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Description
Cells exert forces in their direct environment and cause the deformation of the tissue, which can result in severe consequences in some diseases such as contractures in burn injuries. Various mathematical models --- particularly in various scales ---- have been developed aiming to understand the biomechanics. In most of our work, we used the immerse boundary approach based on a superposition of Dirac Delta functions to describe the forces exerted by individual cells, which results in a finite-element solution that is not in H1. In \cite{Peng2022_pointforces, Peng2022_numerical}, we developed the smoothed particle approach as a replacement of the immersed boundary approach to improve the accuracy of the solution. The smoothed particle approach is categorized as agent-based model, in which cells are considered as individuals. However, once the number of cells is in the order of thousands and the wound scale is large, these models become too expensive from a computational perspective. For the larger scales, continuum-based models are used which are expressed as partial differential equations. We investigated the connections and consistency between these two types of models, regarding the momentum balance equation \cite{Peng2022_upscaling}. In one dimension, we establish the consistency between these approaches in both analytical solutions and finite-element method solutions. In the multi-dimensional case, we have only computationally shown the consistency between the continuum-based and agent-based models.
Bibliography
@article{Peng2022_pointforces,
title = {Point forces in elasticity equation and their alternatives in multi dimensions},
volume = {199},
ISSN = {0378-4754},
url = {http://dx.doi.org/10.1016/j.matcom.2022.03.021},
DOI = {10.1016/j.matcom.2022.03.021},
journal = {Mathematics and Computers in Simulation},
publisher = {Elsevier BV},
author = {Peng, Q. and Vermolen, F.J.},
year = {2022},
month = sep,
pages = {182–201}
}
@article{Peng2022_numerical,
title = {Numerical methods to compute stresses and displacements from cellular forces: Application to the contraction of tissue},
volume = {404},
ISSN = {0377-0427},
DOI = {10.1016/j.cam.2021.113892},
journal = {Journal of Computational and Applied Mathematics},
publisher = {Elsevier BV},
author = {Peng, Q. and Vermolen, F.J.},
year = {2022},
month = apr,
pages = {113892}
}
@article{Peng2022_upscaling,
title = {Upscaling between an agent-based model (smoothed particle approach) and a continuum-based model for skin contractions},
volume = {85},
ISSN = {1432-1416},
DOI = {10.1007/s00285-022-01770-y},
number = {3},
journal = {Journal of Mathematical Biology},
publisher = {Springer Science and Business Media LLC},
author = {Peng, Q. and Vermolen, F. J.},
year = {2022},
month = sep
}
