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A simple model of biofilm growth in a porous medium that accounts for detachment and attachment of suspended biomass and their contribution to substrate degradation

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  12 April 2018

HARRY J. GAEBLER  and
HERMANN J. EBERL
HARRY J. GAEBLER
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, CanadaN1G2W1 email: gaeblerh@uoguelph.ca, heberl@uoguelph.ca
HERMANN J. EBERL
Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, CanadaN1G2W1 email: gaeblerh@uoguelph.ca, heberl@uoguelph.ca
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Abstract

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We derive a macroscopic model for biofilm formation in a porous medium reactor to investigate the role of suspended bacteria on reactor performance. The starting point is the mesoscopic one-dimensional Wanner–Gujer biofilm model. The following processes are included: hydrodynamics and transport of substrate in the reactor, biofilm and suspended bacteria growth in the pore space, attachment of suspended cells to the biofilm, and detachment of biofilm cells. The mesoscopic equations are up-scaled from the biofilm scale to the reactor scale, yielding a stiff system of balance laws, which we study numerically. We find that suspended bacteria and attachment can have a significant effect on biofilm reactor performance.

Keywords

Balance lawsbiofilmporous medium

MSC classification

Secondary: 35Q92: PDEs in connection with biology and other natural sciences 68U20: Simulation 92D25: Population dynamics (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 6: Biofilms , December 2018 , pp. 1110 - 1140
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This study was financially supported through an Ontario Graduate Scholarship and an Arthur D. Latornell Graduate Scholarships awarded to HJG, and through an NSERC Discovery Grant (HJE). Computing resources were made available by an NSERC Research Tool and Infrastructure Grant awarded to HJE.

References

[1] Abbas, F. & Eberl, H. (2013) Investigation of the role of mesoscale detachment rate expressions in a macroscale model of a porous medium biofilm reactor. Int. J. Biomath. Biostat. 2 (1), 123–143.Google Scholar
[2] Abbas, F. & Eberl, H. J. (2011) Analytical substrate flux approximation for the monod boundary value problem. Appl. Math. Comp. 218 (4), 1484–1494.Google Scholar
[3] Abbas, F., Sudarsan, R. & Eberl, H. J. (2012) Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates. Math. Biosci. Eng. 9 (2), 215–239.Google Scholar
[4] Ballyk, M. M., Jones, D. A. & Smith, H. L. (2001) Microbial competition in reactors with wall attachment: A mathematical comparison of chemostat and plug flow models. Microb. Ecol. 41, 210–221.Google Scholar
[5] Burden, R. L. & Faires, J. D. (2010) Numerical Analysis, Brooks/Cole Publishing, Boston, MA.Google Scholar
[6] Carl, S. & Heikkilä, S. (2000) Nonlinear Differential Equations in Ordered Spaces. Chapman & Hall, Boca Raton, FL, USA.Google Scholar
[7] Carpentier, B. & Cerf, O. (1993) Biofilms and their consequences, with particular reference to hygiene in the food industry. J. Appl. Bacteriol. 75 (6), 499–511.Google Scholar
[8] D'Acunto, B., Frunzo, L., Klapper, I. & Mattei, M. R. (2015) Modeling multispecies biofilms including new bacterial species invasion. Math Biosci. 259, 20–26.Google Scholar
[9] Davit, Y., Byrne, H., Osborne, J., Pitt-Francis, J., Gavaghan, D. & Quintard, M. (2013) Hydrodynamic dispersion within porous biofilms. Phys. Rev. E 87, 1–16.Google Scholar
[10] Donlan, R. M. (2002) Biofilms: Microbial life on surfaces. Emerg. Infect. Dis. 82 (1), 881–890.Google Scholar
[11] Duddu, R., Chopp, D. L. & Moran, B. (2009) A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment. Biotechnol. Bioeng. 103 (1), 92–104.Google Scholar
[12] Emerenini, B., Hense, B. A., Kuttler, C. & Eberl, H. J. (2015) A mathematical model of quorum sensing induced biofilm detachment. PloS ONE 10 (7), e0132385.Google Scholar
[13] Epperson, J. F. (2002) An Introduction to Numerical Methods and Analysis. John Wiley & Sons, New York, NY, USA.Google Scholar
[14] Gaebler, H. J. (2017) Mulit-scale model formulation of a porous medium biofilm reactor and the effects of suspended bacteria, mesoscopic attachment, and growth kinetics on the macroscopic reactor. Master's thesis, University of Guelph, ON, Canada.Google Scholar
[15] Golfier, F., Wood, B. D., Orgogozo, L., Quintard, M. & Bues, M. (2009) Biofilms in porous media: Development of macroscopic transport equations via volume averaging with closure for local mass equilibrium conditions. Adv. Water Resour. 32 (3), 463–485.Google Scholar
[16] Huinink, H. (2016) Fluids in Porous Media: Transport and Phase Changes. Morgan & Claypool Publishers, San Rafael, CA, USA.Google Scholar
[17] Jones, D., Kojouharov, H. V., Le, D. & Smith, H. (2003) The Freter model: A simple model of biofilm formation. J. Math. Biol. 47, 137–152.Google Scholar
[18] Kang, Q., Lichtner, P., Viswanathan, H. & Abdel-Fattah, A. (2010) Pore scale modeling of reactive transport involved in geologic CO2 sequestration. Transp. Porous Media 8 (9), 197–213.Google Scholar
[19] Klapper, I. (2012) Productivity and equilibrium in simple biofilm models. Bull. Math. Biol. 74 (12), 2917–2934.Google Scholar
[20] Liotta, S. F., Romano, V. & Russo, G. (2000) Central schemes for balance laws of reaction type. SIAM. J. Numer. Anal. 38 (4), 1337–1356.Google Scholar
[21] Logan, J. D. (2000) Transport Modeling in Hydrogeochemical Systems. Springer-Verlag, New York, NY, USA.Google Scholar
[22] Mašić, A. & Eberl, H. J. (2012) Persistence in a single species CSTR model with suspended flocs and wall attached biofilms. Bull. Math. Biol. 75 (5), 1001–1026.Google Scholar
[23] Mašić, A. & Eberl, H. J. (2016) A chemostat model with wall attachment: The effect of biofilm detachment rates on predicted reactor performance. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R. & Spiteri, R. (editors), Mathematical and Computational Approaches in Advancing Modern Science and Engineering, Springer International Publishing, Cham, pp. 267–276.Google Scholar
[24] Morgenroth, E. (2003) Detachment: An often-overlooked phenomenon in biofilm research and modelling. In: Wuertz, S., Bishop, P. L. & Wilderer, P. A. (editors), Biofilms in Wastewater Treatment, IWA Publishing, London, England, pp. 246–290.Google Scholar
[25] Nieć, J., Spychala, M. & Zawadzki, P. (2016) New approach to modelling of sand filter clogging by septic tank effluent. J. Ecol. Eng. 17 (2), 97–107.Google Scholar
[26] Picioreanu, C., Van Loosdrecht, M. C. M. & Heijnen, J. J. (2000) Effect of diffusive and convective substrate transport on biofilm structure formation: A two-dimensional modeling study. Biotechnol. Bioeng. 69 (5), 504–515.Google Scholar
[27] Rittmann, B. E. (1982) The effect of shear stress on biofilm loss rate. Biotechnol. Bioeng. 24, 501–506.Google Scholar
[28] Rittmann, B. E. & McCarty, P. L. (2001) Environmental Biotechnology: Principles and Applications. McGraw-Hill, Boston, MA, USA.Google Scholar
[29] Rittmann, B. E., Stilwell, D. & Ohashi, A. (2002) The transient-state, multiple-species biofilm model for biofiltration processes. Water Resour. 36, 2342–2356.Google Scholar
[30] Singh, R., Paul, D. & Jain, R. K. (2006) Biofilms: Implications in bioremediation. Trends Microbiol. 14 (9), 388–397.Google Scholar
[31] Tiwari, S. K. & Bowers, K. L. (2001) Modeling biofilm growth for porous media applications. Math. Comput. Model. 33, 299–319.Google Scholar
[32] Vafai, K. (2010) Ed. Porous Media: Applications in Biological Systems and Biotechnology, CNC Press, Boca Raton, FL, USA.Google Scholar
[33] Wanner, O., Cunningham, A. B. & Lundman, R. (1995) Modeling biofilm accumulation and mass transport in a porous medium under high substrate loading. Biotechnol. Bioeng. 47, 703–712.Google Scholar
[34] Wanner, O., Eberl, H. J., Morgenroth, E., Noguera, D. R., Picioreanu, C., Rittmann, B. E. & van Loosdrecht, M. (2006) Mathematical Modeling of Biofilms. IWA Publishing, London, England.Google Scholar
[35] Wanner, O. & Gujer, W. (1986) A multispecies biofilm model. Biotechnol. Bioeng. 28, 314–386.Google Scholar
[36] Wood, B., Radakovich, K. & Golfier, F. (2007) Effective reaction at a fluid-solid interface: Applications to biotransformation in porous media. Adv. Water Resour. 30, 1630–1674.Google Scholar