2013, 10(1): 75-101. doi: 10.3934/mbe.2013.10.75

An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix

1. 

Politecnico di Torino, Torino, 10124, Italy

2. 

Center for Applied Molecular Medicine, Keck School of Medicine, University of Southern California, Los Angeles, 90033, CA, United States

3. 

Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, 10124, Italy

Received  March 2012 Revised  July 2012 Published  December 2012

The basement membrane (BM) and extracellular matrix (ECM) play critical roles in developmental and cancer biology, and are of great interest in biomathematics. We introduce a model of mechanical cell-BM-ECM interactions that extends current (visco)elastic models (e.g. [8,16]), and connects to recent agent-based cell models (e.g. [2,3,20,26]). We model the BM as a linked series of Hookean springs, each with time-varying length, thickness, and spring constant. Each BM spring node exchanges adhesive and repulsive forces with the cell agents using potential functions. We model elastic BM-ECM interactions with analogous ECM springs. We introduce a new model of plastic BM and ECM reorganization in response to prolonged strains, and new constitutive relations that incorporate molecular-scale effects of plasticity into the spring constants. We find that varying the balance of BM and ECM elasticity alters the node spacing along cell boundaries, yielding a nonuniform BM thickness. Uneven node spacing generates stresses that are relieved by plasticity over long times. We find that elasto-viscoplastic cell shape response is critical to relieving uneven stresses in the BM. Our modeling advances and results highlight the importance of rigorously modeling of cell-BM-ECM interactions in clinically important conditions with significant membrane deformations and time-varying membrane properties, such as aneurysms and progression from in situ to invasive carcinoma.
Citation: Gianluca D'Antonio, Paul Macklin, Luigi Preziosi. An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Mathematical Biosciences & Engineering, 2013, 10 (1) : 75-101. doi: 10.3934/mbe.2013.10.75
References:
[1]

M. Aumailley, Structure and function of basement membrane components: laminin, nidogen, collagen IV, and BM-40, Advances in Molecular and Cell Biology, 6 (1993), 183-206. doi: 10.1016/S1569-2558(08)60202-7.

[2]

P. Buske, J. Galle, N. Barker, G. Aust, H. Clevers and M. Loeffler, A comprehensive model of the spatio-temporal stem cell and tissue organisation in the intestinal crypt, PLoS Comput. Biol., 7 (2011), e1001045.

[3]

P. Buske, J. Przybilla, M. Loeffler, N. Sachs, T. Sato, H. Clevers and J. Galle, On the biomechanics of stem cell niche formation in the gut: modelling growing organoids,, FEBS J. (2012, ().  doi: 10.1111/j.1742-4658.2012.08646.x.

[4]

L. M. Coussens and Z. Werb, Matrix metalloproteinases and the development of cancer, Chemistry and Biology, 3 (1996), 895-904.

[5]

L. M. Coussens, C. L. Tinkle, D. Hanahan and Z. Werb, MMP-9 supplied by bone marrow-derived cells contributes to skin carcinogenesis, Cell, 103 (2000), 481-490.

[6]

J. C. Dallon and H. G. Othmer, How cellular movement determines the collective force generated by the dictyostelium discoideum slug, J. Theor. Biol., 231 (2004), 203-222.

[7]

G. D'Antonio, L. Preziosi and P. Macklin, A multiscale hybrid discrete-continuum model of matrix metalloproteinase transport and basement membrane-extracellular matrix degradation, in preparation (2012).

[8]

S. J. Dunn, A. G. Flethcer, S. J. Chapman, D. J. Gavaghan and J. M. Osborne, Modelling the role of the basement membrane beneath a growing epithelial monolayer, J. Theor. Biol., 298 (2012), 82-91.

[9]

S. J. Franks, H. M. Byrne, H. S. Mudhar, J. C. E. Underwood and C. E. Lewis, Mathematical modelling of comedo ductal carcinoma in situ of the breast, Math. Med. Biol., 20 (2003), 277-308.

[10]

S. J. Franks, H. M. Byrne, J. C. E. Underwood and C. E. Lewis, Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast, J. Theor. Biol., 232 (2005), 523-543.

[11]

P. Ghysels, G. Samaey, B. Tijskens, P. Van Liedekerke H. Ramon and D. Roose, Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics, Phys. Biol., 6 (2009).

[12]

J. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E, 47 (1993), 2128-2154. doi: 10.1103/PhysRevE.47.2128.

[13]

F. Graner and J. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 2013-2016. doi: 10.1103/PhysRevLett.69.2013.

[14]

T. Hagemann, S. C. Robinson, M. Schulz, L. Trümper, F. R. Balkwill and C. Binder, Enhanced invasiveness of breast cancer cell lines upon co-cultivation with macrophages is due to TNF-$\alpha$ dependent up-regulation of matrix metalloproteinases, Carcinogenesis, 25 (2004), 1543-1549.

[15]

S. Jodele, L. Blavier, J. M. Yoon and Y. A. DeClerck, Modifying the soil to affect the seed: role of stromal-derived matrix metalloproteinases in cancer progression, Cancer and Metastasis Review, 25 (2006), 35-43.

[16]

Y. Kim, M. A. Stolarska and H.G . Othmer, The role of the microenvironment in tumor growth and invasion, Progress in Biophysics and Molecular Biology, 106 (2011), 353-379. doi: 10.1016/j.pbiomolbio.2011.06.006.

[17]

R. C. Liddington, Mapping out the basement membrane, Natural Structural Biology, 8 (2001), 573-574.

[18]

P. Macklin, Biological background, in "Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach'' (eds. V. Cristini and J. S. Lowengrub), Cambridge University Press (2010), 8-23. doi: 10.1017/CBO9780511781452.003.

[19]

P. Macklin, M. E. Edgerton, J. S. Lowengrub and V. Cristini, Discrete cell modeling, in "Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach'' (eds. V. Cristini and J. S. Lowengrub), Cambridge University Press (2010), 88-122. doi: 10.1017/CBO9780511781452.007.

[20]

P. Macklin, M. E. Edgerton, A. M. Thompson and V. Cristini, Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): From microscopic measurements to macroscopic predictions of clinical progression, J. Theor. Biol., 301 (2012), 122-140. doi: 10.1016/j.jtbi.2012.02.002.

[21]

P. Macklin, J. Kim, G. Tomaiuolo, M. E. Edgerton and V. Cristini, Agent-based modeling of ductal carcinoma in situ: application to patient-specific breast cancer modeling, in "Computational Biology: Issues and Applications in Oncology'' (ed. T. Pham), Springer (2009), 77-112. doi: 10.1007/978-1-4419-0811-7_4.

[22]

P. Macklin, S. Mumenthaler and J. Lowengrub, Modeling multiscale necrotic and calcified tissue biomechanics in cancer patients: application to ductal carcinoma in situ (DCIS), in "Multiscale Computer Modeling in Biomechanics and Biomedical Engineering'' (ed. A. Gefen), Springer (2013), in press. doi: 10.1007/8415_2012_150.

[23]

K. A. Norton, M. Wininger, G. Bhanot, S. Ganesan, N. Barnard and T. Shinbrot, A 2D mechanistic model of breast ductal carcinoma in situ (DCIS). Morphology and progression, J. Theor. Biol., 263 (2010), 393-406.

[24]

N. Poplawski, U. Agero, J. Gens, M. Swat, J. Glazier and A. Anderson, Front instabilities and invasiveness of simulated avascular tumors, Bull. Math. Biol., 71 (2009), 1189-1227. doi: 10.1007/s11538-009-9399-5.

[25]

L. Preziosi, D. Ambrosi and C. Verdier, An elasto-visco-plastic model of cell aggregates, J. Theor. Biol., 262 (2010), 35-47. doi: 10.1016/j.jtbi.2009.08.023.

[26]

I. Ramis-Conde, M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of cancer cell invasion of tissue, Math. Comp. Model., 47 (2006), 533-545.

[27]

B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier and J. P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents, J. Theor. Biol., 243 (2006), 532-541.

[28]

S. A. Sandersius and T. J. Newman, Modeling cell rheology with the subcellular element model, Phys. Biol., 5 (2008), 015002.

[29]

S. A. Sandersius, C. J. Weijer and T. J. Newman, Emergent cell and tissue dynamics from subcellular modeling of active processes, Phys. Biol., 8 (2011), 045007.

[30]

M. Scianna and L. Preziosi, Multiscale developments of cellular Potts models, Multiscale Model. Sim., 10 (2012), 342-382. doi: \%2010.1137/100812951.

[31]

M. Scianna and L. Preziosi, "Cellular Potts Models: Multiscale Developments and Biological Applications,'' CRC/Academic Press, 2012.

[32]

M. Scianna, L. Preziosi and K. Wolf, A Cellular Potts Model simulating cell migration on and in matrix environments,, Math. Biosci. Eng., (). 

[33]

C. Verdier, J. Etienne, A. Duperray and L. Preziosi, Review: rheological properties of biological materials, Comptes Rendus Physique, 10 (2009), 790-811. doi: 10.1016/j.crhy.2009.10.003.

[34]

Z. Zeng, A. M. Cohen and J. G. Guillem, Loss of basement membrane type IV collagen is associated with increased expression of metalloproteinases 2 and 9 (MMP-2 and MMP-9) during human colorectal tumorigenesis, Carcinogenesis, 20 (1999), 749-755. doi: 10.1093/carcin/20.5.749.

show all references

References:
[1]

M. Aumailley, Structure and function of basement membrane components: laminin, nidogen, collagen IV, and BM-40, Advances in Molecular and Cell Biology, 6 (1993), 183-206. doi: 10.1016/S1569-2558(08)60202-7.

[2]

P. Buske, J. Galle, N. Barker, G. Aust, H. Clevers and M. Loeffler, A comprehensive model of the spatio-temporal stem cell and tissue organisation in the intestinal crypt, PLoS Comput. Biol., 7 (2011), e1001045.

[3]

P. Buske, J. Przybilla, M. Loeffler, N. Sachs, T. Sato, H. Clevers and J. Galle, On the biomechanics of stem cell niche formation in the gut: modelling growing organoids,, FEBS J. (2012, ().  doi: 10.1111/j.1742-4658.2012.08646.x.

[4]

L. M. Coussens and Z. Werb, Matrix metalloproteinases and the development of cancer, Chemistry and Biology, 3 (1996), 895-904.

[5]

L. M. Coussens, C. L. Tinkle, D. Hanahan and Z. Werb, MMP-9 supplied by bone marrow-derived cells contributes to skin carcinogenesis, Cell, 103 (2000), 481-490.

[6]

J. C. Dallon and H. G. Othmer, How cellular movement determines the collective force generated by the dictyostelium discoideum slug, J. Theor. Biol., 231 (2004), 203-222.

[7]

G. D'Antonio, L. Preziosi and P. Macklin, A multiscale hybrid discrete-continuum model of matrix metalloproteinase transport and basement membrane-extracellular matrix degradation, in preparation (2012).

[8]

S. J. Dunn, A. G. Flethcer, S. J. Chapman, D. J. Gavaghan and J. M. Osborne, Modelling the role of the basement membrane beneath a growing epithelial monolayer, J. Theor. Biol., 298 (2012), 82-91.

[9]

S. J. Franks, H. M. Byrne, H. S. Mudhar, J. C. E. Underwood and C. E. Lewis, Mathematical modelling of comedo ductal carcinoma in situ of the breast, Math. Med. Biol., 20 (2003), 277-308.

[10]

S. J. Franks, H. M. Byrne, J. C. E. Underwood and C. E. Lewis, Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast, J. Theor. Biol., 232 (2005), 523-543.

[11]

P. Ghysels, G. Samaey, B. Tijskens, P. Van Liedekerke H. Ramon and D. Roose, Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics, Phys. Biol., 6 (2009).

[12]

J. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E, 47 (1993), 2128-2154. doi: 10.1103/PhysRevE.47.2128.

[13]

F. Graner and J. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 2013-2016. doi: 10.1103/PhysRevLett.69.2013.

[14]

T. Hagemann, S. C. Robinson, M. Schulz, L. Trümper, F. R. Balkwill and C. Binder, Enhanced invasiveness of breast cancer cell lines upon co-cultivation with macrophages is due to TNF-$\alpha$ dependent up-regulation of matrix metalloproteinases, Carcinogenesis, 25 (2004), 1543-1549.

[15]

S. Jodele, L. Blavier, J. M. Yoon and Y. A. DeClerck, Modifying the soil to affect the seed: role of stromal-derived matrix metalloproteinases in cancer progression, Cancer and Metastasis Review, 25 (2006), 35-43.

[16]

Y. Kim, M. A. Stolarska and H.G . Othmer, The role of the microenvironment in tumor growth and invasion, Progress in Biophysics and Molecular Biology, 106 (2011), 353-379. doi: 10.1016/j.pbiomolbio.2011.06.006.

[17]

R. C. Liddington, Mapping out the basement membrane, Natural Structural Biology, 8 (2001), 573-574.

[18]

P. Macklin, Biological background, in "Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach'' (eds. V. Cristini and J. S. Lowengrub), Cambridge University Press (2010), 8-23. doi: 10.1017/CBO9780511781452.003.

[19]

P. Macklin, M. E. Edgerton, J. S. Lowengrub and V. Cristini, Discrete cell modeling, in "Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach'' (eds. V. Cristini and J. S. Lowengrub), Cambridge University Press (2010), 88-122. doi: 10.1017/CBO9780511781452.007.

[20]

P. Macklin, M. E. Edgerton, A. M. Thompson and V. Cristini, Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): From microscopic measurements to macroscopic predictions of clinical progression, J. Theor. Biol., 301 (2012), 122-140. doi: 10.1016/j.jtbi.2012.02.002.

[21]

P. Macklin, J. Kim, G. Tomaiuolo, M. E. Edgerton and V. Cristini, Agent-based modeling of ductal carcinoma in situ: application to patient-specific breast cancer modeling, in "Computational Biology: Issues and Applications in Oncology'' (ed. T. Pham), Springer (2009), 77-112. doi: 10.1007/978-1-4419-0811-7_4.

[22]

P. Macklin, S. Mumenthaler and J. Lowengrub, Modeling multiscale necrotic and calcified tissue biomechanics in cancer patients: application to ductal carcinoma in situ (DCIS), in "Multiscale Computer Modeling in Biomechanics and Biomedical Engineering'' (ed. A. Gefen), Springer (2013), in press. doi: 10.1007/8415_2012_150.

[23]

K. A. Norton, M. Wininger, G. Bhanot, S. Ganesan, N. Barnard and T. Shinbrot, A 2D mechanistic model of breast ductal carcinoma in situ (DCIS). Morphology and progression, J. Theor. Biol., 263 (2010), 393-406.

[24]

N. Poplawski, U. Agero, J. Gens, M. Swat, J. Glazier and A. Anderson, Front instabilities and invasiveness of simulated avascular tumors, Bull. Math. Biol., 71 (2009), 1189-1227. doi: 10.1007/s11538-009-9399-5.

[25]

L. Preziosi, D. Ambrosi and C. Verdier, An elasto-visco-plastic model of cell aggregates, J. Theor. Biol., 262 (2010), 35-47. doi: 10.1016/j.jtbi.2009.08.023.

[26]

I. Ramis-Conde, M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of cancer cell invasion of tissue, Math. Comp. Model., 47 (2006), 533-545.

[27]

B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier and J. P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents, J. Theor. Biol., 243 (2006), 532-541.

[28]

S. A. Sandersius and T. J. Newman, Modeling cell rheology with the subcellular element model, Phys. Biol., 5 (2008), 015002.

[29]

S. A. Sandersius, C. J. Weijer and T. J. Newman, Emergent cell and tissue dynamics from subcellular modeling of active processes, Phys. Biol., 8 (2011), 045007.

[30]

M. Scianna and L. Preziosi, Multiscale developments of cellular Potts models, Multiscale Model. Sim., 10 (2012), 342-382. doi: \%2010.1137/100812951.

[31]

M. Scianna and L. Preziosi, "Cellular Potts Models: Multiscale Developments and Biological Applications,'' CRC/Academic Press, 2012.

[32]

M. Scianna, L. Preziosi and K. Wolf, A Cellular Potts Model simulating cell migration on and in matrix environments,, Math. Biosci. Eng., (). 

[33]

C. Verdier, J. Etienne, A. Duperray and L. Preziosi, Review: rheological properties of biological materials, Comptes Rendus Physique, 10 (2009), 790-811. doi: 10.1016/j.crhy.2009.10.003.

[34]

Z. Zeng, A. M. Cohen and J. G. Guillem, Loss of basement membrane type IV collagen is associated with increased expression of metalloproteinases 2 and 9 (MMP-2 and MMP-9) during human colorectal tumorigenesis, Carcinogenesis, 20 (1999), 749-755. doi: 10.1093/carcin/20.5.749.

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