1
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Gallagher K, Strobl MA, Park DS, Spoendlin FC, Gatenby RA, Maini PK, Anderson AR. Mathematical Model-Driven Deep Learning Enables Personalized Adaptive Therapy. Cancer Res 2024:742904. [PMID: 38569183 DOI: 10.1158/0008-5472.can-23-2040] [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: 08/16/2023] [Revised: 01/05/2024] [Accepted: 03/21/2024] [Indexed: 04/05/2024]
Abstract
Standard-of-care treatment regimens have long been designed for maximal cell killing, yet these strategies often fail when applied to metastatic cancers due to the emergence of drug resistance. Adaptive treatment strategies have been developed as an alternative approach, dynamically adjusting treatment to suppress the growth of treatment-resistant populations and thereby delay, or even prevent, tumor progression. Promising clinical results in prostate cancer indicate the potential to optimize adaptive treatment protocols. Here, we applied deep reinforcement learning (DRL) to guide adaptive drug scheduling and demonstrated that these treatment schedules can outperform the current adaptive protocols in a mathematical model calibrated to prostate cancer dynamics, more than doubling the time to progression. The DRL strategies were robust to patient variability, including both tumor dynamics and clinical monitoring schedules. The DRL framework could produce interpretable, adaptive strategies based on a single tumor burden threshold, replicating and informing optimal treatment strategies. The DRL framework had no knowledge of the underlying mathematical tumor model, demonstrating the capability of DRL to help develop treatment strategies in novel or complex settings. Finally, a proposed five-step pathway, which combined mechanistic modeling with the DRL framework and integrated conventional tools to improve interpretability compared to traditional "black-box" DRL models, could allow translation of this approach to the clinic. Overall, the proposed framework generated personalized treatment schedules that consistently outperformed clinical standard-of-care protocols.
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Affiliation(s)
- Kit Gallagher
- University of Oxford, Oxford, Oxfordshire, United Kingdom
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2
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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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3
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Liu Y, Suh K, Maini PK, Cohen DJ, Baker RE. Parameter identifiability and model selection for partial differential equation models of cell invasion. J R Soc Interface 2024; 21:20230607. [PMID: 38442862 PMCID: PMC10914513 DOI: 10.1098/rsif.2023.0607] [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] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2023] [Accepted: 01/31/2024] [Indexed: 03/07/2024] Open
Abstract
When employing mechanistic models to study biological phenomena, practical parameter identifiability is important for making accurate predictions across wide ranges of unseen scenarios, as well as for understanding the underlying mechanisms. In this work, we use a profile-likelihood approach to investigate parameter identifiability for four extensions of the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) model, given experimental data from a cell invasion assay. We show that more complicated models tend to be less identifiable, with parameter estimates being more sensitive to subtle differences in experimental procedures, and that they require more data to be practically identifiable. As a result, we suggest that parameter identifiability should be considered alongside goodness-of-fit and model complexity as criteria for model selection.
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Affiliation(s)
- Yue Liu
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Kevin Suh
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, USA
| | | | - Daniel J. Cohen
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, USA
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA
| | - Ruth E. Baker
- Mathematical Institute, University of Oxford, Oxford, UK
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4
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Stepanova D, Byrne HM, Maini PK, Alarcón T. Computational modeling of angiogenesis: The importance of cell rearrangements during vascular growth. WIREs Mech Dis 2024; 16:e1634. [PMID: 38084799 DOI: 10.1002/wsbm.1634] [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] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2023] [Revised: 11/10/2023] [Accepted: 11/13/2023] [Indexed: 03/16/2024]
Abstract
Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that elongate from the pre-existing vasculature to create new vascular networks. In addition to its essential role in normal development, angiogenesis plays a vital role in pathologies such as cancer, diabetes and atherosclerosis. Mathematical and computational modeling has contributed to unraveling its complexity. Many existing theoretical models of angiogenic sprouting are based on the "snail-trail" hypothesis. This framework assumes that leading ECs positioned at sprout tips migrate toward low-oxygen regions while other ECs in the sprout passively follow the leaders' trails and proliferate to maintain sprout integrity. However, experimental results indicate that, contrary to the snail-trail assumption, ECs exchange positions within developing vessels, and the elongation of sprouts is primarily driven by directed migration of ECs. The functional role of cell rearrangements remains unclear. This review of the theoretical modeling of angiogenesis is the first to focus on the phenomenon of cell mixing during early sprouting. We start by describing the biological processes that occur during early angiogenesis, such as phenotype specification, cell rearrangements and cell interactions with the microenvironment. Next, we provide an overview of various theoretical approaches that have been employed to model angiogenesis, with particular emphasis on recent in silico models that account for the phenomenon of cell mixing. Finally, we discuss when cell mixing should be incorporated into theoretical models and what essential modeling components such models should include in order to investigate its functional role. This article is categorized under: Cardiovascular Diseases > Computational Models Cancer > Computational Models.
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Affiliation(s)
- Daria Stepanova
- Laboratorio Subterráneo de Canfranc, Canfranc-Estación, Huesca, Spain
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
- Ludwig Institute for Cancer Research, Nuffield Department of Medicine, University of Oxford, Oxford, UK
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Tomás Alarcón
- Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain
- Centre de Recerca Matemàtica, Bellaterra, Barcelona, Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Spain
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5
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Browning AP, Lewin TD, Baker RE, Maini PK, Moros EG, Caudell J, Byrne HM, Enderling H. Correction to: Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling. Bull Math Biol 2024; 86:35. [PMID: 38416252 PMCID: PMC10901960 DOI: 10.1007/s11538-024-01268-2] [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: 02/29/2024]
Affiliation(s)
| | - Thomas D Lewin
- Mathematical Institute, University of Oxford, Oxford, UK
- Roche Pharma Research and Early Development, Roche Innovation Center, Basel, Switzerland
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Eduardo G Moros
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA
| | - Jimmy Caudell
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Heiko Enderling
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA.
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA.
- Department of Radiation Oncology, MD Anderson Cancer Center, Houston, TX, USA.
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6
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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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7
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Browning AP, Lewin TD, Baker RE, Maini PK, Moros EG, Caudell J, Byrne HM, Enderling H. Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling. Bull Math Biol 2024; 86:19. [PMID: 38238433 PMCID: PMC10796515 DOI: 10.1007/s11538-023-01246-0] [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] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2023] [Accepted: 12/14/2023] [Indexed: 01/22/2024]
Abstract
Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations.
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Affiliation(s)
| | - Thomas D Lewin
- Mathematical Institute, University of Oxford, Oxford, UK
- Roche Pharma Research and Early Development, Roche Innovation Center, Basel, Switzerland
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Eduardo G Moros
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA
| | - Jimmy Caudell
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Heiko Enderling
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA.
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, USA.
- Department of Radiation Oncology, MD Anderson Cancer Center, Houston, TX, USA.
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8
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Diez A, Krause AL, Maini PK, Gaffney EA, Seirin-Lee S. Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry. Bull Math Biol 2024; 86:13. [PMID: 38170298 PMCID: PMC10764571 DOI: 10.1007/s11538-023-01237-1] [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] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2023] [Accepted: 11/06/2023] [Indexed: 01/05/2024]
Abstract
Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.
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Affiliation(s)
- Antoine Diez
- Institute for the Advanced Study of Human Biology (ASHBi), Kyoto University Institute for Advanced Study (KUIAS), Kyoto University, Yoshida-Konoe-cho, Sakyo-ku, Kyoto, 606-8501, Japan
| | - Andrew L Krause
- Mathematical Sciences Department, Durham University, Upper Mountjoy Campus, Stockton Rd, Durham, DH1 3LE, UK
| | - Philip K Maini
- Wolfson Centre For Mathematical Biology, Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Eamonn A Gaffney
- Wolfson Centre For Mathematical Biology, Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
| | - Sungrim Seirin-Lee
- Institute for the Advanced Study of Human Biology (ASHBi), Kyoto University Institute for Advanced Study (KUIAS), Kyoto University, Yoshida-Konoe-cho, Sakyo-ku, Kyoto, 606-8501, Japan.
- Department of Mathematical Medicine, Graduate School of Medicine, Kyoto University, Yoshida-Konoe-cho, Sakyo-ku, Kyoto, 606-8501, Japan.
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9
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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] [What about the content of this article? (0)] [Affiliation(s)] [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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10
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Colson C, Maini PK, Byrne HM. Investigating the Influence of Growth Arrest Mechanisms on Tumour Responses to Radiotherapy. Bull Math Biol 2023; 85:74. [PMID: 37378740 DOI: 10.1007/s11538-023-01171-2] [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] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2023] [Accepted: 05/24/2023] [Indexed: 06/29/2023]
Abstract
Cancer is a heterogeneous disease and tumours of the same type can differ greatly at the genetic and phenotypic levels. Understanding how these differences impact sensitivity to treatment is an essential step towards patient-specific treatment design. In this paper, we investigate how two different mechanisms for growth control may affect tumour cell responses to fractionated radiotherapy (RT) by extending an existing ordinary differential equation model of tumour growth. In the absence of treatment, this model distinguishes between growth arrest due to nutrient insufficiency and competition for space and exhibits three growth regimes: nutrient limited, space limited (SL) and bistable (BS), where both mechanisms for growth arrest coexist. We study the effect of RT for tumours in each regime, finding that tumours in the SL regime typically respond best to RT, while tumours in the BS regime typically respond worst to RT. For tumours in each regime, we also identify the biological processes that may explain positive and negative treatment outcomes and the dosing regimen which maximises the reduction in tumour burden.
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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.
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK
- Ludwig Institute for Cancer Research, Nuffield Department of Medicine, University of Oxford, Roosevelt Drive, Oxford, OX3 7DQ, UK
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11
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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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12
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Oguma T, Takigawa-Imamura H, Shinoda T, Ogura S, Uemura A, Miyata T, Maini PK, Miura T. Analyzing the effect of cell rearrangement on Delta-Notch pattern formation. Phys Rev E 2023; 107:064404. [PMID: 37464594 DOI: 10.1103/physreve.107.064404] [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: 06/29/2022] [Accepted: 04/24/2023] [Indexed: 07/20/2023]
Abstract
The Delta-Notch system plays a vital role in many areas of biology and typically forms a salt and pepper pattern in which cells strongly expressing Delta and cells strongly expressing Notch are alternately aligned via lateral inhibition. In this study, we consider cell rearrangement events, such as cell mixing and proliferation, that alter the spatial structure itself and affect the pattern dynamics. We model cell rearrangement events by a Poisson process and analyze the model while preserving the discrete properties of the spatial structure. We investigate the effects of the intermittent perturbations arising from these cell rearrangement events on the discrete spatial structure itself in the context of pattern formation and by using an analytical approach, coupled with numerical simulation. We find that the homogeneous expression pattern is stabilized if the frequency of cell rearrangement events is sufficiently large. We analytically obtain the balanced frequencies of the cell rearrangement events where the decrease of the pattern amplitude, as a result of cell rearrangement, is balanced by the increase in amplitude due to the Delta-Notch interaction dynamics. Our framework, while applied here to the specific case of the Delta-Notch system, is applicable more widely to other pattern formation mechanisms.
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Affiliation(s)
- Toshiki Oguma
- Kyushu University Graduate School of Medical Sciences, Fukuoka 812-8582, Japan
| | | | - Tomoyasu Shinoda
- Nagoya University Graduate School of Medicine, Nagoya 466-8550, Japan
| | - Shuntaro Ogura
- Nagoya City University Graduate School of Medical Sciences, Nagoya 467-8601, Japan
| | - Akiyoshi Uemura
- Nagoya City University Graduate School of Medical Sciences, Nagoya 467-8601, Japan
| | - Takaki Miyata
- Nagoya University Graduate School of Medicine, Nagoya 466-8550, Japan
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, Oxford OX2 6GG, United Kingdom
| | - Takashi Miura
- Kyushu University Graduate School of Medical Sciences, Fukuoka 812-8582, Japan
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13
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McLennan R, Giniunaite R, Hildebrand K, Teddy JM, Kasemeier-Kulesa JC, Bolanos L, Baker RE, Maini PK, Kulesa PM. Colec12 and Trail signaling confine cranial neural crest cell trajectories and promote collective cell migration. Dev Dyn 2023; 252:629-646. [PMID: 36692868 DOI: 10.1002/dvdy.569] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [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: 02/18/2022] [Revised: 12/09/2022] [Accepted: 12/14/2022] [Indexed: 01/25/2023] Open
Abstract
BACKGROUND Collective and discrete neural crest cell (NCC) migratory streams are crucial to vertebrate head patterning. However, the factors that confine NCC trajectories and promote collective cell migration remain unclear. RESULTS Computational simulations predicted that confinement is required only along the initial one-third of the cranial NCC migratory pathway. This guided our study of Colec12 (Collectin-12, a transmembrane scavenger receptor C-type lectin) and Trail (tumor necrosis factor-related apoptosis-inducing ligand, CD253) which we show expressed in chick cranial NCC-free zones. NCC trajectories are confined by Colec12 or Trail protein stripes in vitro and show significant and distinct changes in cell morphology and dynamic migratory characteristics when cocultured with either protein. Gain- or loss-of-function of either factor or in combination enhanced NCC confinement or diverted cell trajectories as observed in vivo with three-dimensional confocal microscopy, respectively, resulting in disrupted collective migration. CONCLUSIONS These data provide evidence for Colec12 and Trail as novel NCC microenvironmental factors playing a role to confine cranial NCC trajectories and promote collective cell migration.
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Affiliation(s)
- Rebecca McLennan
- Stowers Institute for Medical Research, Kansas City, Missouri, USA
- Childrens Mercy Kansas City, Kansas City, Missouri, USA
| | - Rasa Giniunaite
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, UK
- Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania
- Faculty of Mathematics and Natural sciences, Kaunas University of Technology, Kaunas, Lithuania
| | - Katie Hildebrand
- University of Kansas School of Medicine, Kansas City, Kansas, USA
| | - Jessica M Teddy
- Stowers Institute for Medical Research, Kansas City, Missouri, USA
| | | | - Lizbeth Bolanos
- Stowers Institute for Medical Research, Kansas City, Missouri, USA
| | - Ruth E Baker
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, UK
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, UK
| | - Paul M Kulesa
- Stowers Institute for Medical Research, Kansas City, Missouri, USA
- University of Kansas School of Medicine, Kansas City, Kansas, USA
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14
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Martinson WD, McLennan R, Teddy JM, McKinney MC, Davidson LA, Baker RE, Byrne HM, Kulesa PM, Maini PK. Dynamic fibronectin assembly and remodeling by leader neural crest cells prevents jamming in collective cell migration. eLife 2023; 12:83792. [PMID: 37073859 DOI: 10.7554/elife.83792] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2022] [Accepted: 04/13/2023] [Indexed: 04/20/2023] Open
Abstract
Collective cell migration plays an essential role in vertebrate development, yet the extent to which dynamically changing microenvironments influence this phenomenon remains unclear. Observations of the distribution of the extracellular matrix (ECM) component fibronectin during the migration of loosely connected neural crest cells (NCCs) lead us to hypothesize that NCC remodeling of an initially punctate ECM creates a scaffold for trailing cells, enabling them to form robust and coherent stream patterns. We evaluate this idea in a theoretical setting by developing an individual-based computational model that incorporates reciprocal interactions between NCCs and their ECM. ECM remodeling, haptotaxis, contact guidance, and cell-cell repulsion are sufficient for cells to establish streams in silico, however additional mechanisms, such as chemotaxis, are required to consistently guide cells along the correct target corridor. Further model investigations imply that contact guidance and differential cell-cell repulsion between leader and follower cells are key contributors to robust collective cell migration by preventing stream breakage. Global sensitivity analysis and simulated gain- and loss-of-function experiments suggest that long-distance migration without jamming is most likely to occur when leading cells specialize in creating ECM fibers, and trailing cells specialize in responding to environmental cues by upregulating mechanisms such as contact guidance.
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Affiliation(s)
| | | | - Jessica M Teddy
- Stowers Institute for Medical Research, Kansas City, United States
| | - Mary C McKinney
- Stowers Institute for Medical Research, Kansas City, United States
| | - Lance A Davidson
- Swanson School of Engineering, University of Pittsburgh, Pittsburgh, United States
| | - Ruth E Baker
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, United Kingdom
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, United Kingdom
| | - Paul M Kulesa
- Stowers Institute for Medical Research, Kansas City, United States
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
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15
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Tseng CC, Woolley TE, Jiang TX, Wu P, Maini PK, Widelitz RB, Chuong CM. Gap junctions in Turing-type periodic feather pattern formation. bioRxiv 2023:2023.04.15.537019. [PMID: 37090608 PMCID: PMC10120740 DOI: 10.1101/2023.04.15.537019] [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] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/25/2023]
Abstract
Periodic patterning requires coordinated cell-cell interactions at the tissue level. Turing showed, using mathematical modeling, how spatial patterns could arise from the reactions of a diffusive activator-inhibitor pair in an initially homogenous two-dimensional field. Most activators and inhibitors studied in biological systems are proteins, and the roles of cell-cell interaction, ions, bioelectricity, etc. are only now being identified. Gap junctions (GJs) mediate direct exchanges of ions or small molecules between cells, enabling rapid long-distance communications in a cell collective. They are therefore good candidates for propagating non-protein-based patterning signals that may act according to the Turing principles. Here, we explore the possible roles of GJs in Turing-type patterning using feather pattern formation as a model. We found seven of the twelve investigated GJ isoforms are highly dynamically expressed in the developing chicken skin. In ovo functional perturbations of the GJ isoform, connexin 30, by siRNA and the dominant-negative mutant applied before placode development led to disrupted primary feather bud formation, including patches of smooth skin and buds of irregular sizes. Later, after the primary feather arrays were laid out, inhibition of gap junctional intercellular communication in the ex vivo skin explant culture allowed the emergence of new feather buds in temporal waves at specific spatial locations relative to the existing primary buds. The results suggest that gap junctional communication may facilitate the propagation of long-distance inhibitory signals. Thus, the removal of GJ activity would enable the emergence of new feather buds if the local environment is competent and the threshold to form buds is reached. We propose Turing-based computational simulations that can predict the appearance of these ectopic bud waves. Our models demonstrate how a Turing activator-inhibitor system can continue to generate patterns in the competent morphogenetic field when the level of intercellular communication at the tissue scale is modulated.
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Affiliation(s)
- Chun-Chih Tseng
- Department of Biochemistry and Molecular Medicine, USC Norris Comprehensive Cancer Center, Keck School of Medicine, University of Southern California, Los Angeles, CA 90033, U.S.A
- Current address: Department of Molecular Biology, Princeton University, Princeton, NJ 08540, U.S.A
| | - Thomas E. Woolley
- School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, U.K
| | - Ting-Xin Jiang
- Department of Pathology, University of Southern California, Los Angeles, CA 90033, U.S.A
| | - Ping Wu
- Department of Pathology, University of Southern California, Los Angeles, CA 90033, U.S.A
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, U.K
| | - Randall B. Widelitz
- Department of Pathology, University of Southern California, Los Angeles, CA 90033, U.S.A
| | - Cheng-Ming Chuong
- Department of Pathology, University of Southern California, Los Angeles, CA 90033, U.S.A
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16
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Strobl MA, Martin AL, Gallagher C, Damaghi M, Robertson-Tessi M, Gatenby R, Wenham RM, Maini PK, Anderson AR. Abstract 5694: Adaptive treatment scheduling of PARP inhibitors in ovarian cancer: Using mathematical modeling to assess clinical feasibility and estimate potential benefits. Cancer Res 2023. [DOI: 10.1158/1538-7445.am2023-5694] [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] [Indexed: 04/07/2023]
Abstract
Abstract
PARP inhibitors (PARPis) are revolutionizing the treatment of ovarian cancer. Yet for many patients these improvements come at the cost of physical and financial toxicity and responses are typically temporary due to emerging drug resistance. A growing body of pre-clinical and clinical work suggests that when cure is unlikely, it is possible to delay progression and reduce drug use through patient-specific drug scheduling. So-called 'adaptive therapy' dynamically adjusts treatment to preserve drug-sensitive cells which interfere with resistant cells through competition. In a prior study, we developed a mathematical model to describe the treatment response of ovarian cancer cells to the PARPi olaparib in vitro, and we proposed a candidate adaptive PARPi algorithm. Here, we extend our model to capture the dynamics in patients and use it to study the feasibility and potential benefit of adaptive PARPi administration in practice.
Our prior model posited that treatment induces cell cycle arrest that moves cells from the proliferating subset of the population to an arrested compartment. The model predicted that while there is scope for treatment reductions, these need to be carefully timed and prolonged treatment breaks should be avoided. Based on these results we proposed an adaptive treatment algorithm in which the olaparib dose is switched between high and low doses, depending on the tumor’s growth rate. To test the translational potential of this strategy, we retrospectively collected data from 53 ovarian cancer patients who received olaparib at the H Lee Moffitt Cancer Center between 2014 and 2021. Using serum CA-125 as a proxy for tumor burden, we first examined whether our mathematical model could capture the patients’ longitudinal dynamics. While the response of some patients was consistent with what we had observed in vitro, in others there was evidence of the emergence of a distinct drug-resistant population, and we extended our mathematical model to account for this. After calibrating and validating the model with the patient data, we tested whether these patients would have benefited from adaptive PARPi treatment. Our simulations suggest that our proposed algorithm is feasible and provides a means for reducing treatment in a patient-specific manner. In addition, in a subset of patients we predict that adaptive therapy could delay progression. Overall, this work corroborates the potential for adaptive PARPi therapy and helps to identify outstanding challenges on the way to clinical translation.
Citation Format: Maximilian A. Strobl, Alexandra L. Martin, Christopher Gallagher, Mehdi Damaghi, Mark Robertson-Tessi, Robert Gatenby, Robert M. Wenham, Philip K. Maini, Alexander R. Anderson. Adaptive treatment scheduling of PARP inhibitors in ovarian cancer: Using mathematical modeling to assess clinical feasibility and estimate potential benefits. [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 5694.
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17
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Dangerfield CE, David Abrahams I, Budd C, Butchers M, Cates ME, Champneys AR, Currie CS, Enright J, Gog JR, Goriely A, Déirdre Hollingsworth T, Hoyle RB, INI Professional Services, Isham V, Jordan J, Kaouri MH, Kavoussanakis K, Leeks J, Maini PK, Marr C, Merritt C, Mollison D, Ray S, Thompson RN, Wakefield A, Wasley D. Getting the most out of maths: How to coordinate mathematical modelling research to support a pandemic, lessons learnt from three initiatives that were part of the COVID-19 response in the UK. J Theor Biol 2023; 557:111332. [PMID: 36323393 PMCID: PMC9618296 DOI: 10.1016/j.jtbi.2022.111332] [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] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2022] [Revised: 10/14/2022] [Accepted: 10/17/2022] [Indexed: 11/16/2022]
Abstract
In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".
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Affiliation(s)
- Ciara E. Dangerfield
- Isaac Newton Institute to Mathematical Sciences, University of Cambridge, United Kingdom,Joint UNIversities Pandemic and Epidemiological Research (JUNIPER) Consortium, United Kingdom1,Corresponding author
| | - I. David Abrahams
- Department for Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
| | - Chris Budd
- Department of Mathematics, University of Bath, United Kingdom
| | - Matt Butchers
- Department of Mathematics, University of Bath, United Kingdom
| | - Michael E. Cates
- Department for Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
| | - Alan R. Champneys
- Department of Engineering Mathematics, University of Bristol, United Kingdom
| | | | - Jessica Enright
- School of Computing Science, University of Glasgow, United Kingdom
| | - Julia R. Gog
- Joint UNIversities Pandemic and Epidemiological Research (JUNIPER) Consortium, United Kingdom1,Department for Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
| | - Alain Goriely
- Mathematical Institute, University of Oxford, United Kingdom
| | - T. Déirdre Hollingsworth
- Joint UNIversities Pandemic and Epidemiological Research (JUNIPER) Consortium, United Kingdom1,Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, University of Oxford, United Kingdom
| | - Rebecca B. Hoyle
- School of Mathematical Sciences, University of Southampton, United Kingdom
| | | | - Valerie Isham
- Department of Statistical Science, University College London, United Kingdom
| | | | - Maha H. Kaouri
- Isaac Newton Institute to Mathematical Sciences, University of Cambridge, United Kingdom
| | | | - Jane Leeks
- Isaac Newton Institute to Mathematical Sciences, University of Cambridge, United Kingdom
| | - Philip K. Maini
- Mathematical Institute, University of Oxford, United Kingdom
| | - Christie Marr
- Isaac Newton Institute to Mathematical Sciences, University of Cambridge, United Kingdom
| | - Clare Merritt
- Isaac Newton Institute to Mathematical Sciences, University of Cambridge, United Kingdom
| | - Denis Mollison
- Department of Actuarial Mathematics and Statistics, Heriot-Watt University, United Kingdom
| | - Surajit Ray
- School of Mathematics and Statistics, University of Glasgow, United Kingdom
| | - Robin N. Thompson
- Mathematics Institute, University of Warwick, United Kingdom,Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of Warwick, United Kingdom
| | | | - Dawn Wasley
- International Centre for Mathematical Sciences, University of Edinburgh & Heriot-Watt University, United Kingdom
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18
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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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19
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Colson C, Byrne HM, Maini PK. Combining Mechanisms of Growth Arrest in Solid Tumours: A Mathematical Investigation. Bull Math Biol 2022; 84:80. [PMID: 35773547 PMCID: PMC9246818 DOI: 10.1007/s11538-022-01034-2] [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: 01/20/2022] [Accepted: 05/26/2022] [Indexed: 11/25/2022]
Abstract
The processes underpinning solid tumour growth involve the interactions between various healthy and tumour tissue components and the vasculature, and can be affected in different ways by cancer treatment. In particular, the growth-limiting mechanisms at play may influence tumour responses to treatment. In this paper, we propose a simple ordinary differential equation model of solid tumour growth to investigate how tumour-specific mechanisms of growth arrest may affect tumour response to different combination cancer therapies. We consider the interactions of tumour cells with the physical space in which they proliferate and a nutrient supplied by the tumour vasculature, with the aim of representing two distinct growth arrest mechanisms. More specifically, we wish to consider growth arrest due to (1) nutrient deficiency, which corresponds to balancing cell proliferation and death rates, and (2) competition for space, which corresponds to cessation of proliferation without cell death. We perform numerical simulations of the model and a steady-state analysis to determine the possible tumour growth scenarios described by the model. We find that there are three distinct growth regimes: the nutrient- and spatially limited regimes and a bi-stable regime, in which both growth arrest mechanisms are simultaneously active. Thus, the proposed model has the features required to investigate and distinguish tumour responses to different cancer treatments.
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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.
| | - 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
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20
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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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21
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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] [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: 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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22
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Hart WS, Abbott S, Endo A, Hellewell J, Miller E, Andrews N, Maini PK, Funk S, Thompson RN. Inference of the SARS-CoV-2 generation time using UK household data. eLife 2022; 11:70767. [PMID: 35138250 PMCID: PMC8967386 DOI: 10.7554/elife.70767] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [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: 05/29/2021] [Accepted: 02/07/2022] [Indexed: 11/17/2022] Open
Abstract
The distribution of the generation time (the interval between individuals becoming infected and transmitting the virus) characterises changes in the transmission risk during SARS-CoV-2 infections. Inferring the generation time distribution is essential to plan and assess public health measures. We previously developed a mechanistic approach for estimating the generation time, which provided an improved fit to data from the early months of the COVID-19 pandemic (December 2019-March 2020) compared to existing models (Hart et al., 2021). However, few estimates of the generation time exist based on data from later in the pandemic. Here, using data from a household study conducted from March to November 2020 in the UK, we provide updated estimates of the generation time. We considered both a commonly used approach in which the transmission risk is assumed to be independent of when symptoms develop, and our mechanistic model in which transmission and symptoms are linked explicitly. Assuming independent transmission and symptoms, we estimated a mean generation time (4.2 days, 95% credible interval 3.3–5.3 days) similar to previous estimates from other countries, but with a higher standard deviation (4.9 days, 3.0–8.3 days). Using our mechanistic approach, we estimated a longer mean generation time (5.9 days, 5.2–7.0 days) and a similar standard deviation (4.8 days, 4.0–6.3 days). As well as estimating the generation time using data from the entire study period, we also considered whether the generation time varied temporally. Both models suggest a shorter mean generation time in September-November 2020 compared to earlier months. Since the SARS-CoV-2 generation time appears to be changing, further data collection and analysis is necessary to continue to monitor ongoing transmission and inform future public health policy decisions.
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Affiliation(s)
| | - Sam Abbott
- Centre for the Mathematical Modelling of Infectious Diseases, London School of Hygiene and Tropical Medicine, London, United Kingdom
| | - Akira Endo
- Centre for the Mathematical Modelling of Infectious Diseases, London School of Hygiene and Tropical Medicine, London, United Kingdom
| | - Joel Hellewell
- Centre for the Mathematical Modelling of Infectious Diseases, London School of Hygiene and Tropical Medicine, London, United Kingdom
| | - Elizabeth Miller
- Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, United Kingdom
| | - Nick Andrews
- Data and Analytical Sciences, UK Health Security Agency, Public Health England, London, United Kingdom
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Sebastian Funk
- Centre for the Mathematical Modelling of Infectious Diseases, London School of Hygiene and Tropical Medicine, London, United Kingdom
| | - Robin N Thompson
- Mathematics Institute, University of Warwick, Coventry, United Kingdom
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23
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Affiliation(s)
- Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
| | - Peter J Hunter
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
| | - Peter J Gawthrop
- Systems Biology Laboratory, Department of Biomedical Engineering, Faculty of Engineering and Information Technology, University of Melbourne, Victoria, 3010, Australia
- Systems Biology Laboratory, School of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia
| | - Nic P Smith
- Auckland Bioengineering Institute, University of Auckland, Auckland, New Zealand
- Queensland University of Technology, Brisbane, Australia
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24
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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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Krause AL, Gaffney EA, Maini PK, Klika V. Introduction to 'Recent progress and open frontiers in Turing's theory of morphogenesis'. Philos Trans A Math Phys Eng Sci 2021; 379:20200280. [PMID: 34743606 PMCID: PMC8580473 DOI: 10.1098/rsta.2020.0280] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/05/2023]
Abstract
Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing's contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing's theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing's mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing's original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Andrew L. Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE, UK
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Václav Klika
- Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova, 13, 120 00 Praha, Czech Republic
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Krause AL, Gaffney EA, Maini PK, Klika V. Modern perspectives on near-equilibrium analysis of Turing systems. Philos Trans A Math Phys Eng Sci 2021; 379:20200268. [PMID: 34743603 PMCID: PMC8580451 DOI: 10.1098/rsta.2020.0268] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/18/2021] [Indexed: 05/02/2023]
Abstract
In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Andrew L. Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE, UK
| | - Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Václav Klika
- Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova, 13, 12000 Praha, Czech Republic
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Affiliation(s)
- Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
| | - Mark A J Chaplain
- School of Mathematics and Statistics, Mathematical Institute, University of St Andrews, St Andrews, KY16 9SS, UK
| | - Mark A Lewis
- Department of Mathematical and Statistical Sciences, CAB 545B, University of Alberta, Edmonton, AB, T6G 2G1, Canada
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Abstract
Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.
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Affiliation(s)
- Andrew L Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
| | - Václav Klika
- Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00, Praha, Czech Republic
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Denis Headon
- The Roslin Institute and Royal (Dick) School of Veterinary Studies, University of Edinburgh, Easter Bush Campus, Midlothian, EH25 9RG, UK
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
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Hart WS, Maini PK, Thompson RN. High infectiousness immediately before COVID-19 symptom onset highlights the importance of continued contact tracing. eLife 2021; 10:e65534. [PMID: 33899740 PMCID: PMC8195606 DOI: 10.7554/elife.65534] [Citation(s) in RCA: 37] [Impact Index Per Article: 12.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 04/25/2021] [Indexed: 01/03/2023] Open
Abstract
Background Understanding changes in infectiousness during SARS-COV-2 infections is critical to assess the effectiveness of public health measures such as contact tracing. Methods Here, we develop a novel mechanistic approach to infer the infectiousness profile of SARS-COV-2-infected individuals using data from known infector-infectee pairs. We compare estimates of key epidemiological quantities generated using our mechanistic method with analogous estimates generated using previous approaches. Results The mechanistic method provides an improved fit to data from SARS-CoV-2 infector-infectee pairs compared to commonly used approaches. Our best-fitting model indicates a high proportion of presymptomatic transmissions, with many transmissions occurring shortly before the infector develops symptoms. Conclusions High infectiousness immediately prior to symptom onset highlights the importance of continued contact tracing until effective vaccines have been distributed widely, even if contacts from a short time window before symptom onset alone are traced. Funding Engineering and Physical Sciences Research Council (EPSRC).
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Affiliation(s)
- William S Hart
- Mathematical Institute, University of OxfordOxfordUnited Kingdom
| | - Philip K Maini
- Mathematical Institute, University of OxfordOxfordUnited Kingdom
| | - Robin N Thompson
- Mathematics Institute, University of WarwickCoventryUnited Kingdom
- Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research, University of WarwickCoventryUnited Kingdom
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Fadai NT, Sachak-Patwa R, Byrne HM, Maini PK, Bafadhel M, Nicolau DV. Infection, inflammation and intervention: mechanistic modelling of epithelial cells in COVID-19. J R Soc Interface 2021; 18:20200950. [PMID: 33593209 PMCID: PMC8086847 DOI: 10.1098/rsif.2020.0950] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [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] [Indexed: 12/15/2022] Open
Abstract
While the pathological mechanisms in COVID-19 illness are still poorly understood, it is increasingly clear that high levels of pro-inflammatory mediators play a major role in clinical deterioration in patients with severe disease. Current evidence points to a hyperinflammatory state as the driver of respiratory compromise in severe COVID-19 disease, with a clinical trajectory resembling acute respiratory distress syndrome, but how this ‘runaway train’ inflammatory response emerges and is maintained is not known. Here, we present the first mathematical model of lung hyperinflammation due to SARS-CoV-2 infection. This model is based on a network of purported mechanistic and physiological pathways linking together five distinct biochemical species involved in the inflammatory response. Simulations of our model give rise to distinct qualitative classes of COVID-19 patients: (i) individuals who naturally clear the virus, (ii) asymptomatic carriers and (iii–v) individuals who develop a case of mild, moderate, or severe illness. These findings, supported by a comprehensive sensitivity analysis, point to potential therapeutic interventions to prevent the emergence of hyperinflammation. Specifically, we suggest that early intervention with a locally acting anti-inflammatory agent (such as inhaled corticosteroids) may effectively blockade the pathological hyperinflammatory reaction as it emerges.
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Affiliation(s)
- Nabil T Fadai
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
| | | | - Helen M Byrne
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
| | - Mona Bafadhel
- Respiratory Medicine Unit, Nuffield Department of Medicine, University of Oxford, Old Road Campus, Oxford OX3 7LF, UK
| | - Dan V Nicolau
- Respiratory Medicine Unit, Nuffield Department of Medicine, University of Oxford, Old Road Campus, Oxford OX3 7LF, UK.,School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4001, Australia
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Strobl MAR, West J, Viossat Y, Damaghi M, Robertson-Tessi M, Brown JS, Gatenby RA, Maini PK, Anderson ARA. Turnover Modulates the Need for a Cost of Resistance in Adaptive Therapy. Cancer Res 2021; 81:1135-1147. [PMID: 33172930 PMCID: PMC8455086 DOI: 10.1158/0008-5472.can-20-0806] [Citation(s) in RCA: 45] [Impact Index Per Article: 15.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2020] [Revised: 08/06/2020] [Accepted: 11/06/2020] [Indexed: 11/16/2022]
Abstract
Adaptive therapy seeks to exploit intratumoral competition to avoid, or at least delay, the emergence of therapy resistance in cancer. Motivated by promising results in prostate cancer, there is growing interest in extending this approach to other neoplasms. As such, it is urgent to understand the characteristics of a cancer that determine whether or not it will respond well to adaptive therapy. A plausible candidate for such a selection criterion is the fitness cost of resistance. In this article, we study a general, but simple, mathematical model to investigate whether the presence of a cost is necessary for adaptive therapy to extend the time to progression beyond that of a standard-of-care continuous therapy. Tumor cells were divided into sensitive and resistant populations and we model their competition using a system of two ordinary differential equations based on the Lotka-Volterra model. For tumors close to their environmental carrying capacity, a cost was not required. However, for tumors growing far from carrying capacity, a cost may be required to see meaningful gains. Notably, it is important to consider cell turnover in the tumor, and we discuss its role in modulating the impact of a resistance cost. To conclude, we present evidence for the predicted cost-turnover interplay in data from 67 patients with prostate cancer undergoing intermittent androgen deprivation therapy. Our work helps to clarify under which circumstances adaptive therapy may be beneficial and suggests that turnover may play an unexpectedly important role in the decision-making process. SIGNIFICANCE: Tumor cell turnover modulates the speed of selection against drug resistance by amplifying the effects of competition and resistance costs; as such, turnover is an important factor in resistance management via adaptive therapy.See related commentary by Strobl et al., p. 811.
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Affiliation(s)
- Maximilian A R Strobl
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida.
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, Oxfordshire, United Kingdom
| | - Jeffrey West
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida
| | - Yannick Viossat
- Ceremade, Université Paris-Dauphine, Université PSL, Paris, France
| | - Mehdi Damaghi
- Department of Cancer Physiology, H. Lee Moffitt Cancer Center, Tampa, Florida
| | - Mark Robertson-Tessi
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida
| | - Joel S Brown
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida
| | - Robert A Gatenby
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida
- Cancer Biology and Evolution Program, H. Lee Moffitt Cancer Center, Tampa, Florida
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford, Oxfordshire, United Kingdom
| | - Alexander R A Anderson
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida.
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33
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Stepanova D, Byrne HM, Maini PK, Alarcón T. A multiscale model of complex endothelial cell dynamics in early angiogenesis. PLoS Comput Biol 2021; 17:e1008055. [PMID: 33411727 PMCID: PMC7817011 DOI: 10.1371/journal.pcbi.1008055] [Citation(s) in RCA: 22] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2020] [Revised: 01/20/2021] [Accepted: 11/19/2020] [Indexed: 12/30/2022] Open
Abstract
We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted a central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis. Angiogenesis, the process by which new blood vessels are formed by sprouting from the pre-existing vascular bed, plays a key role in both physiological and pathological processes, including tumour growth. The structure of a growing vascular network is determined by the coordinated behaviour of endothelial cells in response to various signalling cues. Recent experimental studies have highlighted the importance of cell rearrangements as a driver for sprout elongation. However, the functional role of this phenomenon remains unclear. We formulate a new multiscale model of angiogenesis which, by accounting explicitly for the complex dynamics of endothelial cells within growing angiogenic sprouts, is able to reproduce generic features of angiogenic structures (branching, chemotactic sensitivity, cell mixing, etc.) as emergent properties of its dynamics. We validate our model against experimental data and then use it to quantify the phenomenon of cell mixing in vascular networks generated by endothelial cells of different lineages. Our results show that there is a direct correlation between the time evolution of cell mixing in a growing vascular network and its branching structure, thus paving the way for understanding the functional role of cell rearrangements in angiogenesis.
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Affiliation(s)
- Daria Stepanova
- Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
- * E-mail:
| | - Helen M. Byrne
- 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
| | - Tomás Alarcón
- Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
- Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
- Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
- Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain
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Scott JG, Maini PK, Anderson ARA, Fletcher AG. Inferring Tumor Proliferative Organization from Phylogenetic Tree Measures in a Computational Model. Syst Biol 2021; 69:623-637. [PMID: 31665523 DOI: 10.1093/sysbio/syz070] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2018] [Revised: 09/23/2019] [Accepted: 10/02/2019] [Indexed: 01/15/2023] Open
Abstract
We use a computational modeling approach to explore whether it is possible to infer a solid tumor's cellular proliferative hierarchy under the assumptions of the cancer stem cell hypothesis and neutral evolution. We work towards inferring the symmetric division probability for cancer stem cells, since this is believed to be a key driver of progression and therapeutic response. Motivated by the advent of multiregion sampling and resulting opportunities to infer tumor evolutionary history, we focus on a suite of statistical measures of the phylogenetic trees resulting from the tumor's evolution in different regions of parameter space and through time. We find strikingly different patterns in these measures for changing symmetric division probability which hinge on the inclusion of spatial constraints. These results give us a starting point to begin stratifying tumors by this biological parameter and also generate a number of actionable clinical and biological hypotheses regarding changes during therapy, and through tumor evolutionary time. [Cancer; evolution; phylogenetics.].
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Affiliation(s)
- Jacob G Scott
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK.,Departments of Translational Hematology and Oncology Research and Radiation Oncology, Taussig Cancer Institute, Cleveland Clinic, Cleveland, OH, USA
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Alexander R A Anderson
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL, USA
| | - Alexander G Fletcher
- School of Mathematics and Statistics, University of Sheffield, Sheffield, UK.,Bateson Centre, University of Sheffield, Sheffield, UK
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35
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Martinson WD, Byrne HM, Maini PK. Evaluating snail-trail frameworks for leader-follower behavior with agent-based modeling. Phys Rev E 2020; 102:062417. [PMID: 33466087 DOI: 10.1103/physreve.102.062417] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [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: 06/30/2020] [Accepted: 11/18/2020] [Indexed: 06/12/2023]
Abstract
Branched networks constitute a ubiquitous structure in biology, arising in plants, lungs, and the circulatory system; however, the mechanisms behind their creation are not well understood. A commonly used model for network morphogenesis proposes that sprouts develop through interactions between leader (tip) cells and follower (stalk) cells. In this description, tip cells emerge from existing structures, travel up chemoattractant gradients, and form new networks by guiding the movement of stalk cells. Such dynamics have been mathematically represented by continuum "snail-trail" models in which the tip cell flux contributes to the stalk cell proliferation rate. Although snail-trail models constitute a classical depiction of leader-follower behavior, their accuracy has yet to be evaluated in a rigorous quantitative setting. Here, we extend the snail-trail modeling framework to two spatial dimensions by introducing a novel multiplicative factor to the stalk cell rate equation, which corrects for neglected network creation in directions other than that of the migrating front. Our derivation of this factor demonstrates that snail-trail models are valid descriptions of cell dynamics when chemotaxis dominates cell movement. We confirm that our snail-trail model accurately predicts the dynamics of tip and stalk cells in an existing agent-based model (ABM) for network formation [Pillay et al., Phys. Rev. E 95, 012410 (2017)10.1103/PhysRevE.95.012410]. We also derive conditions for which it is appropriate to use a reduced, one-dimensional snail-trail model to analyze ABM results. Our analysis identifies key metrics for cell migration that may be used to anticipate when simple snail-trail models will accurately describe experimentally observed cell dynamics in network formation.
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Affiliation(s)
- W Duncan Martinson
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
| | - Helen M Byrne
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
| | - Philip K Maini
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom
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36
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Monsivais-Velazquez D, Bhattacharya K, Barrio RA, Maini PK, Kaski KK. Dynamics of hierarchical weighted networks of van der Pol oscillators. Chaos 2020; 30:123146. [PMID: 33380066 DOI: 10.1063/5.0010638] [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] [Subscribe] [Scholar Register] [Received: 04/15/2020] [Accepted: 11/27/2020] [Indexed: 06/12/2023]
Abstract
We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.
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Affiliation(s)
| | - Kunal Bhattacharya
- Department of Industrial Engineering and Management, Aalto University School of Science, 00076 Helsinki, Finland
| | - Rafael A Barrio
- Instituto de Física, Universidad Nacional Autónoma de México, Ap. postal 01000, CDMX, Mexico
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, Oxford University, Oxford OX2 6GG, United Kingdom
| | - Kimmo K Kaski
- Department of Computer Science, Aalto University School of Science, 00076 Helsinki, Finland
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37
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Almet AA, Byrne HM, Maini PK, Moulton DE. The role of mechanics in the growth and homeostasis of the intestinal crypt. Biomech Model Mechanobiol 2020; 20:585-608. [PMID: 33219879 PMCID: PMC7979635 DOI: 10.1007/s10237-020-01402-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [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: 05/21/2020] [Accepted: 11/03/2020] [Indexed: 11/29/2022]
Abstract
We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us to precisely characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous "treadmilling" state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis.
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Affiliation(s)
- A A Almet
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.,NSF-Simons Center for Multiscale Cell Fate Research, University of California, Irvine, Irvine, CA, USA.,Department of Mathematics, University of California, Irvine, Irvine, CA, USA
| | - H M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - P K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - D E Moulton
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
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38
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Ardaševa A, Anderson ARA, Gatenby RA, Byrne HM, Maini PK, Lorenzi T. Comparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environments. Phys Rev E 2020; 102:042404. [PMID: 33212726 PMCID: PMC10900972 DOI: 10.1103/physreve.102.042404] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [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: 04/02/2020] [Accepted: 09/14/2020] [Indexed: 06/11/2023]
Abstract
Deterministic continuum models formulated as nonlocal partial differential equations for the evolutionary dynamics of populations structured by phenotypic traits have been used recently to address open questions concerning the adaptation of asexual species to periodically fluctuating environmental conditions. These models are usually defined on the basis of population-scale phenomenological assumptions and cannot capture adaptive phenomena that are driven by stochastic variability in the evolutionary paths of single individuals. In light of these considerations, in this paper we develop a stochastic individual-based model for the coevolution of two competing phenotype-structured cell populations that are exposed to time-varying nutrient levels and undergo spontaneous, heritable phenotypic changes with different probabilities. Here, the evolution of every cell is described by a set of rules that result in a discrete-time branching random walk on the space of phenotypic states, and nutrient levels are governed by a difference equation in which a sink term models nutrient consumption by the cells. We formally show that the deterministic continuum counterpart of this model comprises a system of nonlocal partial differential equations for the cell population density functions coupled with an ordinary differential equation for the nutrient concentration. We compare the individual-based model and its continuum analog, focusing on scenarios whereby the predictions of the two models differ. The results obtained clarify the conditions under which significant differences between the two models can emerge due to bottleneck effects that bring about both lower regularity of the density functions of the two populations and more pronounced demographic stochasticity. In particular, bottleneck effects emerge in the presence of lower probabilities of phenotypic variation and are more apparent when the two populations are characterized by lower fitness initial mean phenotypes and smaller initial levels of phenotypic heterogeneity. The emergence of these effects, and thus the agreement between the two modeling approaches, is also dependent on the initial proportions of the two populations. As an illustrative example, we demonstrate the implications of these results in the context of the mathematical modeling of the early stage of metastatic colonization of distant organs.
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Affiliation(s)
- Aleksandra Ardaševa
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Alexander R A Anderson
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida 33612, USA
| | - Robert A Gatenby
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida 33612, USA
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Tommaso Lorenzi
- Department of Mathematical Sciences "G. L. Lagrange", Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, Italy
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Weber EL, Woolley TE, Yeh CY, Ou KL, Maini PK, Chuong CM. Self-organizing hair peg-like structures from dissociated skin progenitor cells: New insights for human hair follicle organoid engineering and Turing patterning in an asymmetric morphogenetic field. Exp Dermatol 2020; 28:355-366. [PMID: 30681746 DOI: 10.1111/exd.13891] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2018] [Revised: 01/02/2019] [Accepted: 01/08/2019] [Indexed: 12/11/2022]
Abstract
Human skin progenitor cells will form new hair follicles, although at a low efficiency, when injected into nude mouse skin. To better study and improve upon this regenerative process, we developed an in vitro system to analyse the morphogenetic cell behaviour in detail and modulate physical-chemical parameters to more effectively generate hair primordia. In this three-dimensional culture, dissociated human neonatal foreskin keratinocytes self-assembled into a planar epidermal layer while fetal scalp dermal cells coalesced into stripes, then large clusters, and finally small clusters resembling dermal condensations. At sites of dermal clustering, subjacent epidermal cells protruded to form hair peg-like structures, molecularly resembling hair pegs within the sequence of follicular development. The hair peg-like structures emerged in a coordinated, formative wave, moving from periphery to centre, suggesting that the droplet culture constitutes a microcosm with an asymmetric morphogenetic field. In vivo, hair follicle populations also form in a progressive wave, implying the summation of local periodic patterning events with an asymmetric global influence. To further understand this global patterning process, we developed a mathematical simulation using Turing activator-inhibitor principles in an asymmetric morphogenetic field. Together, our culture system provides a suitable platform to (a) analyse the self-assembly behaviour of hair progenitor cells into periodically arranged hair primordia and (b) identify parameters that impact the formation of hair primordia in an asymmetric morphogenetic field. This understanding will enhance our future ability to successfully engineer human hair follicle organoids.
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Affiliation(s)
- Erin L Weber
- Department of Pathology, Keck School of Medicine of the University of Southern California, Los Angeles, California.,Division of Plastic and Reconstructive Surgery, Keck School of Medicine of the University of Southern California, Los Angeles, California
| | | | - Chao-Yuan Yeh
- Department of Pathology, Keck School of Medicine of the University of Southern California, Los Angeles, California
| | - Kuang-Ling Ou
- Department of Pathology, Keck School of Medicine of the University of Southern California, Los Angeles, California.,Ostrow School of Dentistry of the University of Southern California, Los Angeles, California.,Division of Plastic and Reconstructive Surgery, Department of Surgery, Tri-Service General Hospital, National Defense Medical Center, Taipei, Taiwan
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, Oxford, UK
| | - Cheng-Ming Chuong
- Department of Pathology, Keck School of Medicine of the University of Southern California, Los Angeles, California.,Integrative Stem Cell Center, China Medical University, Taichung, Taiwan
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Almet AA, Maini PK, Moulton DE, Byrne HM. Modeling perspectives on the intestinal crypt, a canonical system for growth, mechanics, and remodeling. Current Opinion in Biomedical Engineering 2020. [DOI: 10.1016/j.cobme.2019.12.012] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
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Affiliation(s)
- Mark Alber
- Department of Mathematics and Center for Quantitative Modeling in Biology, University of California Riverside, Riverside, USA.
| | - Christophe Godin
- Laboratoire de Reproduction et Développement des plantes, Université Lyon, Lyon, France
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Roeland Merks
- Mathematical Institute and Institute of Biology, Leiden University, Leiden, The Netherlands
| | - Eric Mjolsness
- Departments of Computer Science and Mathematics, University of California Irvine, Irvine, USA
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42
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Ardaševa A, Gatenby RA, Anderson ARA, Byrne HM, Maini PK, Lorenzi T. A Mathematical Dissection of the Adaptation of Cell Populations to Fluctuating Oxygen Levels. Bull Math Biol 2020; 82:81. [PMID: 32556703 DOI: 10.1007/s11538-020-00754-7] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2019] [Accepted: 05/27/2020] [Indexed: 12/17/2022]
Abstract
The disordered network of blood vessels that arises from tumour angiogenesis results in variations in the delivery of oxygen into the tumour tissue. This brings about regions of chronic hypoxia (i.e. sustained low oxygen levels) and regions with alternating periods of low and relatively higher oxygen levels, and makes it necessary for cancer cells to adapt to fluctuating environmental conditions. We use a phenotype-structured model to dissect the evolutionary dynamics of cell populations exposed to fluctuating oxygen levels. In this model, the phenotypic state of every cell is described by a continuous variable that provides a simple representation of its metabolic phenotype, ranging from fully oxidative to fully glycolytic, and cells are grouped into two competing populations that undergo heritable, spontaneous phenotypic variations at different rates. Model simulations indicate that, depending on the rate at which oxygen is consumed by the cells, dynamic nonlinear interactions between cells and oxygen can stimulate chronic hypoxia and cycling hypoxia. Moreover, the model supports the idea that under chronic-hypoxic conditions lower rates of phenotypic variation lead to a competitive advantage, whereas higher rates of phenotypic variation can confer a competitive advantage under cycling-hypoxic conditions. In the latter case, the numerical results obtained show that bet-hedging evolutionary strategies, whereby cells switch between oxidative and glycolytic phenotypes, can spontaneously emerge. We explain how these results can shed light on the evolutionary process that may underpin the emergence of phenotypic heterogeneity in vascularised tumours.
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Affiliation(s)
- Aleksandra Ardaševa
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Robert A Gatenby
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, FL, USA
| | - Alexander R A Anderson
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, FL, USA
| | - Helen M Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Tommaso Lorenzi
- School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK. .,Department of Mathematical Sciences "G. L. Lagrange", Dipartimento di Eccellenza, 2018-2022, Politecnico di Torino, 10129, Turin, Italy.
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43
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Hart WS, Maini PK, Yates CA, Thompson RN. A theoretical framework for transitioning from patient-level to population-scale epidemiological dynamics: influenza A as a case study. J R Soc Interface 2020; 17:20200230. [PMID: 32400267 DOI: 10.1098/rsif.2020.0230] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Multi-scale epidemic forecasting models have been used to inform population-scale predictions with within-host models and/or infection data collected in longitudinal cohort studies. However, most multi-scale models are complex and require significant modelling expertise to run. We formulate an alternative multi-scale modelling framework using a compartmental model with multiple infected stages. In the large-compartment limit, our easy-to-use framework generates identical results compared to previous more complicated approaches. We apply our framework to the case study of influenza A in humans. By using a viral dynamics model to generate synthetic patient-level data, we explore the effects of limited and inaccurate patient data on the accuracy of population-scale forecasts. If infection data are collected daily, we find that a cohort of at least 40 patients is required for a mean population-scale forecasting error below 10%. Forecasting errors may be reduced by including more patients in future cohort studies or by increasing the frequency of observations for each patient. Our work, therefore, provides not only an accessible epidemiological modelling framework but also an insight into the data required for accurate forecasting using multi-scale models.
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Affiliation(s)
- W S Hart
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
| | - P K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
| | - C A Yates
- Centre for Mathematical Biology, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - R N Thompson
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK.,Christ Church, University of Oxford, Saint Aldate's, Oxford OX1 1DP, UK
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Ma Z, Niu B, Phan TA, Stensjøen AL, Ene C, Woodiwiss T, Wang T, Maini PK, Holland EC, Tian JP. Stochastic growth pattern of untreated human glioblastomas predicts the survival time for patients. Sci Rep 2020; 10:6642. [PMID: 32313150 PMCID: PMC7171128 DOI: 10.1038/s41598-020-63394-w] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2019] [Accepted: 03/30/2020] [Indexed: 12/01/2022] Open
Abstract
Glioblastomas are highly malignant brain tumors. Knowledge of growth rates and growth patterns is useful for understanding tumor biology and planning treatment logistics. Based on untreated human glioblastoma data collected in Trondheim, Norway, we first fit the average growth to a Gompertz curve, then find a best fitted white noise term for the growth rate variance. Combining these two fits, we obtain a new type of Gompertz diffusion dynamics, which is a stochastic differential equation (SDE). Newly collected untreated human glioblastoma data in Seattle, US, re-verify our model. Instead of growth curves predicted by deterministic models, our SDE model predicts a band with a center curve as the tumor size average and its width as the tumor size variance over time. Given the glioblastoma size in a patient, our model can predict the patient survival time with a prescribed probability. The survival time is approximately a normal random variable with simple formulas for its mean and variance in terms of tumor sizes. Our model can be applied to studies of tumor treatments. As a demonstration, we numerically investigate different protocols of surgical resection using our model and provide possible theoretical strategies.
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Affiliation(s)
- Ziwei Ma
- Department of Mathematical Sciences, New Mexico State University, 1780 E University Ave, Las Cruces, NM, 88003, USA.,College of Sciences, Northwest A&F University, 22 Xinong Rd, Yangling, Shaanxi, 712100, China
| | - Ben Niu
- Department of Mathematical Sciences, New Mexico State University, 1780 E University Ave, Las Cruces, NM, 88003, USA.,Department of Mathematics, Harbin Institute of Technology at Weihai, 2 West Wenhua Road, Weihai, Shandong, 264209, P.R. China
| | - Tuan Anh Phan
- Department of Mathematical Sciences, New Mexico State University, 1780 E University Ave, Las Cruces, NM, 88003, USA
| | - Anne Line Stensjøen
- Department of Circulation and Medical Imaging, Faculty of Medicine and Health sciences, NTNU - Norwegian University of Science and Technology, Post Box 8905, N-7491, Trondheim, Norway
| | - Chibawanye Ene
- Human Biology Division, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue N., PO Box 19024, Seattle, WA, 98109, USA
| | - Timothy Woodiwiss
- Human Biology Division, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue N., PO Box 19024, Seattle, WA, 98109, USA
| | - Tonghui Wang
- Department of Mathematical Sciences, New Mexico State University, 1780 E University Ave, Las Cruces, NM, 88003, USA
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| | - Eric C Holland
- Human Biology Division, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue N., PO Box 19024, Seattle, WA, 98109, USA. .,Solid Tumor Translational Research, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue N., PO Box 19024, Seattle, WA, 98109, USA.
| | - Jianjun Paul Tian
- Department of Mathematical Sciences, New Mexico State University, 1780 E University Ave, Las Cruces, NM, 88003, USA.
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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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Bobadilla AVP, Arévalo J, Sarró E, Byrne HM, Maini PK, Carraro T, Balocco S, Meseguer A, Alarcón T. In vitro cell migration quantification method for scratch assays. J R Soc Interface 2020; 16:20180709. [PMID: 30958186 DOI: 10.1098/rsif.2018.0709] [Citation(s) in RCA: 55] [Impact Index Per Article: 13.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023] Open
Abstract
The scratch assay is an in vitro technique used to assess the contribution of molecular and cellular mechanisms to cell migration. The assay can also be used to evaluate therapeutic compounds before clinical use. Current quantification methods of scratch assays deal poorly with irregular cell-free areas and crooked leading edges which are features typically present in the experimental data. We introduce a new migration quantification method, called 'monolayer edge velocimetry', that permits analysis of low-quality experimental data and better statistical classification of migration rates than standard quantification methods. The new method relies on quantifying the horizontal component of the cell monolayer velocity across the leading edge. By performing a classification test on in silico data, we show that the method exhibits significantly lower statistical errors than standard methods. When applied to in vitro data, our method outperforms standard methods by detecting differences in the migration rates between different cell groups that the other methods could not detect. Application of this new method will enable quantification of migration rates from in vitro scratch assay data that cannot be analysed using existing methods.
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Affiliation(s)
- Ana Victoria Ponce Bobadilla
- 1 Institute for Applied Mathematics, Heidelberg University , 69120 Heidelberg , Germany.,2 Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University , 69120 Heidelberg , Germany
| | - Jazmine Arévalo
- 3 Renal Physiopathology Group, CIBBIM-Nanomedicine, Vall d'Hebron Research Institute , Barcelona , Spain
| | - Eduard Sarró
- 3 Renal Physiopathology Group, CIBBIM-Nanomedicine, Vall d'Hebron Research Institute , Barcelona , Spain
| | - Helen M Byrne
- 4 Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford , Oxford OX2 6GG , UK
| | - Philip K Maini
- 4 Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford , Oxford OX2 6GG , UK
| | - Thomas Carraro
- 1 Institute for Applied Mathematics, Heidelberg University , 69120 Heidelberg , Germany.,2 Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University , 69120 Heidelberg , Germany
| | - Simone Balocco
- 5 Department of Mathematics and Informatics , University of Barcelona , Gran Via 585, 08007 Barcelona , Spain.,6 Computer Vision Center , 08193 Bellaterra , Spain
| | - Anna Meseguer
- 3 Renal Physiopathology Group, CIBBIM-Nanomedicine, Vall d'Hebron Research Institute , Barcelona , Spain.,7 Departament de Bioquímica i Biologia Molecular, Unitat de Bioquímica de Medicina, Universitat Autònoma de Barcelona , Bellaterra , Spain.,8 Red de Investigación Renal (REDINREN), Instituto Carlos III-FEDER , Madrid , Spain
| | - Tomás Alarcón
- 9 ICREA , Pg. Lluís Companys 23, 08010 Barcelona , Spain.,10 Centre de Recerca Matemàtica, Edifici C , Campus de Bellaterra, 08193 Bellaterra (Barcelona) , Spain.,11 Departament de Matemàtiques, Universitat Autònoma de Barcelona , 08193 Bellaterra (Barcelona) , Spain.,12 Barcelona Graduate School of Mathematics (BGSMath) , Barcelona , Spain
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47
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Ardaševa A, Gatenby RA, Anderson ARA, Byrne HM, Maini PK, Lorenzi T. Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments. J Math Biol 2020; 80:775-807. [PMID: 31641842 PMCID: PMC7028828 DOI: 10.1007/s00285-019-01441-5] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Revised: 08/14/2019] [Indexed: 12/20/2022]
Abstract
Living species, ranging from bacteria to animals, exist in environmental conditions that exhibit spatial and temporal heterogeneity which requires them to adapt. Risk-spreading through spontaneous phenotypic variations is a known concept in ecology, which is used to explain how species may survive when faced with the evolutionary risks associated with temporally varying environments. In order to support a deeper understanding of the adaptive role of spontaneous phenotypic variations in fluctuating environments, we consider a system of non-local partial differential equations modelling the evolutionary dynamics of two competing phenotype-structured populations in the presence of periodically oscillating nutrient levels. The two populations undergo heritable, spontaneous phenotypic variations at different rates. The phenotypic state of each individual is represented by a continuous variable, and the phenotypic landscape of the populations evolves in time due to variations in the nutrient level. Exploiting the analytical tractability of our model, we study the long-time behaviour of the solutions to obtain a detailed mathematical depiction of the evolutionary dynamics. The results suggest that when nutrient levels undergo small and slow oscillations, it is evolutionarily more convenient to rarely undergo spontaneous phenotypic variations. Conversely, under relatively large and fast periodic oscillations in the nutrient levels, which bring about alternating cycles of starvation and nutrient abundance, higher rates of spontaneous phenotypic variations confer a competitive advantage. We discuss the implications of our results in the context of cancer metabolism.
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Affiliation(s)
- Aleksandra Ardaševa
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK
| | - Robert A. Gatenby
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, FL USA
| | | | - Helen M. Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK
| | - Tommaso Lorenzi
- School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS UK
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48
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Strobl MAR, Krause AL, Damaghi M, Gillies R, Anderson ARA, Maini PK. Mix and Match: Phenotypic Coexistence as a Key Facilitator of Cancer Invasion. Bull Math Biol 2020; 82:15. [PMID: 31953602 PMCID: PMC6968991 DOI: 10.1007/s11538-019-00675-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2019] [Accepted: 12/03/2019] [Indexed: 01/10/2023]
Abstract
Invasion of healthy tissue is a defining feature of malignant tumours. Traditionally, invasion is thought to be driven by cells that have acquired all the necessary traits to overcome the range of biological and physical defences employed by the body. However, in light of the ever-increasing evidence for geno- and phenotypic intra-tumour heterogeneity, an alternative hypothesis presents itself: could invasion be driven by a collection of cells with distinct traits that together facilitate the invasion process? In this paper, we use a mathematical model to assess the feasibility of this hypothesis in the context of acid-mediated invasion. We assume tumour expansion is obstructed by stroma which inhibits growth and extra-cellular matrix (ECM) which blocks cancer cell movement. Further, we assume that there are two types of cancer cells: (i) a glycolytic phenotype which produces acid that kills stromal cells and (ii) a matrix-degrading phenotype that locally remodels the ECM. We extend the Gatenby-Gawlinski reaction-diffusion model to derive a system of five coupled reaction-diffusion equations to describe the resulting invasion process. We characterise the spatially homogeneous steady states and carry out a simulation study in one spatial dimension to determine how the tumour develops as we vary the strength of competition between the two phenotypes. We find that overall tumour growth is most extensive when both cell types can stably coexist, since this allows the cells to locally mix and benefit most from the combination of traits. In contrast, when inter-species competition exceeds intra-species competition the populations spatially separate and invasion arrests either: (i) rapidly (matrix-degraders dominate) or (ii) slowly (acid-producers dominate). Overall, our work demonstrates that the spatial and ecological relationship between a heterogeneous population of tumour cells is a key factor in determining their ability to cooperate. Specifically, we predict that tumours in which different phenotypes coexist stably are more invasive than tumours in which phenotypes are spatially separated.
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Affiliation(s)
- Maximilian A. R. Strobl
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OX2 6GG Oxford, UK
- Department of Integrated Mathematical Oncology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Andrew L. Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OX2 6GG Oxford, UK
| | - Mehdi Damaghi
- Department of Cancer Physiology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Robert Gillies
- Department of Cancer Physiology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Alexander R. A. Anderson
- Department of Integrated Mathematical Oncology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OX2 6GG Oxford, UK
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Lewin TD, Byrne HM, Maini PK, Caudell JJ, Moros EG, Enderling H. The importance of dead material within a tumour on the dynamics in response to radiotherapy. ACTA ACUST UNITED AC 2020; 65:015007. [DOI: 10.1088/1361-6560/ab4c27] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
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McLennan R, McKinney MC, Teddy JM, Morrison JA, Kasemeier-Kulesa JC, Ridenour DA, Manthe CA, Giniunaite R, Robinson M, Baker RE, Maini PK, Kulesa PM. Neural crest cells bulldoze through the microenvironment using Aquaporin 1 to stabilize filopodia. Development 2020; 147:dev.185231. [PMID: 31826865 DOI: 10.1242/dev.185231] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2019] [Accepted: 12/03/2019] [Indexed: 01/17/2023]
Abstract
Neural crest migration requires cells to move through an environment filled with dense extracellular matrix and mesoderm to reach targets throughout the vertebrate embryo. Here, we use high-resolution microscopy, computational modeling, and in vitro and in vivo cell invasion assays to investigate the function of Aquaporin 1 (AQP-1) signaling. We find that migrating lead cranial neural crest cells express AQP-1 mRNA and protein, implicating a biological role for water channel protein function during invasion. Differential AQP-1 levels affect neural crest cell speed and direction, as well as the length and stability of cell filopodia. Furthermore, AQP-1 enhances matrix metalloprotease activity and colocalizes with phosphorylated focal adhesion kinases. Colocalization of AQP-1 with EphB guidance receptors in the same migrating neural crest cells has novel implications for the concept of guided bulldozing by lead cells during migration.
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Affiliation(s)
- Rebecca McLennan
- Stowers Institute for Medical Research, Kansas City, MO 64110, USA
| | - Mary C McKinney
- Stowers Institute for Medical Research, Kansas City, MO 64110, USA
| | - Jessica M Teddy
- Stowers Institute for Medical Research, Kansas City, MO 64110, USA
| | - Jason A Morrison
- Stowers Institute for Medical Research, Kansas City, MO 64110, USA
| | | | | | - Craig A Manthe
- Stowers Institute for Medical Research, Kansas City, MO 64110, USA
| | - Rasa Giniunaite
- University of Oxford, Wolfson Centre for Mathematical Biology, Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK
| | - Martin Robinson
- University of Oxford, Wolfson Centre for Mathematical Biology, Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK.,Department of Computer Science, Parks Road, Oxford OX1 3QD, UK
| | - Ruth E Baker
- University of Oxford, Wolfson Centre for Mathematical Biology, Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK
| | - Philip K Maini
- University of Oxford, Wolfson Centre for Mathematical Biology, Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK
| | - Paul M Kulesa
- Stowers Institute for Medical Research, Kansas City, MO 64110, USA .,Department of Anatomy and Cell Biology, University of Kansas School of Medicine, Kansas City, KS 66160, USA
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