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Simpson MJ, Murphy KM, McCue SW, Buenzli PR. Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion. ROYAL SOCIETY OPEN SCIENCE 2024; 11:240126. [PMID: 39076824 PMCID: PMC11286127 DOI: 10.1098/rsos.240126] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/22/2024] [Accepted: 02/22/2024] [Indexed: 07/31/2024]
Abstract
Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction-diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub.
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Affiliation(s)
- Matthew J. Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Keeley M. Murphy
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Scott W. McCue
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
| | - Pascal R. Buenzli
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
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2
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Rojek KO, Wrzos A, Żukowski S, Bogdan M, Lisicki M, Szymczak P, Guzowski J. Long-term day-by-day tracking of microvascular networks sprouting in fibrin gels: From detailed morphological analyses to general growth rules. APL Bioeng 2024; 8:016106. [PMID: 38327714 PMCID: PMC10849774 DOI: 10.1063/5.0180703] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2023] [Accepted: 01/04/2024] [Indexed: 02/09/2024] Open
Abstract
Understanding and controlling of the evolution of sprouting vascular networks remains one of the basic challenges in tissue engineering. Previous studies on the vascularization dynamics have typically focused only on the phase of intense growth and often lacked spatial control over the initial cell arrangement. Here, we perform long-term day-by-day analysis of tens of isolated microvasculatures sprouting from endothelial cell-coated spherical beads embedded in an external fibrin gel. We systematically study the topological evolution of the sprouting networks over their whole lifespan, i.e., for at least 14 days. We develop a custom image analysis toolkit and quantify (i) the overall length and area of the sprouts, (ii) the distributions of segment lengths and branching angles, and (iii) the average number of branch generations-a measure of network complexity. We show that higher concentrations of vascular endothelial growth factor (VEGF) lead to earlier sprouting and more branched networks, yet without significantly affecting the speed of growth of individual sprouts. We find that the mean branching angle is weakly dependent on VEGF and typically in the range of 60°-75°, suggesting that, by comparison with the available diffusion-limited growth models, the bifurcating tips tend to follow local VEGF gradients. At high VEGF concentrations, we observe exponential distributions of segment lengths, which signify purely stochastic branching. Our results-due to their high statistical relevance-may serve as a benchmark for predictive models, while our new image analysis toolkit, offering unique features and high speed of operation, could be exploited in future angiogenic drug tests.
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Affiliation(s)
- Katarzyna O. Rojek
- Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland
| | - Antoni Wrzos
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland
| | | | - Michał Bogdan
- Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland
| | - Maciej Lisicki
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland
| | - Piotr Szymczak
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland
| | - Jan Guzowski
- Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland
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3
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Terragni F, Martinson WD, Carretero M, Maini PK, Bonilla LL. Soliton approximation in continuum models of leader-follower behavior. Phys Rev E 2023; 108:054407. [PMID: 38115402 DOI: 10.1103/physreve.108.054407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2023] [Accepted: 10/17/2023] [Indexed: 12/21/2023]
Abstract
Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations (PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equations at leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.
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Affiliation(s)
- F Terragni
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
- Department of Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
| | - W D Martinson
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - M Carretero
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
- Department of Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
| | - P K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - L L Bonilla
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
- Department of Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
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Flandoli F, Leocata M, Ricci C. The Mathematical modeling of Cancer growth and angiogenesis by an individual based interacting system. J Theor Biol 2023; 562:111432. [PMID: 36746298 DOI: 10.1016/j.jtbi.2023.111432] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Revised: 01/23/2023] [Accepted: 01/26/2023] [Indexed: 02/07/2023]
Abstract
We present a mathematical model for the complex system for the growth of a solid tumor. The system embeds proliferation of cells depending on the surrounding oxygen field, hypoxia caused by insufficient oxygen when the tumor reaches a certain size, consequent VEGF release and angiogenic new vasculature growth, re-oxygenation of the tumor and subsequent tumor growth restart. Specifically cancerous cells are represented by individual units, interacting as proliferating particles of a solid body, oxygen, and VEGF are fields with a source and a sink, and new angiogenic vasculature is described by a network of growing curves. The model, as shown by numerical simulations, captures both the time-evolution of the tumor growth before and after angiogenesis and its spatial properties, with different distribution of proliferating and hypoxic cells in the external and deep layers of the tumor, and the spatial structure of the angiogenic network. The microscopic description of the growth opens the possibility of tuning the model to patient-specific scenarios.
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Affiliation(s)
- Franco Flandoli
- Scuola Normale Superiore, P.za dei Cavalieri, 7, Pisa, 56126, Italy.
| | - Marta Leocata
- Scuola Normale Superiore, P.za dei Cavalieri, 7, Pisa, 56126, Italy.
| | - Cristiano Ricci
- University of Pisa, Department of Economics and Management, Via Cosimo Ridolfi, 10, Pisa, 56124, Italy.
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Notch signaling and taxis mechanisms regulate early stage angiogenesis: A mathematical and computational model. PLoS Comput Biol 2020; 16:e1006919. [PMID: 31986145 PMCID: PMC7021322 DOI: 10.1371/journal.pcbi.1006919] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2019] [Revised: 02/14/2020] [Accepted: 10/16/2019] [Indexed: 12/20/2022] Open
Abstract
During angiogenesis, new blood vessels sprout and grow from existing ones. This process plays a crucial role in organ development and repair, in wound healing and in numerous pathological processes such as cancer progression or diabetes. Here, we present a mathematical model of early stage angiogenesis that permits exploration of the relative importance of mechanical, chemical and cellular cues. Endothelial cells proliferate and move over an extracellular matrix by following external gradients of Vessel Endothelial Growth Factor, adhesion and stiffness, which are incorporated to a Cellular Potts model with a finite element description of elasticity. The dynamics of Notch signaling involving Delta-4 and Jagged-1 ligands determines tip cell selection and vessel branching. Through their production rates, competing Jagged-Notch and Delta-Notch dynamics determine the influence of lateral inhibition and lateral induction on the selection of cellular phenotypes, branching of blood vessels, anastomosis (fusion of blood vessels) and angiogenesis velocity. Anastomosis may be favored or impeded depending on the mechanical configuration of strain vectors in the ECM near tip cells. Numerical simulations demonstrate that increasing Jagged production results in pathological vasculatures with thinner and more abundant vessels, which can be compensated by augmenting the production of Delta ligands. Angiogenesis is the process by which new blood vessels grow from existing ones. This process plays a crucial role in organ development, in wound healing and in numerous pathological processes such as cancer growth or in diabetes. Angiogenesis is a complex, multi-step and well regulated process where biochemistry and physics are intertwined. The process entails signaling in vessel cells being driven by both chemical and mechanical mechanisms that result in vascular cell movement, deformation and proliferation. Mathematical models have the ability to bring together these mechanisms in order to explore their relative relevance in vessel growth. Here, we present a mathematical model of early stage angiogenesis that is able to explore the role of biochemical signaling and tissue mechanics. We use this model to unravel the regulating role of Jagged, Notch and Delta dynamics in vascular cells. These membrane proteins have an important part in determining the leading cell in each neo-vascular sprout. Numerical simulations demonstrate that increasing Jagged production results in pathological vasculatures with thinner and more abundant vessels, which can be compensated by augmenting the production of Delta ligands.
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Pillay S, Byrne HM, Maini PK. The impact of exclusion processes on angiogenesis models. J Math Biol 2018; 77:1721-1759. [PMID: 29511857 PMCID: PMC6280878 DOI: 10.1007/s00285-018-1214-1] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2017] [Revised: 01/26/2018] [Indexed: 11/28/2022]
Abstract
Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.
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Affiliation(s)
- Samara Pillay
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK.
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
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Bhattacharya P, Viceconti M. Multiscale modeling methods in biomechanics. WILEY INTERDISCIPLINARY REVIEWS. SYSTEMS BIOLOGY AND MEDICINE 2017; 9:e1375. [PMID: 28102563 PMCID: PMC5412936 DOI: 10.1002/wsbm.1375] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/04/2016] [Revised: 11/09/2016] [Accepted: 11/17/2016] [Indexed: 01/08/2023]
Abstract
More and more frequently, computational biomechanics deals with problems where the portion of physical reality to be modeled spans over such a large range of spatial and temporal dimensions, that it is impossible to represent it as a single space-time continuum. We are forced to consider multiple space-time continua, each representing the phenomenon of interest at a characteristic space-time scale. Multiscale models describe a complex process across multiple scales, and account for how quantities transform as we move from one scale to another. This review offers a set of definitions for this emerging field, and provides a brief summary of the most recent developments on multiscale modeling in biomechanics. Of all possible perspectives, we chose that of the modeling intent, which vastly affect the nature and the structure of each research activity. To the purpose we organized all papers reviewed in three categories: 'causal confirmation,' where multiscale models are used as materializations of the causation theories; 'predictive accuracy,' where multiscale modeling is aimed to improve the predictive accuracy; and 'determination of effect,' where multiscale modeling is used to model how a change at one scale manifests in an effect at another radically different space-time scale. Consistent with how the volume of computational biomechanics research is distributed across application targets, we extensively reviewed papers targeting the musculoskeletal and the cardiovascular systems, and covered only a few exemplary papers targeting other organ systems. The review shows a research subdomain still in its infancy, where causal confirmation papers remain the most common. WIREs Syst Biol Med 2017, 9:e1375. doi: 10.1002/wsbm.1375 For further resources related to this article, please visit the WIREs website.
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Affiliation(s)
- Pinaki Bhattacharya
- Department of Mechanical Engineering and INSIGNEO Institute for in silico MedicineUniversity of SheffieldSheffieldUK
| | - Marco Viceconti
- Department of Mechanical Engineering and INSIGNEO Institute for in silico MedicineUniversity of SheffieldSheffieldUK
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Bonilla LL, Capasso V, Alvaro M, Carretero M, Terragni F. On the mathematical modelling of tumor-induced angiogenesis. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2017; 14:45-66. [PMID: 27879119 DOI: 10.3934/mbe.2017004] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, due to the strong coupling between the kinetic parameters of the relevant branching - growth - anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the ``propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.
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Affiliation(s)
- Luis L Bonilla
- G. Millan Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganes, Spain.
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Bonilla LL, Carretero M, Terragni F. Solitonlike attractor for blood vessel tip density in angiogenesis. Phys Rev E 2016; 94:062415. [PMID: 28085400 DOI: 10.1103/physreve.94.062415] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2016] [Indexed: 12/31/2022]
Abstract
Recently, numerical simulations of a stochastic model have shown that the density of vessel tips in tumor-induced angiogenesis adopts a solitonlike profile [Sci. Rep. 6, 31296 (2016)2045-232210.1038/srep31296]. In this work, we derive and solve the equations for the soliton collective coordinates that indicate how the soliton adapts its shape and velocity to varying chemotaxis and diffusion. The vessel tip density can be reconstructed from the soliton formulas. While the stochastic model exhibits large fluctuations, we show that the location of the maximum vessel tip density for different replicas follows closely the soliton peak position calculated either by ensemble averages or by solving an alternative deterministic description of the density. The simple soliton collective coordinate equations may also be used to ascertain the response of the vessel network to changes in the parameters and thus to control it.
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Affiliation(s)
- L L Bonilla
- G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
| | - M Carretero
- G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
| | - F Terragni
- G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
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Bonilla LL, Carretero M, Terragni F, Birnir B. Soliton driven angiogenesis. Sci Rep 2016; 6:31296. [PMID: 27503562 PMCID: PMC4977512 DOI: 10.1038/srep31296] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2016] [Accepted: 07/12/2016] [Indexed: 11/19/2022] Open
Abstract
Angiogenesis is a multiscale process by which blood vessels grow from existing ones and carry oxygen to distant organs. Angiogenesis is essential for normal organ growth and wounded tissue repair but it may also be induced by tumours to amplify their own growth. Mathematical and computational models contribute to understanding angiogenesis and developing anti-angiogenic drugs, but most work only involves numerical simulations and analysis has lagged. A recent stochastic model of tumour-induced angiogenesis including blood vessel branching, elongation, and anastomosis captures some of its intrinsic multiscale structures, yet allows one to extract a deterministic integropartial differential description of the vessel tip density. Here we find that the latter advances chemotactically towards the tumour driven by a soliton (similar to the famous Korteweg-de Vries soliton) whose shape and velocity change slowly. Analysing these collective coordinates paves the way for controlling angiogenesis through the soliton, the engine that drives this process.
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Affiliation(s)
- L L Bonilla
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics and Department of Materials Science &Engineering, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
| | - M Carretero
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics and Department of Materials Science &Engineering, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
| | - F Terragni
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics and Department of Materials Science &Engineering, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
| | - B Birnir
- Center for Complex and Nonlinear Science and Department of Mathematics, University of California at Santa Barbara, USA
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