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Simpson MJ, Maclaren OJ. Making Predictions Using Poorly Identified Mathematical Models. Bull Math Biol 2024; 86:80. [PMID: 38801489 PMCID: PMC11129983 DOI: 10.1007/s11538-024-01294-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2023] [Accepted: 04/10/2024] [Indexed: 05/29/2024]
Abstract
Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub . Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.
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Affiliation(s)
- Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
| | - Oliver J Maclaren
- Department of Engineering Science and Biomedical Engineering, University of Auckland, Auckland, New Zealand
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2
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Mostofinejad A, Romero DA, Brinson D, Marin-Araujo AE, Bazylak A, Waddell TK, Haykal S, Karoubi G, Amon CH. In silico model development and optimization of in vitro lung cell population growth. PLoS One 2024; 19:e0300902. [PMID: 38748626 PMCID: PMC11095723 DOI: 10.1371/journal.pone.0300902] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2023] [Accepted: 03/04/2024] [Indexed: 05/19/2024] Open
Abstract
Tissue engineering predominantly relies on trial and error in vitro and ex vivo experiments to develop protocols and bioreactors to generate functional tissues. As an alternative, in silico methods have the potential to significantly reduce the timelines and costs of experimental programs for tissue engineering. In this paper, we propose a methodology to formulate, select, calibrate, and test mathematical models to predict cell population growth as a function of the biochemical environment and to design optimal experimental protocols for model inference of in silico model parameters. We systematically combine methods from the experimental design, mathematical statistics, and optimization literature to develop unique and explainable mathematical models for cell population dynamics. The proposed methodology is applied to the development of this first published model for a population of the airway-relevant bronchio-alveolar epithelial (BEAS-2B) cell line as a function of the concentration of metabolic-related biochemical substrates. The resulting model is a system of ordinary differential equations that predict the temporal dynamics of BEAS-2B cell populations as a function of the initial seeded cell population and the glucose, oxygen, and lactate concentrations in the growth media, using seven parameters rigorously inferred from optimally designed in vitro experiments.
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Affiliation(s)
- Amirmahdi Mostofinejad
- Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada
| | - David A. Romero
- Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada
| | - Dana Brinson
- Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
| | - Alba E. Marin-Araujo
- Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
- Latner Research Laboratories, Division of Thoracic Surgery, University Health Network, Toronto, Ontario, Canada
| | - Aimy Bazylak
- Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada
| | - Thomas K. Waddell
- Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
- Latner Research Laboratories, Division of Thoracic Surgery, University Health Network, Toronto, Ontario, Canada
| | - Siba Haykal
- Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
- Division of Plastic Surgery, University Health Network, Toronto, Ontario, Canada
| | - Golnaz Karoubi
- Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
- Latner Research Laboratories, Division of Thoracic Surgery, University Health Network, Toronto, Ontario, Canada
| | - Cristina H. Amon
- Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada
- Institute of Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
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Conte M, Woodall RT, Gutova M, Chen BT, Shiroishi MS, Brown CE, Munson JM, Rockne RC. Structural and practical identifiability of contrast transport models for DCE-MRI. PLoS Comput Biol 2024; 20:e1012106. [PMID: 38748755 PMCID: PMC11132485 DOI: 10.1371/journal.pcbi.1012106] [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: 12/21/2023] [Revised: 05/28/2024] [Accepted: 04/24/2024] [Indexed: 05/28/2024] Open
Abstract
Contrast transport models are widely used to quantify blood flow and transport in dynamic contrast-enhanced magnetic resonance imaging. These models analyze the time course of the contrast agent concentration, providing diagnostic and prognostic value for many biological systems. Thus, ensuring accuracy and repeatability of the model parameter estimation is a fundamental concern. In this work, we analyze the structural and practical identifiability of a class of nested compartment models pervasively used in analysis of MRI data. We combine artificial and real data to study the role of noise in model parameter estimation. We observe that although all the models are structurally identifiable, practical identifiability strongly depends on the data characteristics. We analyze the impact of increasing data noise on parameter identifiability and show how the latter can be recovered with increased data quality. To complete the analysis, we show that the results do not depend on specific tissue characteristics or the type of enhancement patterns of contrast agent signal.
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Affiliation(s)
- Martina Conte
- Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Torino, Italy
- Division of Mathematical Oncology and Computational Systems Biology, Department of Computational and Quantitative Medicine, Beckman Research Institute, City of Hope National Medical Center, Duarte, California, United States of America
| | - Ryan T. Woodall
- Division of Mathematical Oncology and Computational Systems Biology, Department of Computational and Quantitative Medicine, Beckman Research Institute, City of Hope National Medical Center, Duarte, California, United States of America
| | - Margarita Gutova
- Department of Stem Cell Biology and Regenerative Medicine, Beckman Research Institute, City of Hope National Medical Center, Duarte, California, United States of America
| | - Bihong T. Chen
- Department of Diagnostic Radiology, City of Hope National Medical Center, Duarte, California, United States of America
| | - Mark S. Shiroishi
- Department of Radiology, Keck School of Medicine of the University of Southern California, Los Angeles, California, United States of America
| | - Christine E. Brown
- Departments of Hematology & Hematopoietic Cell Transplantation and Immuno-Oncology, Beckman Research Institute, City of Hope National Medical Center Duarte, California, United States of America
| | - Jennifer M. Munson
- Fralin Biomedical Research Institute, Virginia Tech, Roanoke, Virginia, United States of America
| | - Russell C. Rockne
- Division of Mathematical Oncology and Computational Systems Biology, Department of Computational and Quantitative Medicine, Beckman Research Institute, City of Hope National Medical Center, Duarte, California, United States of America
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Wheeler J, Rosengart A, Jiang Z, Tan K, Treutle N, Ionides EL. Informing policy via dynamic models: Cholera in Haiti. PLoS Comput Biol 2024; 20:e1012032. [PMID: 38683863 PMCID: PMC11081515 DOI: 10.1371/journal.pcbi.1012032] [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: 10/07/2023] [Revised: 05/09/2024] [Accepted: 03/29/2024] [Indexed: 05/02/2024] Open
Abstract
Public health decisions must be made about when and how to implement interventions to control an infectious disease epidemic. These decisions should be informed by data on the epidemic as well as current understanding about the transmission dynamics. Such decisions can be posed as statistical questions about scientifically motivated dynamic models. Thus, we encounter the methodological task of building credible, data-informed decisions based on stochastic, partially observed, nonlinear dynamic models. This necessitates addressing the tradeoff between biological fidelity and model simplicity, and the reality of misspecification for models at all levels of complexity. We assess current methodological approaches to these issues via a case study of the 2010-2019 cholera epidemic in Haiti. We consider three dynamic models developed by expert teams to advise on vaccination policies. We evaluate previous methods used for fitting these models, and we demonstrate modified data analysis strategies leading to improved statistical fit. Specifically, we present approaches for diagnosing model misspecification and the consequent development of improved models. Additionally, we demonstrate the utility of recent advances in likelihood maximization for high-dimensional nonlinear dynamic models, enabling likelihood-based inference for spatiotemporal incidence data using this class of models. Our workflow is reproducible and extendable, facilitating future investigations of this disease system.
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Affiliation(s)
- Jesse Wheeler
- Statistics Department, University of Michigan, Ann Arbor, Michigan, United States of America
| | - AnnaElaine Rosengart
- Statistics and Data Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, United States of America
| | - Zhuoxun Jiang
- Statistics Department, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Kevin Tan
- Wharton Statistics and Data Science, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
| | - Noah Treutle
- Statistics Department, University of Michigan, Ann Arbor, Michigan, United States of America
| | - Edward L. Ionides
- Statistics Department, University of Michigan, Ann Arbor, Michigan, United States of America
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Simpson MJ, Murphy RJ, Maclaren OJ. Modelling count data with partial differential equation models in biology. J Theor Biol 2024; 580:111732. [PMID: 38218530 DOI: 10.1016/j.jtbi.2024.111732] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2023] [Revised: 12/03/2023] [Accepted: 01/08/2024] [Indexed: 01/15/2024]
Abstract
Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth-death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction-diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.
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Affiliation(s)
- Matthew J Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
| | - Ryan J Murphy
- School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia
| | - Oliver J Maclaren
- Department of Engineering Science and Biomedical Engineering, University of Auckland, Auckland, New Zealand
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Murphy RJ, Maclaren OJ, Simpson MJ. Implementing measurement error models with mechanistic mathematical models in a likelihood-based framework for estimation, identifiability analysis and prediction in the life sciences. J R Soc Interface 2024; 21:20230402. [PMID: 38290560 PMCID: PMC10827430 DOI: 10.1098/rsif.2023.0402] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2023] [Accepted: 01/03/2024] [Indexed: 02/01/2024] Open
Abstract
Throughout the life sciences, we routinely seek to interpret measurements and observations using parametrized mechanistic mathematical models. A fundamental and often overlooked choice in this approach involves relating the solution of a mathematical model with noisy and incomplete measurement data. This is often achieved by assuming that the data are noisy measurements of the solution of a deterministic mathematical model, and that measurement errors are additive and normally distributed. While this assumption of additive Gaussian noise is extremely common and simple to implement and interpret, it is often unjustified and can lead to poor parameter estimates and non-physical predictions. One way to overcome this challenge is to implement a different measurement error model. In this review, we demonstrate how to implement a range of measurement error models in a likelihood-based framework for estimation, identifiability analysis and prediction, called profile-wise analysis. This frequentist approach to uncertainty quantification for mechanistic models leverages the profile likelihood for targeting parameters and understanding their influence on predictions. Case studies, motivated by simple caricature models routinely used in systems biology and mathematical biology literature, illustrate how the same ideas apply to different types of mathematical models. Open-source Julia code to reproduce results is available on GitHub.
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Affiliation(s)
- Ryan J. Murphy
- School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria, Australia
| | - Oliver J. Maclaren
- Department of Engineering Science and Biomedical Engineering, University of Auckland, Auckland, New Zealand
| | - Matthew J. Simpson
- Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
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Conte M, Woodall RT, Gutova M, Chen BT, Shiroishi MS, Brown CE, Munson JM, Rockne RC. Structural and practical identifiability of contrast transport models for DCE-MRI. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.12.19.572294. [PMID: 38187554 PMCID: PMC10769233 DOI: 10.1101/2023.12.19.572294] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/09/2024]
Abstract
Compartment models are widely used to quantify blood flow and transport in dynamic contrast-enhanced magnetic resonance imaging. These models analyze the time course of the contrast agent concentration, providing diagnostic and prognostic value for many biological systems. Thus, ensuring accuracy and repeatability of the model parameter estimation is a fundamental concern. In this work, we analyze the structural and practical identifiability of a class of nested compartment models pervasively used in analysis of MRI data. We combine artificial and real data to study the role of noise in model parameter estimation. We observe that although all the models are structurally identifiable, practical identifiability strongly depends on the data characteristics. We analyze the impact of increasing data noise on parameter identifiability and show how the latter can be recovered with increased data quality. To complete the analysis, we show that the results do not depend on specific tissue characteristics or the type of enhancement patterns of contrast agent signal.
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Murphy RJ, Gunasingh G, Haass NK, Simpson MJ. Formation and Growth of Co-Culture Tumour Spheroids: New Compartment-Based Mathematical Models and Experiments. Bull Math Biol 2023; 86:8. [PMID: 38091169 DOI: 10.1007/s11538-023-01229-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2023] [Accepted: 10/23/2023] [Indexed: 12/18/2023]
Abstract
Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to quantitatively characterise and interpret co-culture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, such as differing proliferation rates or responses to nutrient availability. At present there is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here, using a range of ordinary differential equation-based mathematical models and parameter estimation, we interpret new co-culture experimental data. We provide new biological insights about spheroid formation, growth, and structure. As part of this analysis we connect Greenspan's seminal mathematical model to co-culture data for the first time. Furthermore, we generalise a class of compartment-based spheroid mathematical models that have previously been restricted to one population so they can be applied to multiple populations. As special cases of the general model, we explore multiple natural two population extensions to Greenspan's seminal model and reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. This mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.
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Affiliation(s)
- Ryan J Murphy
- Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
| | - Gency Gunasingh
- Frazer Institute, The University of Queensland, Brisbane, Australia
| | - Nikolas K Haass
- Frazer Institute, The University of Queensland, Brisbane, Australia
| | - Matthew J Simpson
- Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
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