Bayesian inverse problem for a fractional diffusion model of cell migration

Francisco Julian Ariza-Hernandez; Juan Carlos Najera-Tinoco; Martin Patricio Arciga-Alejandre; Eduardo Castañeda-Saucedo; Jorge Sanchez-Ortiz; Francisco Julian Ariza-Hernandez; Juan Carlos Najera-Tinoco; Martin Patricio Arciga-Alejandre; Eduardo Castañeda-Saucedo; Jorge Sanchez-Ortiz

doi:10.3934/mbe.2024257

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5826-5837. doi: 10.3934/mbe.2024257

Previous Article Next Article

Research article Special Issues

Bayesian inverse problem for a fractional diffusion model of cell migration

1.
Faculty of Mathematics, Autonomous University of Guerrero, Mexico
2.
Laboratory of Cancer Cell Biology, Faculty of Chemical and Biological Sciences, Autonomous University of Guerrero, Mexico

Received: 11 December 2023 Revised: 05 February 2024 Accepted: 20 February 2024 Published: 28 April 2024

In the present work, both direct and inverse problems are considered for a Fisher-type fractional diffusion equation, which is proposed to describe the phenomenon of cell migration. For the direct problem, a solution is given via the Fourier method and the Laplace transform. On the other hand, we solved the inverse problem from a Bayesian statistical framework using a set of data that are the result of a cell migration experiment on a wound closure assay. We estimated the parameters of the mathematical model via Markov Chain Monte Carlo methods.
- cell migration,
- fractional derivative,
- Bayesian estimation
Citation: Francisco Julian Ariza-Hernandez, Juan Carlos Najera-Tinoco, Martin Patricio Arciga-Alejandre, Eduardo Castañeda-Saucedo, Jorge Sanchez-Ortiz. Bayesian inverse problem for a fractional diffusion model of cell migration[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5826-5837. doi: 10.3934/mbe.2024257

Related Papers:

Abstract

In the present work, both direct and inverse problems are considered for a Fisher-type fractional diffusion equation, which is proposed to describe the phenomenon of cell migration. For the direct problem, a solution is given via the Fourier method and the Laplace transform. On the other hand, we solved the inverse problem from a Bayesian statistical framework using a set of data that are the result of a cell migration experiment on a wound closure assay. We estimated the parameters of the mathematical model via Markov Chain Monte Carlo methods.

References

[1]	R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R. E. Baker, P. K. Maini, et al., Multiscale mechanisms of cell migration during development: Theory and experiment, Development, 139 (2012), 2935–2944. http://doi.org/10.1242/dev.081471 doi: 10.1242/dev.081471
[2]	J. A. Sherratt, P. Martin, J. D. Murray, J. Lewis, Mathematical models of wound healing in embryonic and adult epidermis, Math. Med. Biol.: J. IMA, 9 (1992), 175–196. https://doi.org/10.1093/imammb/9.3.177 doi: 10.1093/imammb/9.3.177
[3]	P. Friedl, D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell Biol., 10 (2009), 445–457. https://doi.org/10.1038/nrm2720 doi: 10.1038/nrm2720
[4]	L. Oswald, S. Grosser, D. M. Smith, J. A. Käs, Jamming transitions in cancer, J. Phys. D: Appl. Phys., 50 (2017), 483001. http://doi.org/10.1088/1361-6463/aa8e83 doi: 10.1088/1361-6463/aa8e83
[5]	A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. Biol.: J. IMA, 22 (2005), 163–186. http://doi.org/10.1093/imammb/dqi005 doi: 10.1093/imammb/dqi005
[6]	J. Zahm, H. Kaplan, A. Hérard, F. Doriot, D. Pierrot, P. Somelette, et al., Cell migration and proliferation during the in vitro wound repair of the respiratory epithelium, Cytoskeleton, 37 (1997), 33–43.
[7]	A. Tremel, A. Cai, N. Tirtaatmadja, B. D. Hughes, G. W. Stevens, K. A. Landman, et al., Cell migration and proliferation during monolayer formation and wound healing, Chem. Eng. Sci., 64 (2009), 247–253. https://doi.org/10.1016/j.ces.2008.10.008 doi: 10.1016/j.ces.2008.10.008
[8]	M. C. Robson, D. P. Hill, M. E. Woodske, D. L. Steed, Wound healing trajectories as predictors of effectiveness of therapeutic agents, Arch. Surg., 135 (2000), 773–777. http://doi.org/10.1001/archsurg.135.7.773 doi: 10.1001/archsurg.135.7.773
[9]	J. A. Sherratt, J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. B., 241 (1990), 29–36. https://doi.org/10.1098/rspb.1990.0061 doi: 10.1098/rspb.1990.0061
[10]	P. K Maini, D. L. S. McElwain, D. I. Leavesley, Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells, Tissue Eng., 10 (2004), 475–482. http://doi.org/10.1089/107632704323061834 doi: 10.1089/107632704323061834
[11]	D. P. Stonko, L. Manning, M. Starz-Gaiano, B. E. Peercy, A mathematical model of collective cell migration in a three-dimensional, heterogeneous environment, PloS one, 10 (2015), 0122799. https://doi.org/10.1371/journal.pone.0122799 doi: 10.1371/journal.pone.0122799
[12]	L. Chen, K. Painter, C. Surulescu, A. Zhigun, Mathematical models for cell migration: A non-local perspective. Philos. Trans. R. Soc., B., 375 (2020), 20190379. https://doi.org/10.1098/rstb.2019.0379 doi: 10.1098/rstb.2019.0379
[13]	B. Bonilla, M. Rivero, L. Rodríguez-Germá, J. J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput., 187 (2007), 79–88. https://doi.org/10.1016/j.amc.2006.08.105 doi: 10.1016/j.amc.2006.08.105
[14]	F. J. Ariza-Hernandez, J. Sanchez-Ortiz, M. P. Arciga-Alejandre, L. X. Vivas-Cruz, Bayesian analysis for a fractional population growth model, J. Appl. Math., 2017 (2017), 9654506. https://doi.org/10.1155/2017/9654506 doi: 10.1155/2017/9654506
[15]	M. Du, Z. Wang, H. Hu, Measuring memory with the order of fractional derivative, Sci. Rep., 3 (2013), 3431. https://doi.org/10.1038/srep03431 doi: 10.1038/srep03431
[16]	S. L. Zeger, M. R. Krim, Generalized linear models with random effects; a Gibbs sampling approach, J. Am. Stat. Assoc., 86 (1991), 79–86. http://doi.org/10.1080/01621459.1991.10475006 doi: 10.1080/01621459.1991.10475006
[17]	S. Chib, E. Greenberg, Understanding the metropolis-hastings algorithm, Am. Stat., 49 (1995), 327–335. https://doi.org/10.2307/2684568 doi: 10.2307/2684568
[18]	The R Foundation, The R Project for Statistical Computing, 2024. Available from: https://www.r-project.org/.
[19]	S. Sturtz, U. Ligges, A. Gelman, R2WinBUGS: A package for running WinBUGS from R, J. Stat. Software, 12 (2005), 1–16. http://doi.org/10.18637/jss.v012.i03 doi: 10.18637/jss.v012.i03
[20]	Y. Su, M. Yajima, Package 'R2jags', 2024. Available from: https://cran.r-project.org/web/packages/R2jags/R2jags.pdf.
[21]	M. Plummer, JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling, in Proceedings of the 3rd International Workshop on Distributed Statistical Computing, (2003), 1–10.
[22]	R. M. Neal, Slice sampling, Ann. Statist., 31 (2003), 705–767. http://doi.org/10.1214/aos/1056562461 doi: 10.1214/aos/1056562461

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)