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Mathematical Model of Growth Factor Driven Haptotaxis and Proliferation in a Tissue Engineering Scaffold

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Abstract

Motivated by experimental work (Miller et al. in Biomaterials 27(10):2213–2221, 2006, 32(11):2775–2785, 2011) we investigate the effect of growth factor driven haptotaxis and proliferation in a perfusion tissue engineering bioreactor, in which nutrient-rich culture medium is perfused through a 2D porous scaffold impregnated with growth factor and seeded with cells. We model these processes on the timescale of cell proliferation, which typically is of the order of days. While a quantitative representation of these phenomena requires more experimental data than is yet available, qualitative agreement with preliminary experimental studies (Miller et al. in Biomaterials 27(10):2213–2221, 2006) is obtained, and appears promising. The ultimate goal of such modeling is to ascertain initial conditions (growth factor distribution, initial cell seeding, etc.) that will lead to a final desired outcome.

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References

  • Bussolino, F., DiRenzo, M., Ziche, M., Bocchietto, E., Olivero, M., Naldini, L., Gaudino, G., Tamagnone, L., Coffer, A., & Comoglio, P. (1992). Hepatocyte growth factor is a potent angiogenic factor which stimulates endothelial cell motility and growth. J. Cell Biol., 119(3), 629–641.

    Article  Google Scholar 

  • Campbell, P., Miller, E., Fisher, G., Walker, L., & Weiss, L. (2005). Engineered spatial patterns of FGF-2 immobilized on fibrin direct cell organization. Biomaterials, 26(33), 6762–6770.

    Article  Google Scholar 

  • Chung, C., Chen, C., Chen, C., & Tseng, C. (2007). Enhancement of cell growth in tissue-engineering constructs under direct perfusion: modeling and simulation. Biotechnol. Bioeng., 97, 1603–1616.

    Article  Google Scholar 

  • Chung, C., Chen, C., Lin, T., & Tseng, C. (2008). A compact computational model for cell construct development in perfusion culture. Biotechnol. Bioeng., 99, 1535–1541.

    Article  Google Scholar 

  • Coletti, F., Macchietto, S., & Elvassore, N. (2006). Mathematical modeling of three-dimensional cell cultures in perfusion bioreactors. Ind. Eng. Chem. Res., 45, 8158–8169.

    Article  Google Scholar 

  • Contois, D. (1959). Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures. J. Gen. Microbiol., 21, 40–50.

    Google Scholar 

  • Cooper, G., Miller, E., Decesare, G., Usas, A., Lensie, E., Bykowski, M., Huard, J., Weiss, L., Losee, J., & Campbell, P. (2010). Inkjet-based biopatterning of bone morphogenetic protein-2 to spatially control calvarial bone formation. Tissue Eng., Part A, 16(5), 1749–1759.

    Article  Google Scholar 

  • Curtis, A., & Riehle, M. (2001). Tissue engineering: the biophysical background. Phys. Med. Biol., 46, 47–65.

    Article  Google Scholar 

  • Friedman, A., Aguda, B., Chaplain, M., Kimmel, M., Levine, H., Lolas, G., Matzavinos, A., Nilsen-Hamilton, M., & Swierniak, A. (2010). Lecture notes in mathematics/Mathematical biosciences subseries: Tutorials in mathematical biosciences III: cell cycle, proliferation, and cancer. Berlin: Springer.

    Google Scholar 

  • Galban, C., & Locke, B. (1999). Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold. Biotechnol. Bioeng., 65(2), 121–132.

    Article  Google Scholar 

  • Kapur, S., Baylink, D., & Lau, K.-H. (2003). Fluid flow shear stress stimulates human osteoblast proliferation and differentiation through multiple interacting and competing signal transduction pathways. Bone, 32(3), 241–251.

    Article  Google Scholar 

  • Ker, E., Nain, A., Weiss, L., Wang, J., Suhan, J., Amon, C., & Campbell, P. (2011). Bioprinting of growth factors onto aligned sub-micron fibrous scaffolds for simultaneous control of cell differentiation and alignment. Biomaterials, 32(32), 8097–8107.

    Article  Google Scholar 

  • Lappa, M. (2003). Organic tissues in rotating bioreactors: fluid-mechanical aspects, dynamic growth models, and morphological evolution. Biotechnol. Bioeng., 84(5), 518–532.

    Article  Google Scholar 

  • 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, 607–615.

    Article  Google Scholar 

  • Maini, P. (1989). Spatial and spatio-temporal patterns in a cell-haptotaxis model. J. Math. Biol., 27(5), 507–522.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Miller, E., Fisher, G., Weiss, L., Walker, L., & Campbell, P. (2006). Dose-dependent cell growth in response to concentration modulated patterns of FGF-2 printed on fibrin. Biomaterials, 27(10), 2213–2221.

    Article  Google Scholar 

  • Miller, E., Li, K., Kanade, T., Weiss, L., Walker, L., & Campbell, P. (2011). Spatially directed guidance of stem cell population migration by immobilized patterns of growth factors. Biomaterials, 32(11), 2775–2785.

    Article  Google Scholar 

  • Murray, J. D. (1989). Mathematical biology. Berlin: Springer.

    MATH  Google Scholar 

  • O’Brien, F., Harley, B., Waller, M., Yannas, I., Gibson, L., & Prendergast, P. (2007). The effect of pore size on permeability and cell attachment in collagen scaffolds for tissue engineering. Technol. Health Care, 15(1), 3–17.

    Google Scholar 

  • Obradovic, B., Meldon, J. H., Freed, L. E., & Vunjak-Novakovic, G. (2000). Glycosaminoglycan deposition in engineered cartilage: experiments and mathematical model. AIChE J., 46, 1860–1871.

    Article  Google Scholar 

  • O’Dea, R., Waters, S., & Byrne, H. (2008). A two-fluid model for tissue growth within a dynamic flow environment. Eur. J. Appl. Math., 19(6), 607–634.

    Article  MathSciNet  MATH  Google Scholar 

  • O’Dea, R., Waters, S., & Byrne, H. (2009). A multiphase model for tissue construct growth in a perfusion bioreactor. Math. Med. Biol., 27(2), 95–127.

    Article  MathSciNet  Google Scholar 

  • Orme, M., & Chaplain, M. (1996). A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. Math. Med. Biol., 13(2), 73–98.

    Article  MATH  Google Scholar 

  • Osborne, J., O’Dea, R., Whiteley, J., Byrne, H., & Waters, S. (2010). The influence of bioreactor geometry and the mechanical environment on engineered tissues. J. Biomech. Eng., 132(5).

  • Oster, G., Murray, J., & Harris, A. (1983). Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morphol., 78, 83–125.

    Google Scholar 

  • Perelson, A., Maini, P., & Murray, J. (1986). Nonlinear pattern selection in a mechanical model for morphogenesis. J. Math. Biol., 24(5), 525–541.

    Article  MathSciNet  MATH  Google Scholar 

  • Pohlmeyer, J. (2012). Modeling cell proliferation in a perfusion tissue engineering bioreactor. PhD thesis, New Jersey Institute of Technology.

  • 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. Eng., 38(3), 543–549.

    Article  Google Scholar 

  • Probstein, R. (1994). Physicochemical hydrodynamics, an introduction (2nd ed.). New York: Wiley–Interscience.

    Book  Google Scholar 

  • Raimondi, M., Boschetti, F., Falcone, L., Migliavacca, F., Remuzzi, A., & Dubini, G. (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 Scholar 

  • Shakeel, M., Matthews, P., Waters, S., & Graham, R. (2011). A continuum model of cell proliferation and nutrient transport in a perfusion bioreactor. Math. Med. Biol. doi:10.1093/imammb/dqr022

    Google Scholar 

  • Tao, Y. (2011). Global existence for a haptotaxis model of cancer invasion with tissue remodeling. Nonlinear Anal., Real World Appl., 12(1), 418–435.

    Article  MathSciNet  MATH  Google Scholar 

  • Whittaker, R. J., Booth, R., Dyson, R., Bailey, C., Parsons Chini, L., Naire, S., Payvandi, S., Rong, Z., Woollard, H., Cummings, L., Waters, S., Mawasse, L., Chaudhuri, J., Ellis, M., Michael, V., Kuiper, N., & Cartmell, S. (2009). Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor. J. Theor. Biol., 256(4), 533–546.

    Article  Google Scholar 

  • Yeatts, A., & Fisher, J. (2011). Bone tissue engineering bioreactors: dynamic culture and the influence of shear stress. Bone, 48(2), 171–181.

    Article  Google Scholar 

  • Zelzer, M., Albutt, D., Alexander, M., & Russell, N. (2012). The role of albumin and fibronectin in the adhesion of fibroblasts to plasma polymer surfaces. Plasma Process. Polym., 9(2), 149–156.

    Article  Google Scholar 

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Acknowledgements

This work is supported by Award No. KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST). The authors wish to thank Dr. Lee Weiss and Dr. Phil Campbell for use of experimental images included in this paper. J.P. would like to thank Drs. Treena Arinzeh, Shahriar Afkami, and Michael Siegel for much useful guidance with development and numerical solution of the model S.L.W. is grateful to the ERSRC for funding in the form of an Advanced Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ, 07102, USA

    J. V. Pohlmeyer & L. J. Cummings

  2. Oxford Centre for Industrial and Applied Mathematics, Oxford Centre for Collaborative Applied Mathematics, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, UK

    S. L. Waters

Authors
  1. J. V. Pohlmeyer
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  2. S. L. Waters
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  3. L. J. Cummings
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Correspondence to L. J. Cummings.

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Appendix: Numerical Scheme

Appendix: Numerical Scheme

The first step in solving the system consists of assigning an initial cell seeding, which (via Eq. (32)) determines an initial scaffold permeability.

1.1 A.1 Pressure

Equations (27) and (28) then combine to form

(45)

which is solved, subject to the unit pressure drop boundary conditions (39), using a finite volume method. A sample control volume is shown in Fig. 22. The discretization for solving Eq. (45) is

(46)

where b contains boundary data,

and capital letters refer to points while lower case letters refer to the edge of the control volume. The discretization is set up to find solutions at the centers of boxes created by the prescribed grid, and because of this it allows for simple inclusion of the Dirichlet boundary data at x=0,1 and Neumann boundary data at y=0,1. The built-in MATLAB GMRES program is used to solve the pressure equation at the aforementioned centers of the boxes, and a MATLAB command “TriScatteredInterp” is used to extrapolate the data back onto the desired grid space. From this pressure solution, we determine the fluid velocity corresponding to a unitary pressure drop from Darcy’s law, and calculate the total flux, \(\tilde{Q}_{0}\), as in Eq. (40). We then determine the true fluid velocity in the domain via Eq. (41).

Fig. 22
figure 22

Sample control volume for the finite volume method used

Full size image

1.2 A.2 Nutrient Concentration

We solve for the “initial” nutrient concentration in the scaffold by solving Eq. (29) via an upwind finite difference method from x=0 to x=1. The method is

(47)
(48)

where u=(u,v). This method can be used because in all cases we consider, flow is unidirectional with respect to the x-component of the velocity, thus u i,j is always positive.

1.3 A.3 Cell Density

The advective drag experienced by the cells is then determined as a ratio of the fluid velocity by (δ/ϵ)u and the cell density is calculated at the subsequent time step using a semiimplicit ADI-type method (the nonlinear proliferation term is dealt with explicitly). The ADI-type method is

(49)
(50)

This process is then repeated until the user-defined end time is attained. The solution method described is first order in time and first order in space.

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Pohlmeyer, J.V., Waters, S.L. & Cummings, L.J. Mathematical Model of Growth Factor Driven Haptotaxis and Proliferation in a Tissue Engineering Scaffold. Bull Math Biol 75, 393–427 (2013). https://doi.org/10.1007/s11538-013-9810-0

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