Skip to main content
SpringerLink
Log in

Coherent modelling switch between pointwise and distributed representations of cell aggregates

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Biological systems are typically formed by different cell phenotypes, characterized by specific biophysical properties and behaviors. Moreover, cells are able to undergo differentiation or phenotypic transitions upon internal or external stimuli. In order to take these phenomena into account, we here propose a modelling framework in which cells can be described either as pointwise/concentrated particles or as distributed masses, according to their biological determinants. A set of suitable rules then defines a coherent procedure to switch between the two mathematical representations. The theoretical environment describing cell transition is then enriched by including cell migratory dynamics and duplication/apoptotic processes, as well as the kinetics of selected diffusing chemicals influencing the system evolution. Finally, biologically relevant numerical realizations are presented: in particular, they deal with the growth of a tumor spheroid and with the initial differentiation stages of the formation of the zebrafish posterior lateral line. Both phenomena mainly rely on cell phenotypic transition and differentiated behaviour, thereby constituting biological systems particularly suitable to assess the advantages of the proposed model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2002) Molecular biology of the cell, 3rd edn. Garland Science, New York

    Google Scholar 

  • Anderson ARA, Weaver AM, Cummings PT, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127:905–915

    Article  Google Scholar 

  • Armstrong NJ, Kevin JP, Sherratt JA (2006) A continuum approach to modelling cell–cell adhesion. J. Theor. Biol. 243:98–113

    Article  MathSciNet  Google Scholar 

  • Armstrong NJ, Kevin JP, Sherratt JA (2009) Adding adhesion to a chemical signaling model for somite formation. Bull. Math. Biol. 71:1–24

    Article  MathSciNet  MATH  Google Scholar 

  • Bell HS, Whittle IR, Walker M, Leaver HA, Wharton SB (2001) The development of necrosis and apoptosis in glioma: experimental findings using spheroid culture systems. Neuropathol. Appl. Neurobiol 27:291–304

    Article  Google Scholar 

  • Brazma A, Parkinson H, Schlitt T, Shojatalab M (2001) A Quick Introduction to Elements of Biology-Cells, Molecules, Genes, Functional Genomics, Microarrays. European Bioinformatics Institute, Draft

  • Brown JM (2002) Tumor microenvironment and the response to anticancer therapy. Cancer Biol. Ther. 1:453–458

    Article  Google Scholar 

  • Burleson KM, Boente MP, Parmabuccian SE, Skubitz AP (2006) Disaggregation and invasion of ovarian carcinoma ascites spheroids. J. Transl. Med. 4:1–16

    Article  Google Scholar 

  • Castro MAA, Klamt F, Grieneisen VA, Grivicich I, Moreira JCF (2003) Gompertzian growth pattern correlated with phenotypic organization of colon carcinoma, malignant glioma and non-small cell lung carcinoma cell lines. Cell. Prolif. 36:65–73

    Article  Google Scholar 

  • Cavallaro U, Christofori G (2001) Cell adhesion in tumor invasion and metastasis: loss of the glue is not enough. Biochim. Biophys. Acta 1552:39–45

    Google Scholar 

  • Christofori G (2003) Changing neighbours, changing behaviour: cell adhesion molecule-mediated signalling during tumour progression. Embo. J. 22:2318–2323

    Article  Google Scholar 

  • Colombi A, Scianna M, Tosin A (2014) Differentiated cell behaviour: a multiscale approach using measure theory. J. Math. Biol. 71(5):1049–1079

    Article  MATH  Google Scholar 

  • Colombi A, Scianna M, Preziosi L (2015) A measure-theoretic model for cell migration and aggregation. Math. Model. Nat. Phenom. 1:32–63

    MathSciNet  MATH  Google Scholar 

  • De Luca A, Arena N, Sena LM, Medico E (1999) Met overexpression confers HGF-dependent invasive phenotype to human thyroid carcinoma cells in vitro. J. Cell. Physiol. 180(3):365–371

    Article  Google Scholar 

  • Di Costanzo E, Natalini R, Preziosi L (2015) A hybrid mathematical model for self-organizing cell migration in the zebrafish lateral line. J. Math. Biol. 71:171–214

    Article  MathSciNet  MATH  Google Scholar 

  • Friedl P, Gilmour D (2009) Collective cell migration in morphogenesis, regeneration and cancer. Nat. Rev. Mol. Cell. Biol. 10:445–457

    Article  Google Scholar 

  • Gatenby RA, Smallbone K, Maini PK, Rose F, Averill J, Nagle RB, Worrall L, Gillies RJ (2007) Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer. Br. J. Cancer. 97:646–653

    Article  Google Scholar 

  • Ghysen A, Dambly-Chaudire C (2004) Development of the zebrafish lateral line. Curr. Opin. Neurobiol. 14:67–73

    Article  Google Scholar 

  • Hegedus B, Marga F, Jakab K, Sharpe-Timms KL, Forgacs G (2006) The interplay of cell–cell and cell-matrix interactions in the invasive properties of brain tumors. Biophys. J. 91:2708–2716

    Article  Google Scholar 

  • Ilina O, Friedl P (2009) Mechanisms of collective cell migration at a glance. J. Cell. Sci. 122:3203–3208

    Article  Google Scholar 

  • Khaitan D, Chandna S, Arya MB, Dwarakanath BS (2006) Establishment and characterization of multicellular spheroids from a human glioma cell line: implications for tumor therapy. J. Transl. Med. 4:12–25

    Article  Google Scholar 

  • Khalil AA, Friedl P (2010) Determinants of leader cells in collective cell migration. Integr. Biol. (Camb) 2:568–574

    Article  Google Scholar 

  • Liu ZJ, Shirakawa T, Li Y, Soma A, Oka M, Dotto GP, Fairman RM, Velazquez OC, Herlyn M (2003) Regulation of Notch1 and Dll4 by vascular endothelial growth factor in arterial endothelial cells: implications for modulating arteriogenesis and angiogenesis. Mol. Cell. Biol. 23:14–25

    Article  Google Scholar 

  • Nechipokur A, Raible DW (2008) FGF-dependent mechanosensory organ patterining in zebrafish. Science 320:1774

    Article  Google Scholar 

  • Painter KJ, Bloomfield JM, Sherratt JA, Gerlsch A (2015) A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull. Math. Biol. 77(6):1132–1165

    Article  MathSciNet  MATH  Google Scholar 

  • Puiffe ML, La Page C, Filali-Mouhim A, Zietarska M, Ouellet V, Toniny PN, Chevrette M, Provencher DM, Mes-Masson AM (2007) Characterization of ovarian cancer ascites on cell invasion, proliferation, spheroid formation, and gene expression in an in vitro model of epithelial ovarian cancer. Neoplasia 9:820–829

    Article  Google Scholar 

  • Scianna M, Bassino E, Munaron L (2015) A cellular Potts model analyzing differentiated cell behavior during in vivo vascularization of a hypoxic tissue. Comput. Biol. Med. 63:143–156

    Article  Google Scholar 

  • Scianna M, Bell CG, Preziosi L (2013) A review of mathematical models for the formation of vascular networks. J. Theor. Biol. 333:174–209

    Article  MathSciNet  Google Scholar 

  • Scianna M, Preziosi L (2012) Multiscale developments of cellular Potts models. Multiscale Model. Simul. 10:342–382

    Article  MathSciNet  MATH  Google Scholar 

  • Shield K, Ackland ML, Ahnmed N, Rice GE (2008) Multicellular spheroids in ovarian cancer metastases: biology and pathology. Gynec. Oncol. 113:143–148

    Article  Google Scholar 

  • Smolle J (1998) Fractal tumor stromal border in a non-equilibrium growth model. Anal. Quant. Cytol. Histol. 20:7–13

    Google Scholar 

  • Stein AM, Demuth T, Mobley D, Berens M, Sander LM (2007) A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment. Biophys. J. 92:356–365

    Article  Google Scholar 

  • Sundfeldt K (2003) Cell-cell adhesion in the normal ovary and ovarian tumors of epithelial origin; an exception to the rule. Mol. Cell. Endocrinol. 202:89–96

    Article  Google Scholar 

  • Turner S, Sherratt JA (2002) Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model. J. Theor. Biol. 216:85–100

    Article  MathSciNet  Google Scholar 

  • Weidner KM, Behrens J, Vandekerckhove J, Birchmeier W (1990) Scatter factor: molecular characteristics and effect on the invasiveness of epithelial cells. J. Cell. Biol. 111:2097–2108

    Article  Google Scholar 

  • Williams CK, Li JL, Murga M, Harris AL, Tosato G (2006) Upregulation of the Notch ligand delta-like 4 inhibits VEGF induced endothelial cell function. Blood 107:931–939

    Article  Google Scholar 

  • Wise SM, Lowengrub JS, Frieboes HB, Cristini V (2008) Threedimensional multispecies nonlinear tumor growth: model and numerical method. Int. J. Oncol. 253:524–543

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors extend warm thanks to Prof. Claudio Canuto for many stimulating and fruitful discussions. The research that lead to the present paper was partially supported by a grant of the group GNFM of INdAM.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    A. Colombi, M. Scianna & L. Preziosi

Authors
  1. A. Colombi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Scianna
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Preziosi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Scianna.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Colombi, A., Scianna, M. & Preziosi, L. Coherent modelling switch between pointwise and distributed representations of cell aggregates. J. Math. Biol. 74, 783–808 (2017). https://doi.org/10.1007/s00285-016-1042-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-016-1042-0

Keywords

Mathematics Subject Classification

Search

Navigation