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Cell invasion is a process in which cells degrade surrounding tissue and start populating the newly created space. It occurs in healthy and ill cells, during wound-healing but also during cancer. There are many mathematical models of different modalities and complexity levels that aim to describe and quantify this phenomenon. In this work, we compare the outputs of two partial differential equation (PDE) models \cite{Crossley, Colson} and a hybrid model based on \cite{vanOers} using a naïve data fit, and the relationships of their parameters with a variance-based sensitivity analysis to see how different parameter interactions can or cannot produce similar results. We observe rather superficial parameter relationships in the PDE models, and more deeply intertwined relations in the hybrid model. Despite these differences on a deeper level, we find that the PDE models are suitable to describe the behavior of the hybrid model when restricted to the same, one-dimensional domain. We conclude with a parameter identifiability analysis with the simplest model to understand whether it is reduced enough to allow for a confident parameter estimation. We find that only half of the parameters are practically identifiable, and justify this discovery with the findings from the sensitivity analysis.
Bibliography
@article{Crossley,
abstract = {Many reaction-diffusion models produce travelling wave solutions that can be interpreted as waves of invasion in biological scenarios such as wound healing or tumour growth. These partial differential equation models have since been adapted to describe the interactions between cells and extracellular matrix ({ECM}), using a variety of different underlying assumptions. In this work, we derive a system of reaction-diffusion equations, with cross-species density-dependent diffusion, by coarse-graining an agent-based, volume-filling model of cell invasion into {ECM}. We study the resulting travelling wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic comparison between the behaviours observed in this model and those predicted by simpler models in the literature which do not take into account volume-filling effects in the same way. Our study justifies the use of some of these simpler, more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes, but it also reveals some interesting properties arising from the introduction of cell and {ECM} volume-filling effects, where standard model simplifications might not be appropriate.},
author = {Crossley, Rebecca M. and Maini, Philip K. and Lorenzi, Tommaso and Baker, Ruth E.},
year = {2023},
title = {Travelling waves in a coarse-grained model of volume-filling cell invasion: Simulations and comparisons},
url = {https://doi.org/10.1111/sapm.12635 },
pages = {1471–-1497},
volume = {151},
journal = {Studies in Applied Mathematics},
doi = {https://doi.org/10.1111/sapm.12635 },
keywords = {Mathematics - Analysis of PDEs;Quantitative Biology - Cell Behavior;Quantitative Biology - Populations and Evolution;Quantitative Biology - Tissues and Organs},
file = {Crossley{_}extracellular{_}matrix{_}interactions:Attachments/Crossley{_}extracellular{_}matrix{_}interactions.pdf:application/pdf}
}
@article{Colson,
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.},
author = {Colson, Chlo{\'e} and S{\'a}nchez-Gardu{\~n}o, Faustino and Byrne, Helen M. and Maini, Philip K. and Lorenzi, Tommaso},
year = {2021},
title = {Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion},
pages = {20210593},
volume = {477},
number = {2256},
issn = {1364-5021},
journal = {Proceedings. Mathematical, physical, and engineering sciences},
doi = {10.1098/rspa.2021.0593 },
file = {Colson, S{\'a}nchez-Gardu{\~n}o et al. 2021 - Travelling-wave analysis of a model:Attachments/Colson, S{\'a}nchez-Gardu{\~n}o et al. 2021 - Travelling-wave analysis of a model.pdf:application/pdf}
}
@article{vanOers,
abstract = {In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in an extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the understanding of the mechanisms underlying in vitro angiogenesis is still incomplete. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and do not check if the same model also correctly captures the intermediate scale: the pairwise cell-cell interactions or single cell responses to {ECM} mechanics. Here we show, using a hybrid cellular Potts and finite element computational model, that a single set of biologically plausible rules describing (a) the contractile forces that endothelial cells exert on the {ECM}, (b) the resulting strains in the extracellular matrix, and (c) the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation from scattered cells, and sprouting from endothelial spheroids. Combining the present mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesis.},
author = {{van Oers}, Ren{\'e} F. M. and Rens, Elisabeth G. and LaValley, Danielle J. and Reinhart-King, Cynthia A. and Merks, Roeland M. H.},
year = {2014},
title = {Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro},
url = {https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1003774#s4},
pages = {e1003774},
volume = {10},
number = {8},
issn = {1553-7358},
journal = {PLOS Computational Biology},
doi = {10.1371/journal. pcbi.1003774 },
file = {van Oers, Rens et al. 2014 - Mechanical cell-matrix feedback explains pairwise:Attachments/van Oers, Rens et al. 2014 - Mechanical cell-matrix feedback explains pairwise.pdf:application/pdf}
}
