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A two-fluid model for tissue growth within a dynamic flow environment

Published online by Cambridge University Press:  01 December 2008

R. D. O'DEA ,
S. L. WATERS  and
H. M. BYRNE
R. D. O'DEA
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD email: reuben.odea@nottingham.ac.uk
S. L. WATERS
Affiliation:
Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB
H. M. BYRNE
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD email: reuben.odea@nottingham.ac.uk
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Abstract

We study the growth of a tissue construct in a perfusion bioreactor, focussing on its response to the mechanical environment. The bioreactor system is modelled as a two-dimensional channel containing a tissue construct through which a flow of culture medium is driven. We employ a multiphase formulation of the type presented by G. Lemon, J. King, H. Byrne, O. Jensen and K. Shakesheff in their study (Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52(2), 2006, 571–594) restricted to two interacting fluid phases, representing a cell population (and attendant extracellular matrix) and a culture medium, and employ the simplifying limit of large interphase viscous drag after S. Franks in her study (Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK, 2002) and S. Franks and J. King in their study (Interactions between a uniformly proliferating tumour and its surrounding: Uniform material properties. Math. Med. Biol. 20, 2003, 47–89).

The novel aspects of this study are: (i) the investigation of the effect of an imposed flow on the growth of the tissue construct, and (ii) the inclusion of a mechanotransduction mechanism regulating the response of the cells to the local mechanical environment. Specifically, we consider the response of the cells to their local density and the culture medium pressure. As such, this study forms the first step towards a general multiphase formulation that incorporates the effect of mechanotransduction on the growth and morphology of a tissue construct.

The model is analysed using analytic and numerical techniques, the results of which illustrate the potential use of the model to predict the dominant regulatory stimuli in a cell population.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 6 , December 2008 , pp. 607 - 634
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Araujo, R. & McElwain, D. (2004) A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol. 66 (5), 1039–1091.CrossRefGoogle Scholar
[2]Araujo, R. & McElwain, D. (2005) A mixture theory for the genesis of residual stresses in growing tissues I: A general formulation. SIAM J. Appl. Math. 65 (4), 1261–1284.CrossRefGoogle Scholar
[3]Bhandari, R., Riccalton, L., Lewis, A., Fry, J., Hammond, A., Tendler, S. & Shakesheff, K. (2001) Liver tissue engineering: A role for co-culture systems in modifying hepatocyte function and viability. Tissue Eng. 7 (3), 401–410.CrossRefGoogle ScholarPubMed
[4]Bowen, R. (1976) Theory of mixtures. Continuum Phys. 3, 1–127.Google Scholar
[5]Byrne, H. & Chaplain, M. (1995) Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci. 130 (2), 151–81.CrossRefGoogle ScholarPubMed
[6]Byrne, H., King, J., McElwain, D. & Preziosi, L. (2003) A two-phase model of solid tumour growth. Appl. Math. Lett. 16, 567–573.CrossRefGoogle Scholar
[7]Byrne, H. & Preziosi, L. (2003) Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20 (4), 341.CrossRefGoogle ScholarPubMed
[8]Cartmell, S. & El Haj, A. (2005) Mechanical bioreactors for tissue engineering. In: Chaudhari, J. & Al-Rubeai, M. (Eds) Bioreactors for Tissue Engineering: Principles, Design and Operation, chap. 8, pp. 193–209. Springer, Dordrecht, The Netherlands.CrossRefGoogle Scholar
[9]Chaplain, M. (2000) Mathematical modelling of angiogenesis. J. Neuro-Oncol. 50 (1), 37–51.CrossRefGoogle ScholarPubMed
[10]Chaplain, M., Graziano, L. & Preziosi, L. (2006) Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math. Med. Biol. 23 (3), 197.CrossRefGoogle ScholarPubMed
[11]Chaplain, M., McDougall, S. & Anderson, A. (2006) Mathematical modelling of tumor-induced angiogenesis. Ann. Rev. Biomed. Eng. 8 (1), 233–257.CrossRefGoogle ScholarPubMed
[12]Coletti, F., Macchietto, S. & Elvassore, N. (2006) Mathematical modelling of three-dimensional cell cultures in perfusion bioreactors. Ind. Eng. Chem. Res. 45 (24), 8158–8169.CrossRefGoogle Scholar
[13]Cowin, S. (2004) How is a tissue built? J. Biomech. Eng. 122, 553.Google Scholar
[14]Cowin, S. (2004) Tissue growth and remodelling. Ann. Rev. Biomed. Eng. 6 (1), 77–107.CrossRefGoogle Scholar
[15]Curtis, A. & Riehle, M. (2001) Tissue engineering: The biophysical background. Phys. Med. Biol. 46, 47–65.CrossRefGoogle ScholarPubMed
[16]Drew, D. (1971) Averaged field equations for a two phase media. Stud. Appl. Math. L2, 205–231.CrossRefGoogle Scholar
[17]Drew, D. (1983) Mathematical modelling of two-phase flow. Ann. Rev. Fluid Mech. 15, 261–291.CrossRefGoogle Scholar
[18]Drew, D. & Segel, L. (1971) Averaged equations for two-phase flows. Stud. Appl. Math. 50, 205–231.CrossRefGoogle Scholar
[19]El-Haj, A., Minter, S., Rawlinson, S., Suswillo, R. & Lanyon, L. (1990) Cellular responses to mechanical loading in vitro. J. Bone Miner. Res. 5 (9), 923–32.Google ScholarPubMed
[20]Forgacs, G., Foty, R., Shafrir, Y. & Steinberg, M. (1998) Viscoelastic properties of living embryonic tissues: A quantitative study. Biophys. J. 74 (5), 2227–2234.CrossRefGoogle ScholarPubMed
[21]Franks, S. (2002) Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK.Google Scholar
[22]Franks, S., Byrne, H., King, J., Underwood, J. & Lewis, C. (2003) Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol. 47, 424–452.CrossRefGoogle ScholarPubMed
[23]Franks, S. & King, J. (2003) Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties. Math. Med. Biol. 20, 47–49.CrossRefGoogle ScholarPubMed
[24]Fung, Y. (1991) What are residual stresses doing in our blood vessels? Ann. Biomed. Eng. 19, 237–249.CrossRefGoogle ScholarPubMed
[25]Greenspan, H. (1972) Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 51 (4), 317–340.CrossRefGoogle Scholar
[26]Hamilton, G., Westmoreland, C. & George, E. (2001) Effects of medium composition on the morphology and function of rat hepatocytes cultured as spheroids and monolayers. In Vitro Cell. Develop. Biol. – Animal 37, 656–667.2.0.CO;2>CrossRefGoogle ScholarPubMed
[27]Haskin, C., Cameron, I. & Athanasiou, K. (1993) Physiological levels of hydrostatic pressure alter morphology and organization of cytoskeletal and adhesion proteins in MG-63 osteosarcoma cells. Biochem. Cell Biol. 71 (1–2), 27–35.CrossRefGoogle ScholarPubMed
[28]Higashiyama, S., Noda, M., Muraoka, S., Uyama, N., Kawada, N., Ide, T., Kawase, M. & Kiyohito, Y. (2004) Maintenance of hepatocyte functions in coculture with hepatic stellate cells. Biochem. Eng. J. 20, 113–118.CrossRefGoogle Scholar
[29]Jaecques, S., Van-Oosterwyck, H., Muraru, L., Van-Cleynbreugel, T., De-Smet, E., Wevers, M., Naert, I. & Vander-Sloten, J. (2004) Individualised, microCT-based finite element modelling as a tool for biomechanical analysis relating to tissue engineering of bone. Biomater. 25, 1683–1696.CrossRefGoogle ScholarPubMed
[30]Karp, J., Sarraf, F., Shoichet, M. & Davies, J. (2004) Fibrin-filled scaffolds for bone-tissue engineering. J. Biomed. Mater. Res. 71A, 162–171.CrossRefGoogle Scholar
[31]Keller, E. & Segel, L. (1970) Initiation of slime-mold aggregation viewed as an instability. J. Theor. Biol. 26 (3), 399–415.CrossRefGoogle ScholarPubMed
[32]Keller, E. & Segel, L. (1971) Model for chemotaxis. J. Theor. Biol. 30 (2), 225–234.CrossRefGoogle ScholarPubMed
[33]King, J. & Franks, S. (2004) Mathematical analysis of some multi-dimensional tissue growth models. Euro. J. Appl. Math. 15 (3), 273–295.CrossRefGoogle Scholar
[34]Klein-Nulend, J., Roelofsen, J., Sterck, J., Semeins, C. & Burger, E. (1995) Mechanical loading stimulates the release of transforming growth factor-beta activity by cultured mouse calvariae and periosteal cells. J. Cell Physiol. 163 (1), 115–119.CrossRefGoogle ScholarPubMed
[35]Kolev, N. (2002) Multiphase Flow Dynamics, Vol. 1: Fundamentals, Springer, Berlin, Heidelberg, New York.Google Scholar
[36]Lemon, G. & King, J. (2007) Multiphase modelling of cell behaviour on artificial scaffolds: Effects of nutrient depletion and spatially nonuniform porosity. Math. Med. Biol. 24 (1), 57.CrossRefGoogle ScholarPubMed
[37]Lemon, G., King, J., Byrne, H., Jensen, O. & Shakesheff, K. (2006) Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52 (2), 571–594.CrossRefGoogle Scholar
[38]Lewis, M., Macarthur, B., Malda, J., Pettet, G. & Please, C. (2005) Heterogeneous proliferation within engineered cartilaginous tissue: The role of oxygen tension. Biotechnol. Bioeng. 91 (5), 607–15.CrossRefGoogle ScholarPubMed
[39]Lubkin, S. & Jackson, T.Multiphase mechanics of capsule formation in tumors. J. Biomech. Eng. 124, 237.CrossRefGoogle Scholar
[40]Luca, M., Chavez-Ross, A., Edelstein-Keshet, L. & Mogilner, A. (2003) Chemotactic signalling, microglia and alzheimer's disease senile plaques: Is there a connection? Bull. Math. Biol. 65, 696–730.CrossRefGoogle ScholarPubMed
[41]Ma, M., Xu, J. & Purcell, W. (2003) Biochemical and functional changes of rat liver spheroids during spheroid formation and maintenance in culture. J. Cell. Biochem. 90, 1166–1175.CrossRefGoogle ScholarPubMed
[42]Maggelakis, S. (1990) Mathematical model of prevascular growth of a spherical carcinoma. Math. Comput. Model. 13 (5), 23–38.CrossRefGoogle Scholar
[43]Malda, J., Rouwkema, J., Martens, D., Le Comte, E., Kooy, F., Tramper, J., van Blitterswijk, C. & Riesle, J. (2004) Oxygen gradients in tissue-engineered Pegt/Pbt cartilaginous constructs: Measurement and modelling. Biotechnol. Bioeng. 86 (1), 9–18.CrossRefGoogle Scholar
[44]Marle, C. (1982) On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. Int. J. Eng. Sci. 20 (5), 643–662.CrossRefGoogle Scholar
[45]Martin, I., Wendt, D. & Heberer, M. (2004) The role of bioreactors in tissue engineering. Trends Biotechnol. 22 (2), 80–86.CrossRefGoogle ScholarPubMed
[46]McGarry, J., Klein-Nulend, J., Mullender, M. & Prendergast, P. (2004) A comparison of srain and fluid shear stress in simulating bone cell responses – a computation and experimental study. FASEB J. 18 (15).Google Scholar
[47]Murray, J. (2002) Mathematical Biology, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York.CrossRefGoogle Scholar
[48]Owan, I., Burr, D., Turner, C., Qiu, J., Tu, Y., Onyia, J. & Duncan, R. (1997) Mechanotransduction in bone: Osteoblasts are more responsive to fluid forces than mechanical strain. Am. J. Physiol. – Cell Physiol. 273 (3), 810–815.CrossRefGoogle ScholarPubMed
[49]Palsson, E. (2001) A three-dimensional model of cell movement in multi-cellular systems. Future Generation Comput. Syst. 17, 835–852.CrossRefGoogle Scholar
[50]Palsson, E. & Othmer, H. G. (2000) A model for individual and collective cell movement in Dictyostelium discoideum. PNAS 97 (19), 10, 448–10, 453.CrossRefGoogle Scholar
[51]Porter, B., Zauel, R., Stockman, H., Guldberg, R. & Fyhrie, D. (2005) 3-D computational modelling of media flow through scaffolds in a perfusion bioreactor. J. Biomech. 38 (3), 543–549.CrossRefGoogle Scholar
[52]Powers, M. & Griffith, L. (1998) Adhesion-guided in vitro morphogenesis in pure and mixed cell cultures. Microscopy Res. Tech. 43, 379–384.3.0.CO;2-0>CrossRefGoogle ScholarPubMed
[53]Powers, M., Rodriguez, R. & Griffith, L. (1997) Cell-substratum adhesion strength as a determinant of hepatocyte aggregate morphology. Biotechnol. Bioeng. 20 (4), 15–26.Google Scholar
[54]Raimondi, M. (2004) The effect of media perfusion on three-dimensional cultures of human chondrocytes: Integration of experimental and computational approaches. Biorheology 41 (3), 401–410.Google ScholarPubMed
[55]Riccalton-Banks, L., Liew, C., Bhandari, R., Fry, J. & Shakesheff, K. (2003) Long-term culture of functional liver tissue: Three-dimensional coculture of primary hepatocytes and stellate cells. Tissue Eng. 9 (3), 401–410.CrossRefGoogle ScholarPubMed
[56]Roelofsen, J., Klein-Nulend, J. & Burger, E. (1995) Mechanical stimulation by intermittent hydrostatic compression promotes bone-specific gene expression in vitro. J. Biomech. 28 (12), 1493–1503.CrossRefGoogle ScholarPubMed
[57]Roose, T., Neti, P., Munn, L., Boucher, Y. & Jain, R.Solid stress generated by spheroid growth estimated using a poroelasticity model. Microvascular Res. 66, 204–212.CrossRefGoogle Scholar
[58]Salgado, A., Coutinho, O. & Reis, R.Bone tissue engineering: State of the art and future trends. Macromolecular Biosci. 4, 743–765.CrossRefGoogle Scholar
[59]Sherratt, J. & Dallon, J. (2002) Theoretical models of wound healing: Past successes and future challenges. Comptes Rendus Biologies 325 (5), 557–64.CrossRefGoogle ScholarPubMed
[60]Sipe, J. (2002) Tissue engineering and reparative medicine. Ann. New York Acad. Sci. 961, 1–9.CrossRefGoogle ScholarPubMed
[61]Sutherland, R., Sordat, B., Bamat, J., Gabbert, H., Bourrat, B. & Mueller-Klieser, W. (1986) Oxygenation and differentiation in multicellular spheroids of human colon carcinoma. Cancer Res. 46 (10), 5320–5329.Google ScholarPubMed
[62]Tracqui, P. & Ohayon, J. (2004) Transmission of mechanical stresses within the cytoskeleton of adgerent cells: A theoretical analysis based on a multi-component cell model. Acta Biotheoretica 52, 323–341.CrossRefGoogle Scholar
[63]Whitaker, S. (2000) The transport equations for multi-phase systems. Chem. Eng. Sci. 28, 139–147.CrossRefGoogle Scholar
[64]Wood, M., Marshall, G., Yang, Y. & ElHaj, A. Haj, A. (2003) Seeding efficiency and distribution of primary osteoblasts in 3D porous poly (L-lactide) scaffolds. Euro. Cells Mater. 6, 58.Google Scholar
[65]Yamada, K., Kamihira, M. & Iijima, S. (2001) Self-organisation of liver constitutive cells mediated by artificial matrix and improvement of liver functions in long-term culture. Biochem. Eng. J. 8, 135–143.CrossRefGoogle Scholar