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Pasetto S, Montejo M, Zahid MU, Rosa M, Gatenby R, Schlicke P, Diaz R, Enderling H. Calibrating tumor growth and invasion parameters with spectral spatial analysis of cancer biopsy tissues. NPJ Syst Biol Appl 2024; 10:112. [PMID: 39358360 PMCID: PMC11447233 DOI: 10.1038/s41540-024-00439-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2023] [Accepted: 09/18/2024] [Indexed: 10/04/2024] Open
Abstract
The reaction-diffusion equation is widely used in mathematical models of cancer. The calibration of model parameters based on limited clinical data is critical to using reaction-diffusion equation simulations for reliable predictions on a per-patient basis. Here, we focus on cell-level data as routinely available from tissue biopsies used for clinical cancer diagnosis. We analyze the spatial architecture in biopsy tissues stained with multiplex immunofluorescence. We derive a two-point correlation function and the corresponding spatial power spectral distribution. We show that this data-deduced power spectral distribution can fit the power spectrum of the solution of reaction-diffusion equations that can then identify patient-specific tumor growth and invasion rates. This approach allows the measurement of patient-specific critical tumor dynamical properties from routinely available biopsy material at a single snapshot in time.
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Affiliation(s)
- Stefano Pasetto
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, 12902 Magnolia Drive, Tampa, FL, USA.
| | - Michael Montejo
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, 12902 Magnolia Drive, Tampa, FL, USA
| | - Mohammad U Zahid
- Department of Radiation Oncology, The University of Texas MD Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX, USA
| | - Marilin Rosa
- Department of Pathology, H. Lee Moffitt Cancer Center and Research Institute, 12902 Magnolia Drive, Tampa, FL, USA
| | - Robert Gatenby
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, 12902 Magnolia Drive, Tampa, FL, USA
- Department of Radiology, H. Lee Moffitt Cancer Center and Research Institute, 12902 Magnolia Drive, Tampa, FL, USA
| | - Pirmin Schlicke
- Department of Radiation Oncology, The University of Texas MD Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX, USA
| | - Roberto Diaz
- Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, 12902 Magnolia Drive, Tampa, FL, USA
| | - Heiko Enderling
- Department of Radiation Oncology, The University of Texas MD Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX, USA.
- Institute for Data Science in Oncology, The University of Texas MD Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX, USA.
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Hywood JD, Rice G, Pageon SV, Read MN, Biro M. Detection and characterization of chemotaxis without cell tracking. J R Soc Interface 2021; 18:20200879. [PMID: 33715400 PMCID: PMC8086846 DOI: 10.1098/rsif.2020.0879] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2020] [Accepted: 02/15/2021] [Indexed: 02/06/2023] Open
Abstract
Swarming has been observed in various biological systems from collective animal movements to immune cells. In the cellular context, swarming is driven by the secretion of chemotactic factors. Despite the critical role of chemotactic swarming, few methods to robustly identify and quantify this phenomenon exist. Here, we present a novel method for the analysis of time series of positional data generated from realizations of agent-based processes. We convert the positional data for each individual time point to a function measuring agent aggregation around a given area of interest, hence generating a functional time series. The functional time series, and a more easily visualized swarming metric of agent aggregation derived from these functions, provide useful information regarding the evolution of the underlying process over time. We extend our method to build upon the modelling of collective motility using drift-diffusion partial differential equations (PDEs). Using a functional linear model, we are able to use the functional time series to estimate the drift and diffusivity terms associated with the underlying PDE. By producing an accurate estimate for the drift coefficient, we can infer the strength and range of attraction or repulsion exerted on agents, as in chemotaxis. Our approach relies solely on using agent positional data. The spatial distribution of diffusing chemokines is not required, nor do individual agents need to be tracked over time. We demonstrate our approach using random walk simulations of chemotaxis and experiments investigating cytotoxic T cells interacting with tumouroids.
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Affiliation(s)
- Jack D. Hywood
- Sydney Medical School, The University of Sydney, Sydney, Australia
| | - Gregory Rice
- Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada
| | - Sophie V. Pageon
- EMBL Australia, Single Molecule Science node, School of Medical Sciences, University of New South Wales, Sydney, Australia
| | - Mark N. Read
- School of Computer Science & Charles Perkins Centre, University of Sydney, Sydney, Australia
| | - Maté Biro
- EMBL Australia, Single Molecule Science node, School of Medical Sciences, University of New South Wales, Sydney, Australia
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3
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De Oliveira AL, Binder BJ. Discrete Manhattan and Chebyshev pair correlation functions in k dimensions. Phys Rev E 2020; 102:012130. [PMID: 32795028 DOI: 10.1103/physreve.102.012130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2020] [Accepted: 06/24/2020] [Indexed: 12/22/2022]
Abstract
Pair correlation functions provide a summary statistic which quantifies the amount of spatial correlation between objects in a spatial domain. While pair correlation functions are commonly used to quantify continuous-space point processes, the on-lattice discrete case is less studied. Recent work has brought attention to the discrete case, wherein on-lattice pair correlation functions are formed by normalizing empirical pair distances against the probability distribution of random pair distances in a lattice with Manhattan and Chebyshev metrics. These distance distributions are typically derived on an ad hoc basis as required for specific applications. Here we present a generalized approach to deriving the probability distributions of pair distances in a lattice with discrete Manhattan and Chebyshev metrics, extending the Manhattan and Chebyshev pair correlation functions to lattices in k dimensions. We also quantify the variability of the Manhattan and Chebyshev pair correlation functions, which is important to understanding the reliability and confidence of the statistic.
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Affiliation(s)
| | - Benjamin J Binder
- School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
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5
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Browning AP, Jin W, Plank MJ, Simpson MJ. Identifying density-dependent interactions in collective cell behaviour. J R Soc Interface 2020; 17:20200143. [PMID: 32343933 DOI: 10.1098/rsif.2020.0143] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Scratch assays are routinely used to study collective cell behaviour in vitro. Typical experimental protocols do not vary the initial density of cells, and typical mathematical modelling approaches describe cell motility and proliferation based on assumptions of linear diffusion and logistic growth. Jin et al. (Jin et al. 2016 J. Theor. Biol. 390, 136-145 (doi:10.1016/j.jtbi.2015.10.040)) find that the behaviour of cells in scratch assays is density-dependent, and show that standard modelling approaches cannot simultaneously describe data initiated across a range of initial densities. To address this limitation, we calibrate an individual-based model to scratch assay data across a large range of initial densities. Our model allows proliferation, motility, and a direction bias to depend on interactions between neighbouring cells. By considering a hierarchy of models where we systematically and sequentially remove interactions, we perform model selection analysis to identify the minimum interactions required for the model to simultaneously describe data across all initial densities. The calibrated model is able to match the experimental data across all densities using a single parameter distribution, and captures details about the spatial structure of cells. Our results provide strong evidence to suggest that motility is density-dependent in these experiments. On the other hand, we do not see the effect of crowding on proliferation in these experiments. These results are significant as they are precisely the opposite of the assumptions in standard continuum models, such as the Fisher-Kolmogorov equation and its generalizations.
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Affiliation(s)
- Alexander P Browning
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
| | - Wang Jin
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.,ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia
| | - Michael J Plank
- Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand.,Te Pūnaha Matatini, a New Zealand Centre of Research Excellence, New Zealand
| | - Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
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6
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Johnston ST, Crampin EJ. Corrected pair correlation functions for environments with obstacles. Phys Rev E 2019; 99:032124. [PMID: 30999485 DOI: 10.1103/physreve.99.032124] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2018] [Indexed: 01/30/2023]
Abstract
Environments with immobile obstacles or void regions that inhibit and alter the motion of individuals within that environment are ubiquitous. Correlation in the location of individuals within such environments arises as a combination of the mechanisms governing individual behavior and the heterogeneous structure of the environment. Measures of spatial structure and correlation have been successfully implemented to elucidate the roles of the mechanisms underpinning the behavior of individuals. In particular, the pair correlation function has been used across biology, ecology, and physics to obtain quantitative insight into a variety of processes. However, naively applying standard pair correlation functions in the presence of obstacles may fail to detect correlation, or suggest false correlations, due to a reliance on a distance metric that does not account for obstacles. To overcome this problem, here we present an analytic expression for calculating a corrected pair correlation function for lattice-based domains containing obstacles. We demonstrate that this obstacle pair correlation function is necessary for isolating the correlation associated with the behavior of individuals, rather than the structure of the environment. Using simulations that mimic cell migration and proliferation we demonstrate that the obstacle pair correlation function recovers the short-range correlation known to be present in this process, independent of the heterogeneous structure of the environment. Further, we show that the analytic calculation of the obstacle pair correlation function derived here is significantly faster to implement than the corresponding numerical approach.
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Affiliation(s)
- Stuart T Johnston
- Systems Biology Laboratory, School of Mathematics and Statistics, and Department of Biomedical Engineering, University of Melbourne, Parkville, Victoria 3010, Australia.,ARC Centre of Excellence in Convergent Bio-Nano Science and Technology, Melbourne School of Engineering, University of Melbourne, Parkville, Victoria 3010, Australia
| | - Edmund J Crampin
- Systems Biology Laboratory, School of Mathematics and Statistics, and Department of Biomedical Engineering, University of Melbourne, Parkville, Victoria 3010, Australia.,ARC Centre of Excellence in Convergent Bio-Nano Science and Technology, Melbourne School of Engineering, University of Melbourne, Parkville, Victoria 3010, Australia.,School of Medicine, Faculty of Medicine Dentistry and Health Sciences, University of Melbourne, Parkville, Victoria 3010, Australia
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Gavagnin E, Owen JP, Yates CA. Pair correlation functions for identifying spatial correlation in discrete domains. Phys Rev E 2018; 97:062104. [PMID: 30011502 DOI: 10.1103/physreve.97.062104] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2018] [Indexed: 12/16/2022]
Abstract
Identifying and quantifying spatial correlation are important aspects of studying the collective behavior of multiagent systems. Pair correlation functions (PCFs) are powerful statistical tools that can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation that have not been studied before, including hexagonal, triangular, and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric, allowing investigation of spatial correlation in irregular lattices for which recognizing correlation is less intuitive.
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Affiliation(s)
- Enrico Gavagnin
- Centre for Mathematical Biology, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom
| | - Jennifer P Owen
- Centre for Mathematical Biology, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom
| | - Christian A Yates
- Centre for Mathematical Biology, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom
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Tronnolone H, Gardner JM, Sundstrom JF, Jiranek V, Oliver SG, Binder BJ. Quantifying the dominant growth mechanisms of dimorphic yeast using a lattice-based model. J R Soc Interface 2018; 14:rsif.2017.0314. [PMID: 28954849 DOI: 10.1098/rsif.2017.0314] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2017] [Accepted: 08/31/2017] [Indexed: 12/22/2022] Open
Abstract
A mathematical model is presented for the growth of yeast that incorporates both dimorphic behaviour and nutrient diffusion. The budding patterns observed in the standard and pseudohyphal growth modes are represented by a bias in the direction of cell proliferation. A set of spatial indices is developed to quantify the morphology and compare the relative importance of the directional bias to nutrient concentration and diffusivity on colony shape. It is found that there are three different growth modes: uniform growth, diffusion-limited growth (DLG) and an intermediate region in which the bias determines the morphology. The dimorphic transition due to nutrient limitation is investigated by relating the directional bias to the nutrient concentration, and this is shown to replicate the behaviour observed in vivo Comparisons are made with experimental data, from which it is found that the model captures many of the observed features. Both DLG and pseudohyphal growth are found to be capable of generating observed experimental morphologies.
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Affiliation(s)
- Hayden Tronnolone
- School of Mathematical Sciences, Waite Campus, University of Adelaide, Adelaide, South Australia 5005, Australia
| | - Jennifer M Gardner
- Department of Wine and Food Science, Waite Campus, University of Adelaide, Urrbrae, SA 5064, Australia
| | - Joanna F Sundstrom
- Department of Wine and Food Science, Waite Campus, University of Adelaide, Urrbrae, SA 5064, Australia
| | - Vladimir Jiranek
- Department of Wine and Food Science, Waite Campus, University of Adelaide, Urrbrae, SA 5064, Australia
| | - Stephen G Oliver
- Cambridge Systems Biology Centre and Department of Biochemistry, University of Cambridge, Cambridge CB2 1GA, UK
| | - Benjamin J Binder
- School of Mathematical Sciences, Waite Campus, University of Adelaide, Adelaide, South Australia 5005, Australia
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Ross RJH, Yates CA, Baker RE. Variable species densities are induced by volume exclusion interactions upon domain growth. Phys Rev E 2017; 95:032416. [PMID: 28415323 DOI: 10.1103/physreve.95.032416] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2016] [Indexed: 11/07/2022]
Abstract
In this work we study the effect of domain growth on spatial correlations in agent populations containing multiple species. This is important as heterogenous cell populations are ubiquitous during the embryonic development of many species. We have previously shown that the long-term behavior of an agent population depends on the way in which domain growth is implemented. We extend this work to show that, depending on the way in which domain growth is implemented, different species dominate in multispecies simulations. Continuum approximations of the lattice-based model that ignore spatial correlations cannot capture this behavior, while those that explicitly account for spatial correlations can. The results presented here show that the precise mechanism of domain growth can determine the long-term behavior of multispecies populations and, in certain circumstances, establish spatially varying species densities.
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Affiliation(s)
- Robert J H Ross
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
| | - C A Yates
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom
| | - R E Baker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
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10
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Collective Cell Behaviour with Neighbour-Dependent Proliferation, Death and Directional Bias. Bull Math Biol 2016; 78:2277-2301. [DOI: 10.1007/s11538-016-0222-9] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2016] [Accepted: 10/04/2016] [Indexed: 11/26/2022]
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11
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Binny RN, Plank MJ, James A. Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias. J R Soc Interface 2016; 12:rsif.2015.0228. [PMID: 25904529 DOI: 10.1098/rsif.2015.0228] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023] Open
Abstract
The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and ignores the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual-based model (IBM) that uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. We incorporate a mechanism for local directional bias such that an individual's direction of movement is dependent on the degree of cell crowding in its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias, and we solve this numerically for a spatially homogeneous case.
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Affiliation(s)
- Rachelle N Binny
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand Te Pūnaha Matatini, New Zealand
| | - Michael J Plank
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand Te Pūnaha Matatini, New Zealand
| | - Alex James
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand Te Pūnaha Matatini, New Zealand
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12
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Binny RN, Haridas P, James A, Law R, Simpson MJ, Plank MJ. Spatial structure arising from neighbour-dependent bias in collective cell movement. PeerJ 2016; 4:e1689. [PMID: 26893970 PMCID: PMC4756733 DOI: 10.7717/peerj.1689] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2015] [Accepted: 01/23/2016] [Indexed: 11/20/2022] Open
Abstract
Mathematical models of collective cell movement often neglect the effects of spatial structure, such as clustering, on the population dynamics. Typically, they assume that individuals interact with one another in proportion to their average density (the mean-field assumption) which means that cell-cell interactions occurring over short spatial ranges are not accounted for. However, in vitro cell culture studies have shown that spatial correlations can play an important role in determining collective behaviour. Here, we take a combined experimental and modelling approach to explore how individual-level interactions give rise to spatial structure in a moving cell population. Using imaging data from in vitro experiments, we quantify the extent of spatial structure in a population of 3T3 fibroblast cells. To understand how this spatial structure arises, we develop a lattice-free individual-based model (IBM) and simulate cell movement in two spatial dimensions. Our model allows an individual's direction of movement to be affected by interactions with other cells in its neighbourhood, providing insights into how directional bias generates spatial structure. We consider how this behaviour scales up to the population level by using the IBM to derive a continuum description in terms of the dynamics of spatial moments. In particular, we account for spatial correlations between cells by considering dynamics of the second spatial moment (the average density of pairs of cells). Our numerical results suggest that the moment dynamics description can provide a good approximation to averaged simulation results from the underlying IBM. Using our in vitro data, we estimate parameters for the model and show that it can generate similar spatial structure to that observed in a 3T3 fibroblast cell population.
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Affiliation(s)
- Rachelle N Binny
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand; Te Pūnaha Matatini, New Zealand; Landcare Research-Manaaki Whenua, Lincoln, New Zealand
| | - Parvathi Haridas
- Institute of Health and Biomedical Innovation, Queensland University of Technology , Brisbane , Australia
| | - Alex James
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand; Te Pūnaha Matatini, New Zealand
| | - Richard Law
- York Centre for Complex Systems Analysis, Ron Cooke Hub, University of York , York , United Kingdom
| | - Matthew J Simpson
- Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, Australia; School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Michael J Plank
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand; Te Pūnaha Matatini, New Zealand
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Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection. J Theor Biol 2016; 390:136-45. [DOI: 10.1016/j.jtbi.2015.10.040] [Citation(s) in RCA: 75] [Impact Index Per Article: 9.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2015] [Revised: 09/30/2015] [Accepted: 10/31/2015] [Indexed: 11/18/2022]
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