{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/travelling-wave-analysis-of-substrate","title":"A continuum mathematical model of substrate-mediated tissue growth","arxiv_id":"2111.07559","date":"2021-11-15","proceeding":null,"authors":["Maud El-Hachem","Scott W McCue","Matthew J Simpson"],"abstract":"We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model involves a partial differential equation describing the density of tissue, $\\hat{u}(\\hat{\\mathbf{x}},\\hat{t})$, that is coupled to the concentration of an immobile extracellular substrate, $\\hat{s}(\\hat{\\mathbf{x}},\\hat{t})$. Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $\\hat{s}$, while a logistic growth term models cell proliferation. The extracellular substrate $\\hat{s}$ is produced by cells, and undergoes linear decay. Preliminary numerical simulations show that this mathematical model, which we call the \\textit{substrate model}, is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we then analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, $c = c_{\\rm{min}}$, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $c > c_{\\rm{min}}$. We provide a geometric interpretation that explains the difference between smooth- and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition to exploring the nature of the smooth- and sharp-fronted travelling waves, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted travelling wave solutions, and the smooth-fronted travelling wave solutions. Software to implement all calculations is available on GitHub.","url_abs":"https://arxiv.org/abs/2111.07559v2","url_pdf":"https://arxiv.org/pdf/2111.07559v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"travelling-wave-analysis-of-substrate","repo_url":"https://github.com/profmjsimpson/substrate_mediated_invasion","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"diffusion","method_name":"Diffusion"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}