2012, 9(2): 215-239. doi: 10.3934/mbe.2012.9.215

Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates

1. 

Dept. Mathematics and Statistics, University of Guelph, 50 Stone Rd E, Guelph, ON, N1G 2W1, Canada, Canada, Canada

Received  May 2011 Revised  August 2011 Published  March 2012

We investigate the role of non shear stress and shear stressed based detachment rate functions for the longterm behavior of one-dimensional biofilm models. We find that the particular choice of a detachment rate function can affect the model prediction of persistence or washout of the biofilm. Moreover, by comparing biofilms in three settings: (i) Couette flow reactors, (ii) Poiseuille flow with fixed flow rate and (iii) Poiseuille flow with fixed pressure drop, we find that not only the bulk flow Reynolds number but also the particular mechanism driving the flow can play a crucial role for longterm behavior. We treat primarily the single species-case that can be analyzed with elementary ODE techniques. But we show also how the results, to some extent, can be carried over to multi-species biofilm models, and to biofilm models that are embedded in reactor mass balances.
Citation: Fazal Abbas, Rangarajan Sudarsan, Hermann J. Eberl. Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates. Mathematical Biosciences & Engineering, 2012, 9 (2) : 215-239. doi: 10.3934/mbe.2012.9.215
References:
[1]

F. Abbas and H. J. Eberl, Analytical substrate flux approximation for the Monod boundary value problem,, Appl. Math. Comp., 218 (2011), 1484.  doi: 10.1016/j.amc.2011.05.102.  Google Scholar

[2]

J. P. Boltz, E. Morgenroth, D. Brockmann, C. Bott, W. J. Gellner and P. A. Vanrolleghem, Critical components of biofilm models for engineering practise,, in, (2010), 1072.   Google Scholar

[3]

J. D. Bryers, Biofilm formation and chemostat dynamics: Pure and mixed culture considerations,, Biotech. Bioeng., 26 (1984), 948.  doi: 10.1002/bit.260260820.  Google Scholar

[4]

S. Carl and S. Heikkilä, "Nonlinear Differential Equations in Ordered Spaces,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 111 (2000).   Google Scholar

[5]

M. A. S. Chaudhry and S. A. Beg, A review on the mathematical modeling of biofilm processes: Advances in fundamentals of biofilm modeling,, Chem. Eng. Technol., 21 (1998), 701.   Google Scholar

[6]

B. M. Chen-Charpentier and H. V. Kojouharov, Numerical simulation of dual-species biofilms in porous media,, Appl. Num. Math., 47 (2003), 377.   Google Scholar

[7]

B. M. Chen-Charpentier and H. V. Kojouharov, Mathematical modeling of bioremediation of trichloroenthylene in aquifers,, Comp. Math. Appls., 56 (2008), 645.  doi: 10.1016/j.camwa.2008.01.007.  Google Scholar

[8]

N. Derlon, A. Massé, R. Escudié, N. Bernet and E. Paul, Stratification in the cohesion of biofilms grown under various environmental conditions,, Water Research, 42 (2008), 2102.  doi: 10.1016/j.watres.2007.11.016.  Google Scholar

[9]

B. C. Dunsmore, A. Jacobsen, L. Hall-Stoodley, C. J. Bass, H. M. Lappin-Scott and P. Stoodley, The influence of fluid shear on the structure and material properties of sulphate-reducing bacterial biofilms,, J. Ind. Microbiol. Biotech., 29 (2002), 347.  doi: 10.1038/sj.jim.7000302.  Google Scholar

[10]

H. Horn, T. R. Neu and M. Wulkow, Modelling the structure and function of extracellular polymeric substances in biofilms with new numerical techniques,, Wat. Sci. Tech., 43 (2001), 121.   Google Scholar

[11]

J. C. Kissel, P. L. McCarty and R. L. Street, Numerical simulation of mixed-culture biofilm,, ASCE J. Env. Eng., 110 (1984), 393.  doi: 10.1061/(ASCE)0733-9372(1984)110:2(393).  Google Scholar

[12]

I. Klapper and B. Szomolay, An exclusion principle and the importance of mobility for a class of biofilm models,, Bull. Math. Biol., 73 (2011), 2213.   Google Scholar

[13]

H. Kuchling, "Physik,", 18th edition, (1987).   Google Scholar

[14]

Z. Lewandowski and H. Beyenal, "Fundamentals of Biofilm Research,", CRC Press, (2007).   Google Scholar

[15]

A. Masic, J. Bengtsson and M. Christensson, Measuring and modeling the oxygen profile in a nitrifying moving bed biofilm reactor,, Math. Biosc., 227 (2010), 1.   Google Scholar

[16]

A. Masic and H. J. Eberl, Persistence in a single species CSTR model with suspended flocs and wall attached biofilms,, Bull. Math. Biol., ().  doi: 10.1007/s11538-011-9707-8.  Google Scholar

[17]

E. Morgenroth, Detachment: An often-overlooked phenomenon in biofilm research and modelling,, in, (2003), 246.   Google Scholar

[18]

N. Muhammad and H. J. Eberl, Open MP parallelization of a Mickens time-integration scheme for a mixed-culture biofilm model and its performance on multi-core and multi-processor computers,, LNCS, 5976 (2010), 180.   Google Scholar

[19]

N. Muhammad and H. J. Eberl, Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones,, Math. Biosc., 233 (2011), 1.  doi: 10.1016/j.mbs.2011.05.006.  Google Scholar

[20]

B. R. Munson, D. F. Young and T. H. Okiishi, "Fundamentals of Fluid Mechanics,", John Wiley & Sons, (1990).   Google Scholar

[21]

N. C. Overgaard, Application of variational inequalities to the moving-boundary problem in a fluid model for biofilm growth,, Nonlin. Analysis, 70 (2009), 3658.  doi: 10.1016/j.na.2008.07.021.  Google Scholar

[22]

E. Paramonova, O. J. Kalmykowa, H. C. van der Mei, H. J. Busscher and P. K. Sharma, Impact of hydrodynamics on oral biofilm strength,, J. Dent. Res., 88 (2009), 922.  doi: 10.1177/0022034509344569.  Google Scholar

[23]

B. M. Peyton and W. G. Characklis, A statistical analysis of the effect of substrate utilization and shear stress on the kinetics of biofilm detachment,, Biotech. Bioeng., 41 (1982), 728.  doi: 10.1002/bit.260410707.  Google Scholar

[24]

L. A. Pritchett and J. D. Dockery, Steady state solutions of a one-dimensional biofilm,, Math. Comp. Model., 33 (2001), 255.  doi: 10.1016/S0895-7177(00)00242-9.  Google Scholar

[25]

B. E. Rittmann and P. L. McCarty, Model of steady state biofilm kinetics,, Biotech. Bioeng., 22 (1980), 2343.  doi: 10.1002/bit.260221110.  Google Scholar

[26]

B. E. Rittmann, The effect of shear stress on biofilm loss rate,, Biotech. Bioeng., 24 (1982), 501.   Google Scholar

[27]

B. E. Rittmann, A. O. Schwarz, H. J. Eberl, E. Morgenroth, J. Perez, M. C. M. van Loosdrecht and O. Wanner, Results from the multi-species benchmark problem (BM3) using one-dimensional models,, Wat. Sci. & Tech., 49 (2004), 163.   Google Scholar

[28]

B. E. Rittmann, D. Stilwell and A. Ohashi, The transient-state, multiple-species biofilm model for biofiltration processes,, Water Research, 36 (2002), 2342.  doi: 10.1016/S0043-1354(01)00441-9.  Google Scholar

[29]

A. Rochex, A. Massé, R. Escudié, J. J. Godon and N. Bernet, Influence of abrasion on biofilm detachment: Evidence for stratification of the biofilm,, J. Ind. Microbiol. Biotech., 36 (2009), 467.  doi: 10.1007/s10295-009-0543-x.  Google Scholar

[30]

P. Stoodley, R. Cargo, C. J. Rupp, S. Wilson and I. Klapper, Biofilm material properties as related to shear-induced deformation and detachment phenomena,, J. Ind. Microbiol. Biotech., 29 (2002), 361.  doi: 10.1038/sj.jim.7000282.  Google Scholar

[31]

B. Szomolay, Analysis of a moving boundary value problem arising in biofilm modelling,, Math. Meth. Appl. Sci., 31 (2008), 1835.  doi: 10.1002/mma.1000.  Google Scholar

[32]

U. Telgmann, H. Horn and E. Morgenroth, Influence of growth history on sloughing and erosion from biofilms,, Water Research, 38 (2004), 3671.  doi: 10.1016/j.watres.2004.05.020.  Google Scholar

[33]

L. Tijhuis, M. C. M. van Loosdrecht and J. J. Heijnen, Dynamics of biofilm detachment in biofilm airlift suspension reactors,, Biotech. Bioeng., 45 (1995), 481.  doi: 10.1002/bit.260450604.  Google Scholar

[34]

M. M. Trulear and W. G. Characklis, Dynamics of biofilm processes,, J. Water Pollut. Control Fed., 54 (1982), 1288.   Google Scholar

[35]

W. Walter, "Gewöhnliche Differentialgleichungen," 7th edition,, Springer-Verlag, (2000).   Google Scholar

[36]

O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms,", IWA Publishing, (2006).   Google Scholar

[37]

O. Wanner and W. Gujer, A multispecies biofilm model,, Biotech. Bioeng., 28 (1986), 314.   Google Scholar

show all references

References:
[1]

F. Abbas and H. J. Eberl, Analytical substrate flux approximation for the Monod boundary value problem,, Appl. Math. Comp., 218 (2011), 1484.  doi: 10.1016/j.amc.2011.05.102.  Google Scholar

[2]

J. P. Boltz, E. Morgenroth, D. Brockmann, C. Bott, W. J. Gellner and P. A. Vanrolleghem, Critical components of biofilm models for engineering practise,, in, (2010), 1072.   Google Scholar

[3]

J. D. Bryers, Biofilm formation and chemostat dynamics: Pure and mixed culture considerations,, Biotech. Bioeng., 26 (1984), 948.  doi: 10.1002/bit.260260820.  Google Scholar

[4]

S. Carl and S. Heikkilä, "Nonlinear Differential Equations in Ordered Spaces,", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 111 (2000).   Google Scholar

[5]

M. A. S. Chaudhry and S. A. Beg, A review on the mathematical modeling of biofilm processes: Advances in fundamentals of biofilm modeling,, Chem. Eng. Technol., 21 (1998), 701.   Google Scholar

[6]

B. M. Chen-Charpentier and H. V. Kojouharov, Numerical simulation of dual-species biofilms in porous media,, Appl. Num. Math., 47 (2003), 377.   Google Scholar

[7]

B. M. Chen-Charpentier and H. V. Kojouharov, Mathematical modeling of bioremediation of trichloroenthylene in aquifers,, Comp. Math. Appls., 56 (2008), 645.  doi: 10.1016/j.camwa.2008.01.007.  Google Scholar

[8]

N. Derlon, A. Massé, R. Escudié, N. Bernet and E. Paul, Stratification in the cohesion of biofilms grown under various environmental conditions,, Water Research, 42 (2008), 2102.  doi: 10.1016/j.watres.2007.11.016.  Google Scholar

[9]

B. C. Dunsmore, A. Jacobsen, L. Hall-Stoodley, C. J. Bass, H. M. Lappin-Scott and P. Stoodley, The influence of fluid shear on the structure and material properties of sulphate-reducing bacterial biofilms,, J. Ind. Microbiol. Biotech., 29 (2002), 347.  doi: 10.1038/sj.jim.7000302.  Google Scholar

[10]

H. Horn, T. R. Neu and M. Wulkow, Modelling the structure and function of extracellular polymeric substances in biofilms with new numerical techniques,, Wat. Sci. Tech., 43 (2001), 121.   Google Scholar

[11]

J. C. Kissel, P. L. McCarty and R. L. Street, Numerical simulation of mixed-culture biofilm,, ASCE J. Env. Eng., 110 (1984), 393.  doi: 10.1061/(ASCE)0733-9372(1984)110:2(393).  Google Scholar

[12]

I. Klapper and B. Szomolay, An exclusion principle and the importance of mobility for a class of biofilm models,, Bull. Math. Biol., 73 (2011), 2213.   Google Scholar

[13]

H. Kuchling, "Physik,", 18th edition, (1987).   Google Scholar

[14]

Z. Lewandowski and H. Beyenal, "Fundamentals of Biofilm Research,", CRC Press, (2007).   Google Scholar

[15]

A. Masic, J. Bengtsson and M. Christensson, Measuring and modeling the oxygen profile in a nitrifying moving bed biofilm reactor,, Math. Biosc., 227 (2010), 1.   Google Scholar

[16]

A. Masic and H. J. Eberl, Persistence in a single species CSTR model with suspended flocs and wall attached biofilms,, Bull. Math. Biol., ().  doi: 10.1007/s11538-011-9707-8.  Google Scholar

[17]

E. Morgenroth, Detachment: An often-overlooked phenomenon in biofilm research and modelling,, in, (2003), 246.   Google Scholar

[18]

N. Muhammad and H. J. Eberl, Open MP parallelization of a Mickens time-integration scheme for a mixed-culture biofilm model and its performance on multi-core and multi-processor computers,, LNCS, 5976 (2010), 180.   Google Scholar

[19]

N. Muhammad and H. J. Eberl, Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones,, Math. Biosc., 233 (2011), 1.  doi: 10.1016/j.mbs.2011.05.006.  Google Scholar

[20]

B. R. Munson, D. F. Young and T. H. Okiishi, "Fundamentals of Fluid Mechanics,", John Wiley & Sons, (1990).   Google Scholar

[21]

N. C. Overgaard, Application of variational inequalities to the moving-boundary problem in a fluid model for biofilm growth,, Nonlin. Analysis, 70 (2009), 3658.  doi: 10.1016/j.na.2008.07.021.  Google Scholar

[22]

E. Paramonova, O. J. Kalmykowa, H. C. van der Mei, H. J. Busscher and P. K. Sharma, Impact of hydrodynamics on oral biofilm strength,, J. Dent. Res., 88 (2009), 922.  doi: 10.1177/0022034509344569.  Google Scholar

[23]

B. M. Peyton and W. G. Characklis, A statistical analysis of the effect of substrate utilization and shear stress on the kinetics of biofilm detachment,, Biotech. Bioeng., 41 (1982), 728.  doi: 10.1002/bit.260410707.  Google Scholar

[24]

L. A. Pritchett and J. D. Dockery, Steady state solutions of a one-dimensional biofilm,, Math. Comp. Model., 33 (2001), 255.  doi: 10.1016/S0895-7177(00)00242-9.  Google Scholar

[25]

B. E. Rittmann and P. L. McCarty, Model of steady state biofilm kinetics,, Biotech. Bioeng., 22 (1980), 2343.  doi: 10.1002/bit.260221110.  Google Scholar

[26]

B. E. Rittmann, The effect of shear stress on biofilm loss rate,, Biotech. Bioeng., 24 (1982), 501.   Google Scholar

[27]

B. E. Rittmann, A. O. Schwarz, H. J. Eberl, E. Morgenroth, J. Perez, M. C. M. van Loosdrecht and O. Wanner, Results from the multi-species benchmark problem (BM3) using one-dimensional models,, Wat. Sci. & Tech., 49 (2004), 163.   Google Scholar

[28]

B. E. Rittmann, D. Stilwell and A. Ohashi, The transient-state, multiple-species biofilm model for biofiltration processes,, Water Research, 36 (2002), 2342.  doi: 10.1016/S0043-1354(01)00441-9.  Google Scholar

[29]

A. Rochex, A. Massé, R. Escudié, J. J. Godon and N. Bernet, Influence of abrasion on biofilm detachment: Evidence for stratification of the biofilm,, J. Ind. Microbiol. Biotech., 36 (2009), 467.  doi: 10.1007/s10295-009-0543-x.  Google Scholar

[30]

P. Stoodley, R. Cargo, C. J. Rupp, S. Wilson and I. Klapper, Biofilm material properties as related to shear-induced deformation and detachment phenomena,, J. Ind. Microbiol. Biotech., 29 (2002), 361.  doi: 10.1038/sj.jim.7000282.  Google Scholar

[31]

B. Szomolay, Analysis of a moving boundary value problem arising in biofilm modelling,, Math. Meth. Appl. Sci., 31 (2008), 1835.  doi: 10.1002/mma.1000.  Google Scholar

[32]

U. Telgmann, H. Horn and E. Morgenroth, Influence of growth history on sloughing and erosion from biofilms,, Water Research, 38 (2004), 3671.  doi: 10.1016/j.watres.2004.05.020.  Google Scholar

[33]

L. Tijhuis, M. C. M. van Loosdrecht and J. J. Heijnen, Dynamics of biofilm detachment in biofilm airlift suspension reactors,, Biotech. Bioeng., 45 (1995), 481.  doi: 10.1002/bit.260450604.  Google Scholar

[34]

M. M. Trulear and W. G. Characklis, Dynamics of biofilm processes,, J. Water Pollut. Control Fed., 54 (1982), 1288.   Google Scholar

[35]

W. Walter, "Gewöhnliche Differentialgleichungen," 7th edition,, Springer-Verlag, (2000).   Google Scholar

[36]

O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms,", IWA Publishing, (2006).   Google Scholar

[37]

O. Wanner and W. Gujer, A multispecies biofilm model,, Biotech. Bioeng., 28 (1986), 314.   Google Scholar

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