Abstract
One of the most crucial and lethal characteristics of solid tumors is represented by the increased ability of cancer cells to migrate and invade other organs during the so-called metastatic spread. This is allowed thanks to the production of matrix metalloproteinases (MMPs), enzymes capable of degrading a type of collagen abundant in the basal membrane separating the epithelial tissue from the connective one. In this work, we employ a synergistic experimental and mathematical modelling approach to explore the invasion process of tumor cells. A mathematical model composed of reaction-diffusion equations describing the evolution of the tumor cells density on a gelatin substrate, MMPs enzymes concentration and the degradation of the gelatin is proposed. This is completed with a calibration strategy. We perform a sensitivity analysis and explore a parameter estimation technique both on synthetic and experimental data in order to find the optimal parameters that describe the in vitro experiments. A comparison between numerical and experimental solutions ends the work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availibility
All data supporting the findings of this study are available within the paper and its Supplementary Information.
References
Aceto N, Bardia A, Miyamoto DT, Donaldson MC, Wittner BS, Spencer JA, Yu M, Pely A, Engstrom A, Zhu H et al (2014) Circulating tumor cell clusters are oligoclonal precursors of breast cancer metastasis. Cell 158(5):1110–1122. https://doi.org/10.1016/j.cell.2014.07.013
Aregba-Driollet D, Diele F, Natalini R (2004) A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: Numerical approximation and asymptotic analysis. SIAM J Appl Math 64(5):1636–1667. https://doi.org/10.1137/S003613990342829X
Artym VV, Zhang Y, Seillier-Moiseiwitsch F, Yamada KM, Mueller SC (2006) Dynamic interactions of cortactin and membrane type 1 matrix metalloproteinase at invadopodia: defining the stages of invadopodia formation and function. Cancer Res 66(6):3034–3043. https://doi.org/10.1158/0008-5472.CAN-05-2177
Braun EC (2021) Organs-On-Chips: mathematical modelling and parameter estimation. PhD thesis, Università degli Studi di Roma Tre. http://www.matfis.uniroma3.it/Allegati/Dottorato/TESI/ecbraun/Thesis%20Braun%20Revision.pdf
Braun EC, Bretti G, Natalini R (2022) Parameter estimation techniques for a chemotaxis model inspired by cancer-on-chip (coc) experiments. Int J Non Linear Mech 140:103895. https://doi.org/10.1016/j.ijnonlinmec.2021.103895
Bubba F, Pouchol C, Ferrand N, Vidal G, Almeida L, Perthame B, Sabbah M (2019) A chemotaxis-based explanation of spheroid formation in 3D cultures of breast cancer cells. J Theor Biol 479:73–80. https://doi.org/10.1016/j.jtbi.2019.07.002
Chaplain MAJ, Giverso C, Lorenzi T, Preziosi L (2019) Derivation and application of effective interface conditions for continuum mechanical models of cell invasion through thin membranes. SIAM J Appl Math 79(5):2011–2031. https://doi.org/10.1137/19M124263X
Chen W-T, Singer S (1980) Fibronectin is not present in the focal adhesions formed between normal cultured fibroblasts and their substrata. Proc Natl Acad Sci 77(12):7318–7322. https://doi.org/10.1073/pnas.77.12.7318
Chen W-T, Olden K, Bernard BA, Chu FF (1984) Expression of transformation-associated protease (s) that degrade fibronectin at cell contact sites. J Cell Biol 98(4):1546–1555. https://doi.org/10.1083/jcb.98.4.1546
Chen W-T, Chen J-M, Parsons SJ, Parsons JT (1985) Local degradation of fibronectin at sites of expression of the transforming gene product pp60src. Nature 316(6024):156–158. https://doi.org/10.1038/316156a0
Ciavolella G, David N, Poulain A (2023) Effective interface conditions for a porous medium type problem. To appear in Interface Free Bound. https://arxiv.org/abs/2105.02063
Connolly L, Maxwell P (2002) Image analysis of transwell assays in the assessment of invasion by malignant cell lines. Br J Biomed Sci 59(1):11–14. https://doi.org/10.1080/09674845.2002.11783627
Di Costanzo E, Ingangi V, Angelini C, Carfora MF, Carriero MV, Natalini R (2016) A macroscopic mathematical model for cell migration assays using a real-time cell analysis. PLoS One 11(9):e0162553. https://doi.org/10.1371/journal.pone.0162553
Dillekås H, Rogers MS, Straume O (2019) Are 90% of deaths from cancer caused by metastases? Cancer Med 8(12):5574–5576. https://doi.org/10.1002/cam4.2474
Franssen LC, Lorenzi T, Burgess AE, Chaplain MA (2019) A mathematical framework for modelling the metastatic spread of cancer. Bull Math Biol 81(6):1965–2010. https://doi.org/10.1007/s11538-019-00597-x
Friedl P, Wolf K (2010) Plasticity of cell migration: a multiscale tuning model. J Cell Biol 188(1):11–19. https://doi.org/10.1083/jcb.200909003
Gallinato O, Colin T, Saut O, Poignard C (2017) Tumor growth model of ductal carcinoma: from in situ phase to stroma invasion. J Theoret Biol 429:253–266. https://doi.org/10.1016/j.jtbi.2017.06.022
Giverso C, Lorenzi T, Preziosi L (2022) Effective interface conditions for continuum mechanical models describing the invasion of multiple cell populations through thin membranes. Appl Math Lett 125:107708. https://doi.org/10.1016/j.aml.2021.107708
Guarguaglini F, Natalini R (2007) Fast reaction limit and large time behavior of solutions to a nonlinear model of sulphation phenomena. Commun. Part. Diff. Equ. 32(2):163–189. https://doi.org/10.1080/03605300500361438
Harbeck N, Penault-Llorca F, Cortes J, Gnant M, Houssami N, Poortmans P, Ruddy K, Tsang J, Cardoso F (2019) Breast cancer. Nat Rev Dis Primers 5(1):1–31. https://doi.org/10.1038/s41572-019-0111-2
Hong Y, Fang F, Zhang Q (2016) Circulating tumor cell clusters: what we know and what we expect. Int J Oncol 49(6):2206–2216. https://doi.org/10.3892/ijo.2016.3747
Kachanov L (1986) Introduction to continuum damage mechanics. Springer Sci Bus Med. https://doi.org/10.1007/978-94-017-1957-5
Ke N, Wang X, Xu X, Abassi YA (2011) The xcelligence system for real-time and label-free monitoring of cell viability. In: Mammalian cell viability, pp 33–43. Springer. https://doi.org/10.1007/978-1-61779-108-6_6
Malaquin N, Vercamer C, Bouali F, Martien S, Deruy E, Wernert N, Chwastyniak M, Pinet F, Abbadie C, Pourtier A (2013) Senescent fibroblasts enhance early skin carcinogenic events via a paracrine MMP-par-1 axis. PloS One 8(5):e63607. https://doi.org/10.1371/journal.pone.0063607
Martin KH, Hayes KE, Walk EL, Ammer AG, Markwell SM, Weed SA (2012) Quantitative measurement of invadopodia-mediated extracellular matrix proteolysis in single and multicellular contexts. J Vis Exp 66:e4119. https://doi.org/10.3791/4119
Martinez-Serra J, Gutierrez A, Muñoz-Capó S, Navarro-Palou M, Ros T, Amat JC, Lopez B, Marcus TF, Fueyo L, Suquia AG et al (2014) xcelligence system for real-time label-free monitoring of growth and viability of cell lines from hematological malignancies. OncoTargets Ther 7:985–994. https://doi.org/10.2147/OTT.S62887
Morton KW, Mayers DF (2005) Numerical solution of partial differential equations: an introduction. Cambridge University Press. https://doi.org/10.1017/CBO9780511812248
Murray JD (2002) Mathematical biology I. An introduction. Springer. https://doi.org/10.1007/b98868
Obr A, Röselová P, Grebeňová D, Kuželová K (2013) Real-time monitoring of hematopoietic cell interaction with fibronectin fragment: The effect of histone deacetylase inhibitors. Cell Adh Migr 7(3):275–282. https://doi.org/10.4161/cam.24531
Quarteroni A, Sacco R, Saleri F (2010) Numerical mathematics. Springer Sci Bus Med. https://doi.org/10.1007/b98885
Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. WileyWiley, Hoboken. https://doi.org/10.1002/9780470725184
Segel LA (1972) Simplification and scaling. SIAM Rev 14(4):547–571. https://doi.org/10.1137/1014099
Sung H, Ferlay J, Siegel RL, Laversanne M, Soerjomataram I, Jemal A, Bray F (2021) Global cancer statistics 2020: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA Cancer J Clin 71(3):209–249. https://doi.org/10.3322/caac.21660
Türker Şener L, Albeniz G, Dinç B, Albeniz I (2017) Icelligence real-time cell analysis system for examining the cytotoxicity of drugs to cancer cell lines. Exp Ther Med 14(3):1866–1870. https://doi.org/10.3892/etm.2017.4781
Waks AG, Winer EP (2019) Breast cancer treatment: a review. JAMA 321(3):288–300. https://doi.org/10.1001/jama.2018.19323
Zaoui M, Morel M, Ferrand N, Fellahi S, Bastard J-P, Lamazière A, Larsen AK, Béréziat V, Atlan M, Sabbah M (2012) Breast-associated adipocytes secretome induce fatty acid uptake and invasiveness in breast cancer cells via cd36 independently of body mass index, menopausal status and mammary density. Cancers 11(12):2019. https://doi.org/10.3390/cancers11122012
Acknowledgements
G.C. has received fundings from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740623). The work of G.C. was also partially supported by GNAMPA-INdAM. The authors are grateful to Elishan Christian Braun for fruitful discussions, concerning the numerical implementation of the parameter estimation method. Finally, authors are very grateful to reviewers for their appropriate and constructive remarks.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ciavolella, G., Ferrand, N., Sabbah, M. et al. A Model for Membrane Degradation Using a Gelatin Invadopodia Assay. Bull Math Biol 86, 30 (2024). https://doi.org/10.1007/s11538-024-01260-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-024-01260-w
Keywords
- Reaction-diffusion equations
- Finite difference methods
- Tumour degradation and invasion models
- Parameter estimation
- Sensitivity analysis