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Liao Q, Zhu M, Wu L, Wang D, Wang Z, Zhang S, Cao W, Pan X, Li J, Tang X, Xin J, Sun Y, Zhu J, Wang Z. Probing the capacity of a spatiotemporal deep learning model for short-term PM 2.5 forecasts in a coastal urban area. THE SCIENCE OF THE TOTAL ENVIRONMENT 2024; 950:175233. [PMID: 39102955 DOI: 10.1016/j.scitotenv.2024.175233] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2024] [Revised: 07/22/2024] [Accepted: 07/31/2024] [Indexed: 08/07/2024]
Abstract
Accurate forecast of fine particulate matter (PM2.5) is crucial for city air pollution control, yet remains challenging due to the complex urban atmospheric chemical and physical processes. Recently deep learning has been routinely applied for better urban PM2.5 forecasts. However, their capacity to represent the spatiotemporal urban atmospheric processes remains underexplored, especially compared with traditional approaches such as chemistry-transport models (CTMs) and shallow statistical methods other than deep learning. Here we probe such urban-scale representation capacity of a spatiotemporal deep learning (STDL) model for 24-hour short-term PM2.5 forecasts at six urban stations in Rizhao, a coastal city in China. Compared with two operational CTMs and three statistical models, the STDL model shows its superiority with improvements in all five evaluation metrics, notably in root mean square error (RMSE) for forecasts at lead times within 12 h with reductions of 49.8 % and 47.8 % respectively. This demonstrates the STDL model's capacity to represent nonlinear small-scale phenomena such as street-level emissions and urban meteorology that are in general not well represented in either CTMs or shallow statistical models. This gain of small-scale representation in forecast performance decreases at increasing lead times, leading to similar RMSEs to the statistical methods (linear shallow representations) at about 12 h and to the CTMs (mesoscale representations) at 24 h. The STDL model performs especially well in winter, when complex urban physical and chemical processes dominate the frequent severe air pollution, and in moisture conditions fostering hygroscopic growth of particles. The DL-based PM2.5 forecasts align with observed trends under various humidity and wind conditions. Such investigation into the potential and limitations of deep learning representation for urban PM2.5 forecasting could hopefully inspire further fusion of distinct representations from CTMs and deep networks to break the conventional limits of short-term PM2.5 forecasts.
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Affiliation(s)
- Qi Liao
- College of Electronic Engineering, Chengdu University of Information Technology, Chengdu 610225, China
| | - Mingming Zhu
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China.
| | - Lin Wu
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; Carbon Neutrality Research Center (CNRC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Dawei Wang
- State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Zixi Wang
- State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Si Zhang
- Carbon Neutrality Research Center (CNRC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Wudi Cao
- Carbon Neutrality Research Center (CNRC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Xiaole Pan
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Jie Li
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Xiao Tang
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Jinyuan Xin
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Yele Sun
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Jiang Zhu
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; International Center for Climate and Environment Science (ICCES), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
| | - Zifa Wang
- Key Laboratory of Atmospheric Environment and Extreme Meteorology, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry (LAPC), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China; College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China; Center for Excellence in Urban Atmospheric Environment, Institute of Urban Environment, Chinese Academy of Sciences, Xiamen 361021, China
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Clark LP, Zilber D, Schmitt C, Fargo DC, Reif DM, Motsinger-Reif AA, Messier KP. A review of geospatial exposure models and approaches for health data integration. JOURNAL OF EXPOSURE SCIENCE & ENVIRONMENTAL EPIDEMIOLOGY 2024:10.1038/s41370-024-00712-8. [PMID: 39251872 DOI: 10.1038/s41370-024-00712-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2024] [Revised: 08/01/2024] [Accepted: 08/05/2024] [Indexed: 09/11/2024]
Abstract
BACKGROUND Geospatial methods are common in environmental exposure assessments and increasingly integrated with health data to generate comprehensive models of environmental impacts on public health. OBJECTIVE Our objective is to review geospatial exposure models and approaches for health data integration in environmental health applications. METHODS We conduct a literature review and synthesis. RESULTS First, we discuss key concepts and terminology for geospatial exposure data and models. Second, we provide an overview of workflows in geospatial exposure model development and health data integration. Third, we review modeling approaches, including proximity-based, statistical, and mechanistic approaches, across diverse exposure types, such as air quality, water quality, climate, and socioeconomic factors. For each model type, we provide descriptions, general equations, and example applications for environmental exposure assessment. Fourth, we discuss the approaches used to integrate geospatial exposure data and health data, such as methods to link data sources with disparate spatial and temporal scales. Fifth, we describe the landscape of open-source tools supporting these workflows.
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Affiliation(s)
- Lara P Clark
- National Institute of Environmental Health Sciences, Office of the Scientific Director, Office of Data Science, Durham, NC, USA
| | - Daniel Zilber
- National Institute of Environmental Health Sciences, Division of Translational Toxicology, Predictive Toxicology Branch, Durham, NC, USA
| | - Charles Schmitt
- National Institute of Environmental Health Sciences, Office of the Scientific Director, Office of Data Science, Durham, NC, USA
| | - David C Fargo
- National Institute of Environmental Health Sciences, Office of the Director, Office of Environmental Science Cyberinfrastructure, Durham, NC, USA
| | - David M Reif
- National Institute of Environmental Health Sciences, Division of Translational Toxicology, Predictive Toxicology Branch, Durham, NC, USA
| | - Alison A Motsinger-Reif
- National Institute of Environmental Health Sciences, Division of Intramural Research, Biostatistics and Computational Biology Branch, Durham, NC, USA
| | - Kyle P Messier
- National Institute of Environmental Health Sciences, Division of Translational Toxicology, Predictive Toxicology Branch, Durham, NC, USA.
- National Institute of Environmental Health Sciences, Division of Intramural Research, Biostatistics and Computational Biology Branch, Durham, NC, USA.
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3
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Tayewo R, Septier F, Nevat I, Peters GW. Graph Regression Model for Spatial and Temporal Environmental Data-Case of Carbon Dioxide Emissions in the United States. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1272. [PMID: 37761572 PMCID: PMC10529149 DOI: 10.3390/e25091272] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2023] [Revised: 08/18/2023] [Accepted: 08/24/2023] [Indexed: 09/29/2023]
Abstract
We develop a new model for spatio-temporal data. More specifically, a graph penalty function is incorporated in the cost function in order to estimate the unknown parameters of a spatio-temporal mixed-effect model based on a generalized linear model. This model allows for more flexible and general regression relationships than classical linear ones through the use of generalized linear models (GLMs) and also captures the inherent structural dependencies or relationships of the data through this regularization based on the graph Laplacian. We use a publicly available dataset from the National Centers for Environmental Information (NCEI) in the United States of America and perform statistical inferences of future CO2 emissions in 59 counties. We empirically show how the proposed method outperforms widely used methods, such as the ordinary least squares (OLS) and ridge regression for this challenging problem.
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Affiliation(s)
- Roméo Tayewo
- Univ Bretagne Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France;
| | | | - Ido Nevat
- TUMCREATE, 1 Create Way, #10-02 CREATE Tower, Singapore 138602, Singapore;
| | - Gareth W. Peters
- Department of Statistics and Applied Probability, University of California Santa Barbara, Santa Barbara, CA 93106, USA;
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4
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Bayesian spatio-temporal models for stream networks. Comput Stat Data Anal 2022. [DOI: 10.1016/j.csda.2022.107446] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Lockout, Lockdown and Land Use: Exploring the Spatio-Temporal Evolution Patterns of Licensed Venues in Sydney, Australia between 2012 and 2021 in the Context of NSW Public Policy. BUILDINGS 2022. [DOI: 10.3390/buildings12010035] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
The hospitality industry in Sydney, Australia, has been subject to several regulatory interventions in the last decade, including lockout laws, COVID-19 lockdowns and land use planning restrictions. This study has sought to explore the spatial implications of these policies in Inner Sydney between 2012 to 2021. Methods based in spatial analysis were applied to a database of over 40,000 licensed venues. Point pattern analysis and spatial autocorrelation methods were used to identify spatially significant venue clusters. Space-time cube and emerging-hot-spot methods were used to explore clusters over time. The results indicate that most venues are located in the Sydney CBD on business-zoned land and show a high degree of spatial clustering. Spatio-temporal analysis reveals this clustering to be consistent over time, with variations between venue types. Venue numbers declined following the introduction of the lockout laws, with numbers steadily recovering in the following years. There was no discernible change in the number of venues following the COVID-19 lockdowns; however, economic data suggest that there has been a decline in revenue. Some venues were identified as having temporarily ceased trading, with these clustered in the Sydney CBD. The findings of this study provide a data-driven approach to assist policymakers and industry bodies in better understanding the spatial implications of policies targeting the hospitality sector and will assist with recovery following the COVID-19 pandemic. Further research utilising similar methods could assess the impacts of further COVID-19 lockdowns as experienced in Sydney in 2021.
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6
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Development of Roadmap for Photovoltaic Solar Technologies and Market in Poland. ENERGIES 2021. [DOI: 10.3390/en15010174] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Poland is dynamically changing its energy mix. As a result of this process, solar energy is increasing its share in energy production. The development of the solar energy market is determined by numerous factors. This paper aims to develop a roadmap for further development of the photovoltaic (PV) energy market in Poland. The scope of the research covers five areas of PV technology and market development in Poland: (i) technology; (ii) power grids; (iii) law; (iv) economic conditions; and (v) social conditions. With the use of a Technology Roadmapping Methodology (TRM), for each of the determined areas, several factors were analyzed, and their development paths were described. In addition, the article focuses on technological challenges (regarding PV cells, modules, components, power conversion and monitoring and management system, optimizers, batteries, and other energy storage systems), grid efficiency, recycling, production costs, subsidies, public awareness and education, and the energy exclusion problem. The main result of the research is the roadmap of the photovoltaic solar energy technology and market development in Poland. Further development of the PV market and technology requires parallel progress in all the identified areas. This study offers implications for policymakers, investors, managers, and technology and infrastructure developers regarding their involvement in photovoltaic market.
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7
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Broitman D, Portnov BA. Forecasting health effects potentially associated with the relocation of a major air pollution source. ENVIRONMENTAL RESEARCH 2020; 182:109088. [PMID: 31901630 DOI: 10.1016/j.envres.2019.109088] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2019] [Revised: 12/22/2019] [Accepted: 12/23/2019] [Indexed: 06/10/2023]
Abstract
Epidemiological studies often focus on risk assessments associated with exposures to specific air pollutants or proximity to different air pollution sources. Although this information is essential for devising informed health policies, it is not always helpful when it comes to the estimation of potential health effects associated with the introduction or relocation of local health hazards. In this paper, we suggest a novel approach to forecasting the morbidity-reduction impact of hypothetical removal of a major air pollution source from a densely populated urban area. The proposed approach is implemented in three stages. First, we identify and measure the strength of association of individual environmental factors with local morbidity patterns. Next, we use the estimated models to simulate the impact of removal of the pollution source under analysis and its replacement by green areas. Using this assessment, we then estimate potential changes in the local morbidity rates by mutually comparing the observed risk surface of disease with the risk surface simulated by modelling. To validate the proposed approach empirically, we use childhood asthma morbidity data available for a major metropolitan area in Israel, which hosts a large petrochemical complex. According to our estimates, relocation of the petrochemical complex in question is expected to result in about 70% drop in the childhood asthma morbidity rate area-wide. To the best of our knowledge, the present study is the first that suggests an operational approach to incorporating epidemiological assessments as an input for urban development plans related to local sources of air pollution.
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Affiliation(s)
- Dani Broitman
- Faculty of Architecture and Town Planning, Technion - Israel Institute of Technology, Technion City, Haifa, 32000, Israel.
| | - Boris A Portnov
- Department of Natural Resources & Environmental Management, University of Haifa, Mount Carmel, Haifa, 31905, Israel.
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9
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Simonis JL, Merz JE. Prey availability, environmental constraints, and aggregation dictate population distribution of an imperiled fish. Ecosphere 2019. [DOI: 10.1002/ecs2.2634] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
Affiliation(s)
| | - Joseph E. Merz
- Department of Ecology and Evolutionary Biology University of California 100 Shaffer Road Santa Cruz California 95060 USA
- Cramer Fish Sciences 3300 Industrial Boulevard, Suite 100 West Sacramento California 95691 USA
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10
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Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data. ENTROPY 2019; 21:e21020184. [PMID: 33266899 PMCID: PMC7514666 DOI: 10.3390/e21020184] [Citation(s) in RCA: 41] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/29/2018] [Revised: 02/03/2019] [Accepted: 02/12/2019] [Indexed: 11/20/2022]
Abstract
Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications.
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12
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Tensor Cubic Smoothing Splines in Designed Experiments Requiring Residual Modelling. JOURNAL OF AGRICULTURAL, BIOLOGICAL AND ENVIRONMENTAL STATISTICS 2018. [DOI: 10.1007/s13253-018-0334-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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13
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McDermott PL, Wikle CK, Millspaugh J. A hierarchical spatiotemporal analog forecasting model for count data. Ecol Evol 2018; 8:790-800. [PMID: 29321914 PMCID: PMC5756884 DOI: 10.1002/ece3.3621] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2016] [Revised: 07/26/2017] [Accepted: 10/11/2017] [Indexed: 11/25/2022] Open
Abstract
Analog forecasting is a mechanism‐free nonlinear method that forecasts a system forward in time by examining how past states deemed similar to the current state moved forward. Previous applications of analog forecasting has been successful at producing robust forecasts for a variety of ecological and physical processes, but it has typically been presented in an empirical or heuristic procedure, rather than as a formal statistical model. The methodology presented here extends the model‐based analog method of McDermott and Wikle (Environmetrics, 27, 2016, 70) by placing analog forecasting within a fully hierarchical statistical framework that can accommodate count observations. Using a Bayesian approach, the hierarchical analog model is able to quantify rigorously the uncertainty associated with forecasts. Forecasting waterfowl settling patterns in the northwestern United States and Canada is conducted by applying the hierarchical analog model to a breeding population survey dataset. Sea surface temperature (SST) in the Pacific Ocean is used to help identify potential analogs for the waterfowl settling patterns.
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14
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McDermott PL, Wikle CK. An ensemble quadratic echo state network for non-linear spatio-temporal forecasting. Stat (Int Stat Inst) 2017. [DOI: 10.1002/sta4.160] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Affiliation(s)
- Patrick L. McDermott
- Department of Statistics; University of Missouri; 146 Middlebush Hall Columbia 65211 MO USA
| | - Christopher K. Wikle
- Department of Statistics; University of Missouri; 146 Middlebush Hall Columbia 65211 MO USA
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15
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Maclaren OJ, Parker A, Pin C, Carding SR, Watson AJM, Fletcher AG, Byrne HM, Maini PK. A hierarchical Bayesian model for understanding the spatiotemporal dynamics of the intestinal epithelium. PLoS Comput Biol 2017; 13:e1005688. [PMID: 28753601 PMCID: PMC5550005 DOI: 10.1371/journal.pcbi.1005688] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2016] [Revised: 08/09/2017] [Accepted: 07/18/2017] [Indexed: 01/13/2023] Open
Abstract
Our work addresses two key challenges, one biological and one methodological. First, we aim to understand how proliferation and cell migration rates in the intestinal epithelium are related under healthy, damaged (Ara-C treated) and recovering conditions, and how these relations can be used to identify mechanisms of repair and regeneration. We analyse new data, presented in more detail in a companion paper, in which BrdU/IdU cell-labelling experiments were performed under these respective conditions. Second, in considering how to more rigorously process these data and interpret them using mathematical models, we use a probabilistic, hierarchical approach. This provides a best-practice approach for systematically modelling and understanding the uncertainties that can otherwise undermine the generation of reliable conclusions-uncertainties in experimental measurement and treatment, difficult-to-compare mathematical models of underlying mechanisms, and unknown or unobserved parameters. Both spatially discrete and continuous mechanistic models are considered and related via hierarchical conditional probability assumptions. We perform model checks on both in-sample and out-of-sample datasets and use them to show how to test possible model improvements and assess the robustness of our conclusions. We conclude, for the present set of experiments, that a primarily proliferation-driven model suffices to predict labelled cell dynamics over most time-scales.
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Affiliation(s)
- Oliver J. Maclaren
- Department of Engineering Science, University of Auckland, Auckland, New Zealand
| | - Aimée Parker
- Gut Health and Food Safety Research Programme, Institute of Food Research, Norwich, United Kingdom
| | - Carmen Pin
- Gut Health and Food Safety Research Programme, Institute of Food Research, Norwich, United Kingdom
| | - Simon R. Carding
- Gut Health and Food Safety Research Programme, Institute of Food Research, Norwich, United Kingdom
- Norwich Medical School, University of East Anglia, Norwich, United Kingdom
| | - Alastair J. M. Watson
- Gut Health and Food Safety Research Programme, Institute of Food Research, Norwich, United Kingdom
- Norwich Medical School, University of East Anglia, Norwich, United Kingdom
| | - Alexander G. Fletcher
- School of Mathematics and Statistics, University of Sheffield, Sheffield, United Kingdom
- Bateson Centre, University of Sheffield, Sheffield, United Kingdom
| | - Helen M. Byrne
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
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16
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Hu X, Steinsland I. Spatial modeling with system of stochastic partial differential equations. ACTA ACUST UNITED AC 2016. [DOI: 10.1002/wics.1378] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Affiliation(s)
- Xiangping Hu
- Department of Mathematics; University of Oslo; Oslo Norway
| | - Ingelin Steinsland
- Department of Mathematical Sciences; Norwegian University of Science and Technology; Trondheim Norway
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