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Two-Fluid Model of Biofilm Disinfection

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Abstract

We consider a dynamic model of biofilm disinfection in two dimensions. The biofilm is treated as a viscous fluid immersed in a fluid of less viscosity. The bulk fluid moves due to an imposed external parabolic flow. The motion of the fluid is coupled to the biofilm inducing motion of the biofilm. Both the biofilm and the bulk fluid are dominated by viscous forces, hence the Reynolds number is negligible and the appropriate equations are Stokes equations.

The governing partial differential equations are recast as boundary integral equations using a version of the Lorenz reciprocal relationship. This allows for robust treatment of the simplified fluid/biofilm motion. The transport of nutrients and antimicrobials, which depends directly on the velocities of the fluid and biofilm, is also included. Disinfection of the bacteria is considered under the assumption that the biofilm growth is overwhelmed by disinfection.

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References

  • Anderl, J.N., Franklin, M.J., Stewart, P.S., 2000. Role of antibiotic penetration limitation in klebsiella pneumoniae biofilm resistance to ampicillin and ciprofloxacin. Antimicrob. Agents Chemother. 44, 1818–1824.

    Article  Google Scholar 

  • Balaban, N.Q., Merrin, J., Chait, R., Kowalik, L., Leibler, S., 2005. Bacterial persistence as a phenotypic switch. Science 305, 1622–1625.

    Article  Google Scholar 

  • Banin, E., Brady, K.M., Greenberg, E.P., 2006. Chelator-induced dispersal and killing of Pseudomonas aeruginosa cells in a biofilm. Appl. Environ. Microbiol. 72, 2064–2069.

    Article  Google Scholar 

  • Chen, X., Stewart, P., 2002. Role of electrostatic interactions in cohesion of bacterial biofilms. Appl. Microbiol. Biotechnol. 59, 718–720.

    Article  Google Scholar 

  • Cogan, N.G., 2005. Effects of persister formation on bacterial response to dosing. J. Theor. Biol. 88, 2525–2529.

    Google Scholar 

  • Cogan, N.G., 2007. Hybrid numerical treatment of two-fluid problems with passive interfaces. Commun. Appl. Comput. Sci. 2(1).

  • Cogan, N.G., Cortez, R., Fauci, L.J., 2005. Modeling physiological resistence in bacterial biofilms. Bull. Math. Biol. 67, 831–853.

    Article  MathSciNet  Google Scholar 

  • Cortez, R., 2001. The method of regularized stokeslets. SIAM J. Sci. Comput. 23, 1204–1225.

    Article  MATH  MathSciNet  Google Scholar 

  • Cortez, R., Fauci, L., Medovikov, A., 2005. The method of regularized stokeslets in three dimensions: analysis, validation, and application to helical swimming. Phys. Fluids 17, 1–14.

    Article  MathSciNet  Google Scholar 

  • Costerton, J., 2001. Cystic fibrosis pathogenesis and the role of biofilms in persistent infection. Trends Microbiol. 9, 50–52.

    Article  Google Scholar 

  • Davies, D., 2003. Understanding biofilm resistance to antibacterial agents. Nat. Rev. Drug Discov. 2, 114–122.

    Article  Google Scholar 

  • Desai, M., Buhler, T., Weller, P., Brown, M., 1998. Increasing resistance of planktonic and biofilm cultures of Burkholderia cepeciax to ciproflaxacin and ceftazidime during exponential growth. J. Antimicrob. Chemother. 42, 153–160.

    Article  Google Scholar 

  • Dodds, M.G., Grobe, K.J., Stewart, P.S., 2000. Modeling biofilm antimicrobial resistance. Biotechnol. Bioeng. 68, 456–465.

    Article  Google Scholar 

  • Eberl, H., Picioreanu, C., Heijnen, J., van Loosdrecht, M., 2000. A three-dimensional numerical study on the correlation of spatial structure, hydrodynamic conditions, and mass transfer and conversion in biofilms. Chem. Eng. Sci. 55, 6209–6222.

    Article  Google Scholar 

  • Grobe, K., Zahler, J., Stewart, P., 2002. Role of dose concentration in biocide efficacy against Pseudomonas aeruginosa biofilms. J. Ind. Microbiol. Biotechnol. 29, 10–15.

    Article  Google Scholar 

  • Hentzer, M., Wu, H., Andersen, J.B., Riedel, K., Rasmussen, T.B., Bagge, N., Kumar, N., nd Zhijun Song, M.A.S., Kristoffersen, P., Manefield, M., Costerton, J.W., Molin, S., Eberl, L., Steinberg, P., Kjelleberg, S., Hoiby, N., Givskov, M., 2003. Attenuation of Pseudomonas aeruginosa virulence by quorum sensing inhibitors. EMBO J. 22, 3803–3815.

    Article  Google Scholar 

  • Hou, T.Y., Lowengrub, J.S., Shelley, M.J., 2001. Boundary integral methods for multicomponent fluids and multiphase fluids. J. Comput. Phys. 169, 302–362.

    Article  MATH  MathSciNet  Google Scholar 

  • Keren, I., Kaldalu, N., Spoering, A., Wang, Y., Lewis, K., 2004. Persister cells and tolerance to antimicrobials. FEMS Microbiol. Lett. 230, 13–18.

    Article  Google Scholar 

  • Klapper, I., Rupp, C., Cargo, R., Purvedorj, B., Stoodley, P., 2002. Viscoelastic fluid description of bacterial biofilm material properties. Biotechnol. Bioeng. 80, 289–296.

    Article  Google Scholar 

  • Lappin-Scott, H.M., Costerton, J.W. (Eds.), 1995. Microbial Biofilms. Cambridge University Press, Cambridge. Ch. Mechanisms of the Protection of Bacterial Biofilms from Antimicrobial Agents, pp. 118–130.

    Google Scholar 

  • Layton, A.T., 2007. Modeling water transport across elastic boundaries using an explicit jump method. SIAM J. Sci. Comput. 28(6), 2189–2207.

    Article  MathSciNet  Google Scholar 

  • Leveque, R.J., Li, Z., 1994. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044.

    Article  MATH  MathSciNet  Google Scholar 

  • Leveque, R.J., Li, Z., 1997. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18, 709–735.

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, K., 2001. Riddle of biofilm resistance. Antimicrob. Agents Chemother. 45, 999–1007.

    Article  Google Scholar 

  • Lorenz, H.A., 1907. Ein allgemeiner satz, die bewegung einer reibenden flussugkeit betreffend, nebst einigen anwendungen desselben. Abhand. Theor. Phys. 1, 23–42.

    Google Scholar 

  • Mayo, A., 1985. Fast high order accurate solution of Laplace’s equation on irregular regions. SIAM J. Sci. Comput. 6, 144–157.

    Article  MATH  MathSciNet  Google Scholar 

  • Mittal, R., Iaccarino, G., 2005. Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261.

    Article  MathSciNet  Google Scholar 

  • Morgenroth, E., Eberl, H., van Loosdrecht, M., 2000. Evaluating 3d and 1d mathematical models for mass transport in heterogenous biofilms. Water Sci. Technol. 41, 347–356.

    Google Scholar 

  • Picioreanu, C., van Loosdrecht, M.C.M., Heijnen, J.J., 2000. A theoretical study on the effect of surface roughness on mass transport and transformation in biofilms. Biotechnol. Bioeng. 68, 355–369.

    Article  Google Scholar 

  • Pozrikidis, C., 1992. Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Pozrikidis, C., 2001. Interfacial dynamics for Stokes flow. J. Comput. Phys. 169, 250–301.

    Article  MATH  Google Scholar 

  • Prakash, B., Veeregowda, B., Krishnappa, G., 2003. Biofilms: a survival strategy of bacteria. Curr. Sci. India 85, 1299–1307.

    Google Scholar 

  • Roberts, M.E., Stewart, P.S., 2004. Modeling antibiotic tolerance in biofilms by accounting for nutrient limitation. Antimicrob. Agents Chemother. 48, 48–52.

    Article  Google Scholar 

  • Sanderson, S.S., Stewart, P.S., 1997. Evidence of bacterial adaption to monochloramine in Pseudomonas aeruginosa biofilms and evaluation of biocide action model. Biotechnol. Bioeng. 56, 201–209.

    Article  Google Scholar 

  • Shaw, T., Winston, M., Rupp, C.J., Klapper, I., Stoodley, P., 2005. Commanality of elastic relaxation times in biofilm. Phys. Rev. Lett. 93(098102), 1–4.

    Google Scholar 

  • Stewart, P.S., 1996. Theoretical aspects of antibiotic diffusion into microbial biofilms. Antimicrob. Agents Chemother. 40, 2517–2522.

    Google Scholar 

  • Stewart, P., Rayner, J., Roe, F., Rees, W., 2001. Biofilm penetration and disinfection efficacy of alkaline hypochlorite and chlorosulfamates. J. Appl. Microbiol. 91, 525–532.

    Article  Google Scholar 

  • Sufya, N., Allison, D., Gilbert, P., 2003. Clonal variation in maximum specific growth rate and susceptibility towards antimicrobials. J. Appl. Microbiol. 95, 1261–1267.

    Article  Google Scholar 

  • Vetterling, W.T., Teukolsky, S.A., Press, W.H., Flannerly, B., 2002. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Florida State University, 208 Love Building, Tallahassee, FL, 32306, USA

    N. G. Cogan

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  1. N. G. Cogan
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Correspondence to N. G. Cogan.

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Supported by NSF award DMS-0612467.

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Cogan, N.G. Two-Fluid Model of Biofilm Disinfection. Bull. Math. Biol. 70, 800–819 (2008). https://doi.org/10.1007/s11538-007-9280-3

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  • DOI: https://doi.org/10.1007/s11538-007-9280-3

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