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March  2011, 15(2): 417-456. doi: 10.3934/dcdsb.2011.15.417

## A multicomponent model for biofilm-drug interaction

 1 US Naval Research Laboratory, 4555 Overlook Ave. Southwest, Washington, DC 20375, United States 2 Department of Mathematics & NanoCenter, University of South Carolina, Columbia, SC 29208 3 Department of Mathematical Sciences, Montana State University, P.O. Box 172400, Bozeman, MT 59717-2400, United States

Received  December 2009 Revised  March 2010 Published  December 2010

We develop a tri-component model for the biofilm and solvent mixture, in which the extracellular polymeric substance (EPS) network, bacteria and effective solvent consisting of the solvent, nutrient, drugs etc. are modeled explicitly. The tri-component mixture is assumed incompressible as a whole while inter-component mixing, dissipation, and conversion are allowed. A linear stability analysis is conducted on constant equilibria revealing up to two unstable modes corresponding to possible bacterial growth induced by the bacterial and EPS production and dependent upon the regime of the model parameters. A 1-D transient simulation is carried out to investigate the nonlinear dynamics of the EPS network, bacteria distribution, drug and nutrient distribution in a channel with and without shear. Finally, the transient biofilm dynamics are studied with respect to a host of diffusive properties of the drug and nutrient present in the biofilm.
Citation: Brandon Lindley, Qi Wang, Tianyu Zhang. A multicomponent model for biofilm-drug interaction. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 417-456. doi: 10.3934/dcdsb.2011.15.417
##### References:
 [1] G. C. Barker and M. J. Grimson, A cellular automaton model of microbial growth, Binary Comput. Microbiol., 5 (1993), 132-137. [2] E. Ben-Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok and V. Tamas, Generic modelling of cooperative growth patterns in bacterial colonies, Nature(London), 368 (1994), 46-49. doi: 10.1038/368046a0. [3] A. N. Beris and B. Edwards, "Thermodynamics of Flowing System," Oxford University Press, 1994. [4] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [5] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system-iii: Nucleation in a 2-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447. [6] N. G. Cogan and J. P. Keener, The role of the biofilm matrix in structural development, Math. Med. Biol., 21 (2004), 147-166. doi: 10.1093/imammb/21.2.147. [7] R. L. Colasanti, Cellular automata models of microbial colonies, Binary Comput. Microbiol., 4 (1992), 191. [8] J. D. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM. J. Appl. Math., 62 (2001), 853-869. [9] H. J. Eberl, D. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Mde., 3 (2001), 161-175. doi: 10.1080/10273660108833072. [10] P. J. Flory, "Principles of Polymer Chemistry," Cornell University Press, Ithaca, NY, 1953. [11] D. R. Noguera G. Pizarro and D. Griffeath, Quantative cellular automaton model for biofilms, J. Environ. Eng., 127 (2001), 782-789. doi: 10.1061/(ASCE)0733-9372(2001)127:9(782). [12] S. W. Hermanowicz, A simple 2d biofilm model yields a variety of morphological features, Math. Biosci., 169 (2001), 1-14. doi: 10.1016/S0025-5564(00)00049-3. [13] R. K. Hinson and W. M. Kocher, Model for effective diffusivities in aerobic biofilms, Journal of Environmental Engineering, (1996), 1023-1030. doi: 10.1061/(ASCE)0733-9372(1996)122:11(1023). [14] J. C. Kissel, P. L. McCarty and R. L. Street, Numerical simulation of mixed-culture biofilm, J. Environ. Eng., 110 (1984), 393-411. doi: 10.1061/(ASCE)0733-9372(1984)110:2(393). [15] I. Klapper and J. Dockery, Role of cohesion in material description of biofilms, Phys. Rev. E, 74 (2006), 031902. doi: 10.1103/PhysRevE.74.031902. [16] C. Picioreanu, M. C. Loosdrecht and J. J. Heijnen, Discrete-differential modelling of biofilm structure, Wat. Sci. Tech., 39 (1999), 115-122. doi: 10.1016/S0273-1223(99)00158-4. [17] C. Picioreanu, M. C. Loosdrecht and J. J. Heijnen, Effect of diffusive and convective substrate transport on biofilm structure formation: a two-dimensional modeling study, Biotech. Bioeng., 69 (2000), 504-515. doi: 10.1002/1097-0290(20000905)69:5<504::AID-BIT5>3.0.CO;2-S. [18] C. Picioreanu, M. C. Loosdrecht and J. J. Heijnen, Two-dimensional model of biofilm detachment caused by internal stress from liquid flow, Biotech. Bioeng., 72 (2001), 205-218. doi: 10.1002/1097-0290(20000120)72:2<205::AID-BIT9>3.0.CO;2-L. [19] C. Picioreanu, M. C. M. Loosdrecht and J. J. Heijnen, A new combined differential-discrete cellular automaton approach for biofilm modeling: Application for growth in gel beads, Biotech. Bioeng., 57 (1998), 718-731. doi: 10.1002/(SICI)1097-0290(19980320)57:6<718::AID-BIT9>3.0.CO;2-O. [20] G. Pizarro, R. Moreno C. Garcia and M. E. Sepulveda, Two-dimensional cellular automaton model for mixed-culture biofilm, Wat. Sci. Tech., 49 (2004), 193-198. [21] B. E. Rittmann, The effect of shear stress on biofilm loss rate, Biotech. Bioeng., 24 (1982), 501-506. doi: 10.1002/bit.260240219. [22] B. E. Rittmann and P. L. McCarty, Evaluation of steady-state biofilm kinetics, Biotech. Bioeng., 22 (1980), 2359-2373. doi: 10.1002/bit.260221111. [23] B. E. Rittmann and P. L. McCarty, Model of steady-state-biofilm kinetics, Biotech. Bioeng., 22 (1980), 2243-2357. [24] Q. Wang and T. Zhang, Review of mathematical models for biofilms, Communication in Solid State Physics, 2010. [25] O. Wanner and W. Gujer, Competition in biofilms, Wat. Sci. Tech., 17 (1984), 27-44. [26] O. Wanner and W. Gujer, A multispecies biofilm model, Wat. Sci. Tech., 28 (1986), 314-328. [27] J. W. T. Wimpenny and R. Colasanti, A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models, FEMS Micro. Ecol., 22 (1997), 1-16. doi: 10.1111/j.1574-6941.1997.tb00351.x. [28] T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms I. theory and simulations, SIAM, J. Appl. Math., 69 (2008), 641-669. doi: 10.1137/070691966. [29] T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms II. 2-d numerical simulations of biofilm-flow interaction, Commun. Comput. Phys., 4 (2008), 72-101.

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##### References:
 [1] G. C. Barker and M. J. Grimson, A cellular automaton model of microbial growth, Binary Comput. Microbiol., 5 (1993), 132-137. [2] E. Ben-Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok and V. Tamas, Generic modelling of cooperative growth patterns in bacterial colonies, Nature(London), 368 (1994), 46-49. doi: 10.1038/368046a0. [3] A. N. Beris and B. Edwards, "Thermodynamics of Flowing System," Oxford University Press, 1994. [4] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [5] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system-iii: Nucleation in a 2-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447. [6] N. G. Cogan and J. P. Keener, The role of the biofilm matrix in structural development, Math. Med. Biol., 21 (2004), 147-166. doi: 10.1093/imammb/21.2.147. [7] R. L. Colasanti, Cellular automata models of microbial colonies, Binary Comput. Microbiol., 4 (1992), 191. [8] J. D. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM. J. Appl. Math., 62 (2001), 853-869. [9] H. J. Eberl, D. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Mde., 3 (2001), 161-175. doi: 10.1080/10273660108833072. [10] P. J. Flory, "Principles of Polymer Chemistry," Cornell University Press, Ithaca, NY, 1953. [11] D. R. Noguera G. Pizarro and D. Griffeath, Quantative cellular automaton model for biofilms, J. Environ. Eng., 127 (2001), 782-789. doi: 10.1061/(ASCE)0733-9372(2001)127:9(782). [12] S. W. Hermanowicz, A simple 2d biofilm model yields a variety of morphological features, Math. Biosci., 169 (2001), 1-14. doi: 10.1016/S0025-5564(00)00049-3. [13] R. K. Hinson and W. M. Kocher, Model for effective diffusivities in aerobic biofilms, Journal of Environmental Engineering, (1996), 1023-1030. doi: 10.1061/(ASCE)0733-9372(1996)122:11(1023). [14] J. C. Kissel, P. L. McCarty and R. L. Street, Numerical simulation of mixed-culture biofilm, J. Environ. Eng., 110 (1984), 393-411. doi: 10.1061/(ASCE)0733-9372(1984)110:2(393). [15] I. Klapper and J. Dockery, Role of cohesion in material description of biofilms, Phys. Rev. E, 74 (2006), 031902. doi: 10.1103/PhysRevE.74.031902. [16] C. Picioreanu, M. C. Loosdrecht and J. J. Heijnen, Discrete-differential modelling of biofilm structure, Wat. Sci. Tech., 39 (1999), 115-122. doi: 10.1016/S0273-1223(99)00158-4. [17] C. Picioreanu, M. C. Loosdrecht and J. J. Heijnen, Effect of diffusive and convective substrate transport on biofilm structure formation: a two-dimensional modeling study, Biotech. Bioeng., 69 (2000), 504-515. doi: 10.1002/1097-0290(20000905)69:5<504::AID-BIT5>3.0.CO;2-S. [18] C. Picioreanu, M. C. Loosdrecht and J. J. Heijnen, Two-dimensional model of biofilm detachment caused by internal stress from liquid flow, Biotech. Bioeng., 72 (2001), 205-218. doi: 10.1002/1097-0290(20000120)72:2<205::AID-BIT9>3.0.CO;2-L. [19] C. Picioreanu, M. C. M. Loosdrecht and J. J. Heijnen, A new combined differential-discrete cellular automaton approach for biofilm modeling: Application for growth in gel beads, Biotech. Bioeng., 57 (1998), 718-731. doi: 10.1002/(SICI)1097-0290(19980320)57:6<718::AID-BIT9>3.0.CO;2-O. [20] G. Pizarro, R. Moreno C. Garcia and M. E. Sepulveda, Two-dimensional cellular automaton model for mixed-culture biofilm, Wat. Sci. Tech., 49 (2004), 193-198. [21] B. E. Rittmann, The effect of shear stress on biofilm loss rate, Biotech. Bioeng., 24 (1982), 501-506. doi: 10.1002/bit.260240219. [22] B. E. Rittmann and P. L. McCarty, Evaluation of steady-state biofilm kinetics, Biotech. Bioeng., 22 (1980), 2359-2373. doi: 10.1002/bit.260221111. [23] B. E. Rittmann and P. L. McCarty, Model of steady-state-biofilm kinetics, Biotech. Bioeng., 22 (1980), 2243-2357. [24] Q. Wang and T. Zhang, Review of mathematical models for biofilms, Communication in Solid State Physics, 2010. [25] O. Wanner and W. Gujer, Competition in biofilms, Wat. Sci. Tech., 17 (1984), 27-44. [26] O. Wanner and W. Gujer, A multispecies biofilm model, Wat. Sci. Tech., 28 (1986), 314-328. [27] J. W. T. Wimpenny and R. Colasanti, A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models, FEMS Micro. Ecol., 22 (1997), 1-16. doi: 10.1111/j.1574-6941.1997.tb00351.x. [28] T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms I. theory and simulations, SIAM, J. Appl. Math., 69 (2008), 641-669. doi: 10.1137/070691966. [29] T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms II. 2-d numerical simulations of biofilm-flow interaction, Commun. Comput. Phys., 4 (2008), 72-101.
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