1
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Crossley RM, Painter KJ, Lorenzi T, Maini PK, Baker RE. Phenotypic switching mechanisms determine the structure of cell migration into extracellular matrix under the 'go-or-grow' hypothesis. Math Biosci 2024; 374:109240. [PMID: 38906525 DOI: 10.1016/j.mbs.2024.109240] [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: 01/14/2024] [Revised: 06/10/2024] [Accepted: 06/11/2024] [Indexed: 06/23/2024]
Abstract
A fundamental feature of collective cell migration is phenotypic heterogeneity which, for example, influences tumour progression and relapse. While current mathematical models often consider discrete phenotypic structuring of the cell population, in-line with the 'go-or-grow' hypothesis (Hatzikirou et al., 2012; Stepien et al., 2018), they regularly overlook the role that the environment may play in determining the cells' phenotype during migration. Comparing a previously studied volume-filling model for a homogeneous population of generalist cells that can proliferate, move and degrade extracellular matrix (ECM) (Crossley et al., 2023) to a novel model for a heterogeneous population comprising two distinct sub-populations of specialist cells that can either move and degrade ECM or proliferate, this study explores how different hypothetical phenotypic switching mechanisms affect the speed and structure of the invading cell populations. Through a continuum model derived from its individual-based counterpart, insights into the influence of the ECM and the impact of phenotypic switching on migrating cell populations emerge. Notably, specialist cell populations that cannot switch phenotype show reduced invasiveness compared to generalist cell populations, while implementing different forms of switching significantly alters the structure of migrating cell fronts. This key result suggests that the structure of an invading cell population could be used to infer the underlying mechanisms governing phenotypic switching.
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Affiliation(s)
- Rebecca M Crossley
- Mathematical Institute, University of Oxford, OX2 6GG, Oxford, United Kingdom.
| | - Kevin J Painter
- Dipartimento di Scienze, Progetto e Politiche del Territorio (DIST), Politecnico di Torino, 10129, Torino, Italy.
| | - Tommaso Lorenzi
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, 10129, Torino, Italy.
| | - Philip K Maini
- Mathematical Institute, University of Oxford, OX2 6GG, Oxford, United Kingdom.
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, OX2 6GG, Oxford, United Kingdom.
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2
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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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3
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Strobl MAR, Gallaher J, Robertson-Tessi M, West J, Anderson ARA. Treatment of evolving cancers will require dynamic decision support. Ann Oncol 2023; 34:867-884. [PMID: 37777307 PMCID: PMC10688269 DOI: 10.1016/j.annonc.2023.08.008] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2023] [Revised: 08/01/2023] [Accepted: 08/21/2023] [Indexed: 10/02/2023] Open
Abstract
Cancer research has traditionally focused on developing new agents, but an underexplored question is that of the dose and frequency of existing drugs. Based on the modus operandi established in the early days of chemotherapies, most drugs are administered according to predetermined schedules that seek to deliver the maximum tolerated dose and are only adjusted for toxicity. However, we believe that the complex, evolving nature of cancer requires a more dynamic and personalized approach. Chronicling the milestones of the field, we show that the impact of schedule choice crucially depends on processes driving treatment response and failure. As such, cancer heterogeneity and evolution dictate that a one-size-fits-all solution is unlikely-instead, each patient should be mapped to the strategy that best matches their current disease characteristics and treatment objectives (i.e. their 'tumorscape'). To achieve this level of personalization, we need mathematical modeling. In this perspective, we propose a five-step 'Adaptive Dosing Adjusted for Personalized Tumorscapes (ADAPT)' paradigm to integrate data and understanding across scales and derive dynamic and personalized schedules. We conclude with promising examples of model-guided schedule personalization and a call to action to address key outstanding challenges surrounding data collection, model development, and integration.
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Affiliation(s)
- M A R Strobl
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa; Translational Hematology and Oncology Research, Lerner Research Institute, Cleveland Clinic Foundation, Cleveland, USA
| | - J Gallaher
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa
| | - M Robertson-Tessi
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa
| | - J West
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa
| | - A R A Anderson
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa.
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4
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Falcó C, Cohen DJ, Carrillo JA, Baker RE. Quantifying tissue growth, shape and collision via continuum models and Bayesian inference. J R Soc Interface 2023; 20:20230184. [PMID: 37464804 DOI: 10.1098/rsif.2023.0184] [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: 03/29/2023] [Accepted: 06/27/2023] [Indexed: 07/20/2023] Open
Abstract
Although tissues are usually studied in isolation, this situation rarely occurs in biology, as cells, tissues and organs coexist and interact across scales to determine both shape and function. Here, we take a quantitative approach combining data from recent experiments, mathematical modelling and Bayesian parameter inference, to describe the self-assembly of multiple epithelial sheets by growth and collision. We use two simple and well-studied continuum models, where cells move either randomly or following population pressure gradients. After suitable calibration, both models prove to be practically identifiable, and can reproduce the main features of single tissue expansions. However, our findings reveal that whenever tissue-tissue interactions become relevant, the random motion assumption can lead to unrealistic behaviour. Under this setting, a model accounting for population pressure from different cell populations is more appropriate and shows a better agreement with experimental measurements. Finally, we discuss how tissue shape and pressure affect multi-tissue collisions. Our work thus provides a systematic approach to quantify and predict complex tissue configurations with applications in the design of tissue composites and more generally in tissue engineering.
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Affiliation(s)
- Carles Falcó
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Daniel J Cohen
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA
| | - José A Carrillo
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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5
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AI-Powered Diagnosis of Skin Cancer: A Contemporary Review, Open Challenges and Future Research Directions. Cancers (Basel) 2023; 15:cancers15041183. [PMID: 36831525 PMCID: PMC9953963 DOI: 10.3390/cancers15041183] [Citation(s) in RCA: 16] [Impact Index Per Article: 16.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2022] [Revised: 02/07/2023] [Accepted: 02/08/2023] [Indexed: 02/15/2023] Open
Abstract
Skin cancer continues to remain one of the major healthcare issues across the globe. If diagnosed early, skin cancer can be treated successfully. While early diagnosis is paramount for an effective cure for cancer, the current process requires the involvement of skin cancer specialists, which makes it an expensive procedure and not easily available and affordable in developing countries. This dearth of skin cancer specialists has given rise to the need to develop automated diagnosis systems. In this context, Artificial Intelligence (AI)-based methods have been proposed. These systems can assist in the early detection of skin cancer and can consequently lower its morbidity, and, in turn, alleviate the mortality rate associated with it. Machine learning and deep learning are branches of AI that deal with statistical modeling and inference, which progressively learn from data fed into them to predict desired objectives and characteristics. This survey focuses on Machine Learning and Deep Learning techniques deployed in the field of skin cancer diagnosis, while maintaining a balance between both techniques. A comparison is made to widely used datasets and prevalent review papers, discussing automated skin cancer diagnosis. The study also discusses the insights and lessons yielded by the prior works. The survey culminates with future direction and scope, which will subsequently help in addressing the challenges faced within automated skin cancer diagnosis.
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6
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Murphy RJ, Gunasingh G, Haass NK, Simpson MJ. Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability. PLoS Comput Biol 2023; 19:e1010833. [PMID: 36634128 PMCID: PMC9876349 DOI: 10.1371/journal.pcbi.1010833] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2022] [Revised: 01/25/2023] [Accepted: 12/21/2022] [Indexed: 01/13/2023] Open
Abstract
Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.
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Affiliation(s)
- Ryan J. Murphy
- Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
- * E-mail:
| | - Gency Gunasingh
- Frazer Institute, The University of Queensland, Brisbane, Queensland, Australia
| | - Nikolas K. Haass
- Frazer Institute, The University of Queensland, Brisbane, Queensland, Australia
| | - Matthew J. Simpson
- Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland, Australia
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7
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Browning AP, Simpson MJ. Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates. PLoS Comput Biol 2023; 19:e1010844. [PMID: 36662831 PMCID: PMC9891533 DOI: 10.1371/journal.pcbi.1010844] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2022] [Revised: 02/01/2023] [Accepted: 12/26/2022] [Indexed: 01/22/2023] Open
Abstract
An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.
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Affiliation(s)
- Alexander P. Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Matthew J. Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- QUT Centre for Data Science, Queensland University of Technology, Brisbane, Australia
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8
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Jepson JM, Fadai NT, O'Dea RD. Travelling-Wave and Asymptotic Analysis of a Multiphase Moving Boundary Model for Engineered Tissue Growth. Bull Math Biol 2022; 84:87. [PMID: 35821278 PMCID: PMC9276621 DOI: 10.1007/s11538-022-01044-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2021] [Accepted: 06/15/2022] [Indexed: 11/13/2022]
Abstract
We derive a multiphase, moving boundary model to represent the development of tissue in vitro in a porous tissue engineering scaffold. We consider a cell, extra-cellular liquid and a rigid scaffold phase, and adopt Darcy's law to relate the velocity of the cell and liquid phases to their respective pressures. Cell-cell and cell-scaffold interactions which can drive cellular motion are accounted for by utilising relevant constitutive assumptions for the pressure in the cell phase. We reduce the model to a nonlinear reaction-diffusion equation for the cell phase, coupled to a moving boundary condition for the tissue edge, the diffusivity being dependent on the cell and scaffold volume fractions, cell and liquid viscosities and parameters that relate to cellular motion. Numerical simulations reveal that the reduced model admits three regimes for the evolution of the tissue edge at large time: linear, logarithmic and stationary. Employing travelling-wave and asymptotic analysis, we characterise these regimes in terms of parameters related to cellular production and motion. The results of our investigation allow us to suggest optimal values for the governing parameters, so as to stimulate tissue growth in an engineering scaffold.
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Affiliation(s)
- Jacob M Jepson
- School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK.
| | - Nabil T Fadai
- School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
| | - Reuben D O'Dea
- School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
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9
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El-Hachem M, McCue SW, Simpson MJ. A Continuum Mathematical Model of Substrate-Mediated Tissue Growth. Bull Math Biol 2022; 84:49. [PMID: 35237899 PMCID: PMC8891221 DOI: 10.1007/s11538-022-01005-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Accepted: 02/09/2022] [Indexed: 11/30/2022]
Abstract
We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, \documentclass[12pt]{minimal}
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\begin{document}$${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$\end{document}u^(x^,t^) that is coupled to the concentration of an immobile extracellular substrate, \documentclass[12pt]{minimal}
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\begin{document}$${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$\end{document}s^(x^,t^). Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to \documentclass[12pt]{minimal}
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\begin{document}$${\hat{s}}$$\end{document}s^, while a logistic growth term models cell proliferation. The extracellular substrate \documentclass[12pt]{minimal}
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\begin{document}$${\hat{s}}$$\end{document}s^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, \documentclass[12pt]{minimal}
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\begin{document}$$c = c_{\mathrm{min}}$$\end{document}c=cmin, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, \documentclass[12pt]{minimal}
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\begin{document}$$c > c_{\mathrm{min}}$$\end{document}c>cmin. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.
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Affiliation(s)
- Maud El-Hachem
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Scott W McCue
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
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10
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Colson C, Sánchez-Garduño F, Byrne HM, Maini PK, Lorenzi T. Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. Proc Math Phys Eng Sci 2022; 477:20210593. [PMID: 35153606 PMCID: PMC8791052 DOI: 10.1098/rspa.2021.0593] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2021] [Accepted: 11/14/2021] [Indexed: 12/15/2022] Open
Abstract
In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling-wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling-wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.
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Affiliation(s)
- Chloé Colson
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford OX2 6GG, UK
| | - Faustino Sánchez-Garduño
- Departamento de Matemáticas, Facultad de Ciencias, UNAM, Ciudad Universitaria, Circuito Exterior, Cd. de México, C.P. 04510, Mexico
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford OX2 6GG, UK
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford OX2 6GG, UK
| | - Tommaso Lorenzi
- Department of Mathematical Sciences 'G. L. Lagrange', Politecnico di Torino, 10129 Torino, Italy
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11
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Abstract
AbstractTumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal sampling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.
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12
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Warne DJ, Baker RE, Simpson MJ. Rapid Bayesian Inference for Expensive Stochastic Models. J Comput Graph Stat 2021. [DOI: 10.1080/10618600.2021.2000419] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- David J. Warne
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Ruth E. Baker
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Matthew J. Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
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13
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Browning AP, Sharp JA, Murphy RJ, Gunasingh G, Lawson B, Burrage K, Haass NK, Simpson M. Quantitative analysis of tumour spheroid structure. eLife 2021; 10:e73020. [PMID: 34842141 PMCID: PMC8741212 DOI: 10.7554/elife.73020] [Citation(s) in RCA: 29] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2021] [Accepted: 11/26/2021] [Indexed: 11/25/2022] Open
Abstract
Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.
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Affiliation(s)
- Alexander P Browning
- School of Mathematical Sciences, Queensland University of TechnologyBrisbaneAustralia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of TechnologyMelbourneAustralia
| | - Jesse A Sharp
- School of Mathematical Sciences, Queensland University of TechnologyBrisbaneAustralia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of TechnologyMelbourneAustralia
| | - Ryan J Murphy
- School of Mathematical Sciences, Queensland University of TechnologyBrisbaneAustralia
| | - Gency Gunasingh
- The University of Queensland Diamantina Institute, The University of QueenslandBrisbaneAustralia
| | - Brodie Lawson
- School of Mathematical Sciences, Queensland University of TechnologyBrisbaneAustralia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of TechnologyMelbourneAustralia
| | - Kevin Burrage
- School of Mathematical Sciences, Queensland University of TechnologyBrisbaneAustralia
- ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of TechnologyMelbourneAustralia
- Department of Computer Science, University of OxfordOxfordUnited Kingdom
| | - Nikolas K Haass
- The University of Queensland Diamantina Institute, The University of QueenslandBrisbaneAustralia
| | - Matthew Simpson
- School of Mathematical Sciences, Queensland University of TechnologyBrisbaneAustralia
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14
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Browning AP, Maclaren OJ, Buenzli PR, Lanaro M, Allenby MC, Woodruff MA, Simpson MJ. Model-based data analysis of tissue growth in thin 3D printed scaffolds. J Theor Biol 2021; 528:110852. [PMID: 34358535 DOI: 10.1016/j.jtbi.2021.110852] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Revised: 07/08/2021] [Accepted: 07/26/2021] [Indexed: 10/24/2022]
Abstract
Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in more biologically realistic geometries than that allowed by traditional 2D cell culture. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that were deliberately increased between experiments. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds.
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Affiliation(s)
- Alexander P Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia; ARC Centre of Excellence for Mathematical and Statistical Frontiers, QUT, Australia.
| | - Oliver J Maclaren
- Department of Engineering Science, University of Auckland, Auckland 1142, New Zealand
| | - Pascal R Buenzli
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Matthew Lanaro
- School of Mechanical, Medical & Process Engineering, Centre for Biomedical Technologies, Queensland University of Technology, Brisbane, Australia
| | - Mark C Allenby
- School of Mechanical, Medical & Process Engineering, Centre for Biomedical Technologies, Queensland University of Technology, Brisbane, Australia
| | - Maria A Woodruff
- School of Mechanical, Medical & Process Engineering, Centre for Biomedical Technologies, Queensland University of Technology, Brisbane, Australia
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia; ARC Centre of Excellence for Mathematical and Statistical Frontiers, QUT, Australia
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15
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Simpson MJ, Browning AP, Drovandi C, Carr EJ, Maclaren OJ, Baker RE. Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media. Proc Math Phys Eng Sci 2021; 477:20210214. [PMID: 34248392 PMCID: PMC8262525 DOI: 10.1098/rspa.2021.0214] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2021] [Accepted: 05/13/2021] [Indexed: 12/31/2022] Open
Abstract
We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether these data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second, we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods, we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.
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Affiliation(s)
- Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Alexander P Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Christopher Drovandi
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Elliot J Carr
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Oliver J Maclaren
- Department of Engineering Science, University of Auckland, Auckland 1142, New Zealand
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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16
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Invading and Receding Sharp-Fronted Travelling Waves. Bull Math Biol 2021; 83:35. [PMID: 33611673 DOI: 10.1007/s11538-021-00862-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2020] [Accepted: 01/20/2021] [Indexed: 02/03/2023]
Abstract
Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.
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17
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Browning AP, Jin W, Plank MJ, Simpson MJ. Identifying density-dependent interactions in collective cell behaviour. J R Soc Interface 2020; 17:20200143. [PMID: 32343933 DOI: 10.1098/rsif.2020.0143] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Scratch assays are routinely used to study collective cell behaviour in vitro. Typical experimental protocols do not vary the initial density of cells, and typical mathematical modelling approaches describe cell motility and proliferation based on assumptions of linear diffusion and logistic growth. Jin et al. (Jin et al. 2016 J. Theor. Biol. 390, 136-145 (doi:10.1016/j.jtbi.2015.10.040)) find that the behaviour of cells in scratch assays is density-dependent, and show that standard modelling approaches cannot simultaneously describe data initiated across a range of initial densities. To address this limitation, we calibrate an individual-based model to scratch assay data across a large range of initial densities. Our model allows proliferation, motility, and a direction bias to depend on interactions between neighbouring cells. By considering a hierarchy of models where we systematically and sequentially remove interactions, we perform model selection analysis to identify the minimum interactions required for the model to simultaneously describe data across all initial densities. The calibrated model is able to match the experimental data across all densities using a single parameter distribution, and captures details about the spatial structure of cells. Our results provide strong evidence to suggest that motility is density-dependent in these experiments. On the other hand, we do not see the effect of crowding on proliferation in these experiments. These results are significant as they are precisely the opposite of the assumptions in standard continuum models, such as the Fisher-Kolmogorov equation and its generalizations.
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Affiliation(s)
- Alexander P Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
| | - Wang Jin
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
| | - Michael J Plank
- Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand.,Te Pūnaha Matatini, a New Zealand Centre of Research Excellence, New Zealand
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
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18
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Warne DJ, Baker RE, Simpson MJ. Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art. J R Soc Interface 2020; 16:20180943. [PMID: 30958205 DOI: 10.1098/rsif.2018.0943] [Citation(s) in RCA: 33] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.
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Affiliation(s)
- David J Warne
- 1 School of Mathematical Sciences, Queensland University of Technology , Brisbane, Queensland 4001 , Australia
| | - Ruth E Baker
- 2 Mathematical Institute, University of Oxford , Oxford OX2 6GG , UK
| | - Matthew J Simpson
- 1 School of Mathematical Sciences, Queensland University of Technology , Brisbane, Queensland 4001 , Australia
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19
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Simpson MJ, Baker RE, Vittadello ST, Maclaren OJ. Practical parameter identifiability for spatio-temporal models of cell invasion. J R Soc Interface 2020; 17:20200055. [PMID: 32126193 DOI: 10.1098/rsif.2020.0055] [Citation(s) in RCA: 41] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
We examine the practical identifiability of parameters in a spatio-temporal reaction-diffusion model of a scratch assay. Experimental data involve fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction-diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.
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Affiliation(s)
- Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Sean T Vittadello
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Oliver J Maclaren
- Department of Engineering Science, University of Auckland, Auckland 1142, New Zealand
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20
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El-Hachem M, McCue SW, Jin W, Du Y, Simpson MJ. Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy. Proc Math Phys Eng Sci 2019; 475:20190378. [PMID: 31611732 DOI: 10.1098/rspa.2019.0378] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2019] [Accepted: 07/30/2019] [Indexed: 11/12/2022] Open
Abstract
The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a spreading-extinction dichotomy. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, c≪1.
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Affiliation(s)
- Maud El-Hachem
- School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia
| | - Scott W McCue
- School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia
| | - Wang Jin
- School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia
| | - Yihong Du
- School of Science and Technology, University of New England, Armidale, Australia
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia
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21
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Using Experimental Data and Information Criteria to Guide Model Selection for Reaction–Diffusion Problems in Mathematical Biology. Bull Math Biol 2019; 81:1760-1804. [DOI: 10.1007/s11538-019-00589-x] [Citation(s) in RCA: 44] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2018] [Accepted: 02/20/2019] [Indexed: 12/20/2022]
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