By contrast, mathematical models can be used to predict the response to a wide range of cyclic dynamics. Accordingly, in this paper we develop a mechanistic model of the cell-cycle that can be combined with in vitro experiments, to better understand the link between cyclic hypoxia and cell-cycle dysregulation. A distinguishing feature of our model is the inclusion of impaired DNA synthesis and cell-cycle arrest due to periodic exposure to severely low oxygen levels. Our model decomposes the cell population into five compartments and a time-dependent delay accounts for the variability in the duration of the S phase which increases in severe hypoxia due to reduced rates of DNA synthesis. We calibrate our model against experimental data and show that it recapitulates the observed cell-cycle dynamics. We use the calibrated model to investigate the response of cells to oxygen cycles not yet tested experimentally. When the re-oxygenation phase is sufficiently long, our model predicts that cyclic hypoxia simply slows cell proliferation since cells spend more time in the S phase. On the contrary, cycles with short periods of re-oxygenation are predicted to lead to inhibition of proliferation, with cells arresting from the cell-cycle in the G2 phase. While model predictions on short time scales (about a day) are fairly accurate (i.e, confidence intervals are small), the predictions become more uncertain over longer periods. Hence, we use our model to inform experimental design that can lead to improved model parameter estimates and validate model predictions.
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Affiliation(s)
| | - Samuel B Bader
- Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, UK
| | - Ester M Hammond
- Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, UK
| | | | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford, UK
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6
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Köry J, Maini PK, Pitt-Francis JM, Byrne HM. Dependence of cell-free-layer width on rheological parameters: Combining empirical data on flow separation at microvascular bifurcations with geometrical considerations. Phys Rev E 2022; 105:014414. [PMID: 35193324 DOI: 10.1103/physreve.105.014414] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2021] [Accepted: 12/13/2021] [Indexed: 06/14/2023]
Abstract
When blood flows through vessel networks, red blood cells (RBCs) are typically concentrated close to the vessel center line, creating a lubrication layer near the vessel wall. As RBCs bind oxygen, the width of this cell-free layer (CFL) impacts not only the blood rheology inside the vasculature, but also oxygen delivery to the tissues they perfuse and, hence, their function. Existing attempts to relate the width of the CFL to the rheological properties of the blood and the geometrical properties of the vessel are based on an analysis of the forces acting on RBCs suspended in the blood. However, the complexity of interactions in the blood makes this a challenging task. Here, we propose an alternative, two-step approach to derive such a functional relationship. First, we extend widely accepted empirical fits describing the minimum flow fraction needed for RBCs to enter a daughter vessel downstream of a microvascular bifurcation so that it depends not only on the diameter and discharge haematocrit of the parent vessel, but also on its average shear rate. Second, we propose a simple geometrical model for the minimum flow fraction based on the cross-sectional blood flow profile within the parent vessel upstream of the bifurcation-considering uniform, parabolic, and blunt velocity profiles-and derive the leading-order approximation to this model for small CFL widths. By equating the functional relationships obtained using these two approaches, we derive expressions relating the CFL width to the vessel diameter, discharge haematocrit, and mean shear rate. The resulting expressions are in good agreement with available in vivo data and represent a promising basis for future research.
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Affiliation(s)
- Jakub Köry
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Joe M Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
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7
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Cooper FR, Baker RE, Bernabeu MO, Bordas R, Bowler L, Bueno-Orovio A, Byrne HM, Carapella V, Cardone-Noott L, Jonatha C, Dutta S, Evans BD, Fletcher AG, Grogan JA, Guo W, Harvey DG, Hendrix M, Kay D, Kursawe J, Maini PK, McMillan B, Mirams GR, Osborne JM, Pathmanathan P, Pitt-Francis JM, Robinson M, Rodriguez B, Spiteri RJ, Gavaghan DJ. Chaste: Cancer, Heart and Soft Tissue Environment. J Open Source Softw 2020; 5:1848. [PMID: 37192932 PMCID: PMC7614534 DOI: 10.21105/joss.01848] [Citation(s) in RCA: 32] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 05/18/2023]
Abstract
Chaste (Cancer, Heart And Soft Tissue Environment) is an open source simulation package for the numerical solution of mathematical models arising in physiology and biology. To date, Chaste development has been driven primarily by applications that include continuum modelling of cardiac electrophysiology ('Cardiac Chaste'), discrete cell-based modelling of soft tissues ('Cell-based Chaste'), and modelling of ventilation in lungs ('Lung Chaste'). Cardiac Chaste addresses the need for a high-performance, generic, and verified simulation framework for cardiac electrophysiology that is freely available to the scientific community. Cardiac chaste provides a software package capable of realistic heart simulations that is efficient, rigorously tested, and runs on HPC platforms. Cell-based Chaste addresses the need for efficient and verified implementations of cell-based modelling frameworks, providing a set of extensible tools for simulating biological tissues. Computational modelling, along with live imaging techniques, plays an important role in understanding the processes of tissue growth and repair. A wide range of cell-based modelling frameworks have been developed that have each been successfully applied in a range of biological applications. Cell-based Chaste includes implementations of the cellular automaton model, the cellular Potts model, cell-centre models with cell representations as overlapping spheres or Voronoi tessellations, and the vertex model. Lung Chaste addresses the need for a novel, generic and efficient lung modelling software package that is both tested and verified. It aims to couple biophysically-detailed models of airway mechanics with organ-scale ventilation models in a package that is freely available to the scientific community. Chaste is designed to be modular and extensible, providing libraries for common scientific computing infrastructure such as linear algebra operations, finite element meshes, and ordinary and partial differential equation solvers. This infrastructure is used by libraries for specific applications, such as continuum mechanics, cardiac models, and cell-based models. The software engineering techniques used to develop Chaste are intended to ensure code quality, re-usability and reliability. Primary applications of the software include cardiac and respiratory physiology, cancer and developmental biology.
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Affiliation(s)
- Fergus R Cooper
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Ruth E Baker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Miguel O Bernabeu
- Centre for Medical Informatics, Usher Institute, University of Edinburgh, Edinburgh, United Kingdom
| | - Rafel Bordas
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Louise Bowler
- Department of Computer Science, University of Oxford, Oxford, UK
| | | | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Valentina Carapella
- Division of Cardiovascular Medicine, Radcliffe Department of Medicine, University of Oxford, Oxford, UK
| | | | - Cooper Jonatha
- Research IT Services, University College London, London, UK
| | - Sara Dutta
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Benjamin D Evans
- Centre for Biomedical Modelling and Analysis, Living Systems Institute, University of Exeter, Exeter, UK
- School of Psychological Science, University of Bristol, Bristol, UK
| | - Alexander G Fletcher
- School of Mathematics & Statistics, University of Sheffield, Sheffield, UK
- Bateson Centre, University of Sheffield, Sheffield, UK
| | - James A Grogan
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Wenxian Guo
- Department of Computer Science, University of Saskatchewan, Canada
| | - Daniel G Harvey
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Maurice Hendrix
- Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
- Digital Research Service, University of Nottingham, Nottingham, UK
| | - David Kay
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Jochen Kursawe
- Faculty of Biology, Medicine and Health, University of Manchester, Manchester, UK
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Beth McMillan
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Gary R Mirams
- Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
| | - James M Osborne
- School of Mathematics and Statistics, University of Melbourne, Victoria, Australia
| | - Pras Pathmanathan
- Office of Science and Engineering Laboratories (OSEL), Center for Devices and Radiological Health (CDRH), U.S. Food and Drug Administration (FDA), Silver Spring, MD 20993, USA
| | | | - Martin Robinson
- Department of Computer Science, University of Oxford, Oxford, UK
| | - Blanca Rodriguez
- Department of Computer Science, University of Oxford, Oxford, UK
| | | | - David J Gavaghan
- Department of Computer Science, University of Oxford, Oxford, UK
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8
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Frost I, Smith WPJ, Mitri S, Millan AS, Davit Y, Osborne JM, Pitt-Francis JM, MacLean RC, Foster KR. Cooperation, competition and antibiotic resistance in bacterial colonies. ISME J 2018; 12:1582-1593. [PMID: 29563570 DOI: 10.1038/s41396-018-0090-4] [Citation(s) in RCA: 105] [Impact Index Per Article: 17.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2017] [Revised: 01/09/2018] [Accepted: 01/26/2018] [Indexed: 12/22/2022]
Abstract
Bacteria commonly live in dense and genetically diverse communities associated with surfaces. In these communities, competition for resources and space is intense, and yet we understand little of how this affects the spread of antibiotic-resistant strains. Here, we study interactions between antibiotic-resistant and susceptible strains using in vitro competition experiments in the opportunistic pathogen Pseudomonas aeruginosa and in silico simulations. Selection for intracellular resistance to streptomycin is very strong in colonies, such that resistance is favoured at very low antibiotic doses. In contrast, selection for extracellular resistance to carbenicillin is weak in colonies, and high doses of antibiotic are required to select for resistance. Manipulating the density and spatial structure of colonies reveals that this difference is partly explained by the fact that the local degradation of carbenicillin by β-lactamase-secreting cells protects neighbouring sensitive cells from carbenicillin. In addition, we discover a second unexpected effect: the inducible elongation of cells in response to carbenicillin allows sensitive cells to better compete for the rapidly growing colony edge. These combined effects mean that antibiotic treatment can select against antibiotic-resistant strains, raising the possibility of treatment regimes that suppress sensitive strains while limiting the rise of antibiotic resistance. We argue that the detailed study of bacterial interactions will be fundamental to understanding and overcoming antibiotic resistance.
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Affiliation(s)
- Isabel Frost
- Department of Zoology, University of Oxford, Oxford, OX1 3PS, UK.,Center for Disease Dynamics, Economics & Policy, New Delhi, 110020, India
| | - William P J Smith
- Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK
| | - Sara Mitri
- Département de Microbiologie Fondamentale (DMF), Université de Lausanne, Lausanne, 1015, Switzerland
| | - Alvaro San Millan
- Department of Microbiology, Hospital Universitario Ramón y Cajal (IRYCIS), Madrid, 28034, Spain
| | - Yohan Davit
- Institut de Mécanique des Fluides de Toulouse (IMFT)-Université de Toulouse, CNRS-INPT-UPS, Toulouse, France
| | - James M Osborne
- School of Mathematics and Statistics, University of Melbourne, Melbourne, VIC, 3010, Australia
| | - Joe M Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK
| | - R Craig MacLean
- Department of Zoology, University of Oxford, Oxford, OX1 3PS, UK.
| | - Kevin R Foster
- Department of Zoology, University of Oxford, Oxford, OX1 3PS, UK.
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Grogan JA, Connor AJ, Pitt-Francis JM, Maini PK, Byrne HM. The importance of geometry in the corneal micropocket angiogenesis assay. PLoS Comput Biol 2018; 14:e1006049. [PMID: 29522527 PMCID: PMC5862519 DOI: 10.1371/journal.pcbi.1006049] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2017] [Revised: 03/21/2018] [Accepted: 02/22/2018] [Indexed: 11/19/2022] Open
Abstract
The corneal micropocket angiogenesis assay is an experimental protocol for studying vessel network formation, or neovascularization, in vivo. The assay is attractive due to the ease with which the developing vessel network can be observed in the same animal over time. Measurements from the assay have been used in combination with mathematical modeling to gain insights into the mechanisms of angiogenesis. While previous modeling studies have adopted planar domains to represent the assay, the hemispherical shape of the cornea and asymmetric positioning of the angiogenic source can be seen to affect vascular patterning in experimental images. As such, we aim to better understand: i) how the geometry of the assay influences vessel network formation and ii) how to relate observations from planar domains to those in the hemispherical cornea. To do so, we develop a three-dimensional, off-lattice mathematical model of neovascularization in the cornea, using a spatially resolved representation of the assay for the first time. Relative to the detailed model, we predict that the adoption of planar geometries has a noticeable impact on vascular patterning, leading to increased vessel 'merging', or anastomosis, in particular when circular geometries are adopted. Significant differences in the dynamics of diffusible aniogenesis simulators are also predicted between different domains. In terms of comparing predictions across domains, the 'distance of the vascular front to the limbus' metric is found to have low sensitivity to domain choice, while metrics such as densities of tip cells and vessels and 'vascularized fraction' are sensitive to domain choice. Given the widespread adoption and attractive simplicity of planar tissue domains, both in silico and in vitro, the differences identified in the present study should prove useful in relating the results of previous and future theoretical studies of neovascularization to in vivo observations in the cornea.
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Affiliation(s)
- James A. Grogan
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Anthony J. Connor
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
| | - Joe M. Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Helen M. Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
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10
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Grogan JA, Connor AJ, Markelc B, Muschel RJ, Maini PK, Byrne HM, Pitt-Francis JM. Microvessel Chaste: An Open Library for Spatial Modeling of Vascularized Tissues. Biophys J 2017; 112:1767-1772. [PMID: 28494948 PMCID: PMC5425404 DOI: 10.1016/j.bpj.2017.03.036] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2016] [Revised: 02/22/2017] [Accepted: 03/27/2017] [Indexed: 11/29/2022] Open
Abstract
Spatial models of vascularized tissues are widely used in computational physiology. We introduce a software library for composing multiscale, multiphysics models for applications including tumor growth, angiogenesis, osteogenesis, coronary perfusion, and oxygen delivery. Composition of such models is time consuming, with many researchers writing custom software. Recent advances in imaging have produced detailed three-dimensional (3D) datasets of vascularized tissues at the scale of individual cells. To fully exploit such data there is an increasing need for software that allows user-friendly composition of efficient, 3D models of vascularized tissues, and comparison of predictions with in vivo or in vitro experiments and alternative computational formulations. Microvessel Chaste can be used to build simulations of vessel growth and adaptation in response to mechanical and chemical stimuli; intra- and extravascular transport of nutrients, growth factors and drugs; and cell proliferation in complex 3D geometries. In addition, it can be used to develop custom software for integrating modeling with experimental data processing workflows, facilitated by a comprehensive Python interface to solvers implemented in C++. This article links to two reproducible example problems, showing how the library can be used to build simulations of tumor growth and angiogenesis with realistic vessel networks.
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Affiliation(s)
- James A Grogan
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom.
| | - Anthony J Connor
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom; Department of Computer Science, University of Oxford, Oxford, United Kingdom
| | - Bostjan Markelc
- CRUK/MRC Oxford Institute for Radiation Oncology, University of Oxford, Oxford, United Kingdom
| | - Ruth J Muschel
- CRUK/MRC Oxford Institute for Radiation Oncology, University of Oxford, Oxford, United Kingdom
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Joe M Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
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11
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Grogan JA, Markelc B, Connor AJ, Muschel RJ, Pitt-Francis JM, Maini PK, Byrne HM. Predicting the Influence of Microvascular Structure On Tumor Response to Radiotherapy. IEEE Trans Biomed Eng 2017; 64:504-511. [PMID: 27623567 DOI: 10.1109/tbme.2016.2606563] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2024]
Abstract
OBJECTIVE The purpose of this study is to investigate how theoretical predictions of tumor response to radiotherapy (RT) depend on the morphology and spatial representation of the microvascular network. METHODS A hybrid multiscale model, which couples a cellular automaton model of tumor growth with a model for oxygen transport from blood vessels, is used to predict the viable fraction of cells following one week of simulated RT. Both artificial and biologically derived three-dimensional (3-D) vessel networks of well vascularized tumors are considered and predictions compared with 2-D descriptions. RESULTS For literature-derived values of the cellular oxygen consumption rate there is little difference in predicted viable fraction when 3-D network representations of biological or artificial vessel networks are employed. Different 2-D representations are shown to either over- or under-estimate viable fractions relative to the 3-D cases, with predictions based on point-wise descriptions shown to have greater sensitivity to vessel network morphology. CONCLUSION The predicted RT response is relatively insensitive to the morphology of the microvessel network when 3-D representations are adopted, however, sensitivity is greater in certain 2-D representations. SIGNIFICANCE By using realistic 3-D vessel network geometries this study shows that real and artificial network descriptions and assumptions of spatially uniform oxygen distributions lead to similar RT response predictions in relatively small tissue volumes. This suggests that either a more detailed description of oxygen transport in the microvasculature is required or that the oxygen enhancement ratio used in the well known linear-quadratic RT response model is relatively insensitive to microvascular structure.
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Affiliation(s)
- James A Grogan
- Mathematical Institute, University of Oxford, Oxford, U.K
| | - Bostjan Markelc
- Department of OncologyCRUK/MRC Oxford Institute for Radiation Oncology, University of Oxford
| | | | - Ruth J Muschel
- Department of OncologyCRUK/MRC Oxford Institute for Radiation Oncology, University of Oxford
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Osborne JM, Fletcher AG, Pitt-Francis JM, Maini PK, Gavaghan DJ. Comparing individual-based approaches to modelling the self-organization of multicellular tissues. PLoS Comput Biol 2017; 13:e1005387. [PMID: 28192427 PMCID: PMC5330541 DOI: 10.1371/journal.pcbi.1005387] [Citation(s) in RCA: 107] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2016] [Revised: 02/28/2017] [Accepted: 01/28/2017] [Indexed: 12/28/2022] Open
Abstract
The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.
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Affiliation(s)
- James M. Osborne
- School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, Australia
| | - Alexander G. Fletcher
- School of Mathematics and Statistics, University of Sheffield, Sheffield, United Kingdom
- Bateson Centre, University of Sheffield, Sheffield, United Kingdom
| | - Joe M. Pitt-Francis
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - David J. Gavaghan
- Department of Computer Science, University of Oxford, Oxford, United Kingdom
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