Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T12:31:52.133Z Has data issue: false hasContentIssue false
  • English
  • Français

Simulating Biofilm Deformation and Detachment with the Immersed Boundary Method

Part of: Basic methods in fluid mechanics Coupling of solid mechanics with other effects Incompressible viscous fluids Materials of strain-rate type and history type, other materials with memory

Published online by Cambridge University Press:  16 March 2016

Rangarajan Sudarsan ,
Sudeshna Ghosh ,
John M. Stockie  and
Hermann J. Eberl
Rangarajan Sudarsan*
Affiliation:
Department of Mathematics and Statistics and Biophysics Interdepartmental Graduate Program, University of Guelph, ON, N1G 2W1, Canada
Sudeshna Ghosh
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
John M. Stockie
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
Hermann J. Eberl
Affiliation:
Department of Mathematics and Statistics and Biophysics Interdepartmental Graduate Program, University of Guelph, ON, N1G 2W1, Canada
*
*Corresponding author. Email addresses:[email protected] (R. Sudarsan), [email protected] (S. Ghosh), [email protected] (J. M. Stockie), [email protected] (H. J. Eberl)
Get access

Abstract

We apply the immersed boundary (or IB) method to simulate deformation and detachment of a periodic array of wall-bounded biofilm colonies in response to a linear shear flow. The biofilm material is represented as a network of Hookean springs that are placed along the edges of a triangulation of the biofilm region. The interfacial shear stress, lift and drag forces acting on the biofilm colony are computed by using fluid stress jump method developed by Williams, Fauci and Gaver [Disc. Con-tin. Dyn. Sys. B 11(2):519–540, 2009], with a modified version of their exclusion filter. Our detachment criterion is based on the novel concept of an averaged equivalent continuum stress tensor defined at each IB point in the biofilm which is then used to determine a corresponding von Mises yield stress; wherever this yield stress exceeds a given critical threshold the connections to that node are severed, thereby signalling the onset of a detachment event. In order to capture the deformation and detachment behaviour of a biofilm colony at different stages of growth, we consider a family of four biofilm shapes with varying aspect ratio. For each aspect ratio, we varied the spacing between colonies to investigate role of spatial clustering in offering protection against detachment. Our numerical simulations focus on the behaviour of weak biofilms (with relatively low yield stress threshold) and investigate features of the fluid-structure interaction such as locations of maximum shear and increased drag. The most important conclusions of this work are: (a) reducing the spacing between colonies reduces drag by from 50 to 100% and alters the interfacial shear stress profile, suggesting that even weak biofilms may be able to grow into tall structures because of the protection they gain from spatial proximity with other colonies; (b) the commonly employed detachment strategy in biofilm models based only on interfacial shear stress can lead to incorrect or inaccurate results when applied to the study of shear induced detachment of weak biofilms. Our detachment strategy based on equivalent continuum stresses provides a unified and consistent IB framework that handles both sloughing and erosion modes of biofilm detachment, and is consistent with strategies employed in many other continuum based biofilm models.

Keywords

Immersed boundarymethodbiofilmsdetachmentvon Mises yield stressinterfacial shear stressdrag and lift force

MSC classification

Secondary: 74D10: Nonlinear constitutive equations 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76D05: Navier-Stokes equations 76M20: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 682 - 732
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alpkvist, E. and Klapper, I.. Description of mechanical response including detachment using a novel particle model of biofilm/flow interaction. Water Sci. Technol., 55(8-9):265, 2007.CrossRefGoogle ScholarPubMed
[2]Alpkvist, E. and Klapper, I.. A multidimensional multispecies continuum model for heterogeneous biofilm development. Bull. Math. Biol., 69(2):765–789, 2007.Google Scholar
[3]Boresi, A.P. and Schmidt, R.J.. Advanced Mechanics of Materials. Wiley New York, 6th edition, 2003.Google Scholar
[4]Arvo, J.. Graphics Gems II, volume 2. Morgan Kaufmann, 1991.Google Scholar
[5]Bagi, K.. Stress and strain in granular assemblies. Mech. Mater., 22(3):165–177, 1996.CrossRefGoogle Scholar
[6]Balevičius, R., Sielamowicz, I., Mroz, Z., and Kačianauskas, R.. Investigation of wall stress and outflow rate in a flat-bottomed bin: A comparison of the DEM model results with the experimental measurements. Powder Technol., 214(3):322–336, 2011.CrossRefGoogle Scholar
[7]Boaventura, R. A. and Rodrigues, A. E.. Denitrification kinetics in a rotating disk biofilm reactor. Chem. Eng. J., 65(3):227–235, 1997.CrossRefGoogle Scholar
[8]Böl, M., Möhle, R. B., Haesner, M., Neu, T. R., Horn, H., and Krull, R.. 3D finite element model of biofilm detachment using real biofilm structures from CLSM data. Biotechnol. Bioeng., 103(1):177–186, 2009.Google Scholar
[9]Bottino, D. C. and Fauci, L. J.. A computational model of ameboid deformation and locomotion. Euro. Biophys. J., 27(5):532–539, 1998.CrossRefGoogle ScholarPubMed
[10]Byrne, E., Dzul, S., Solomon, M., Younger, J., and Bortz, D. M.. Postfragmentation density function for bacterial aggregates in laminar flow. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, 83(401):041911, 2011.Google Scholar
[11]Costerton, J. W., Stewart, P. S., and Greenberg, E. P.. Bacterial biofilms: A common cause of persistent infections. Science, 284(5418):1318–1322, 1999.CrossRefGoogle ScholarPubMed
[12]Cotin, Stéphane, Delingette, Hervé, and Ayache, Nicholas. Real-time elastic deformations of soft tissues for surgery simulation. IEEE Trans. Vis. Comput. Graphics, 5(1):62–73, 1999.Google Scholar
[13]Dillon, R. and Fauci, L.. A microscale model of bacterial and biofilm dynamics in porous media. Biotechnol. Bioeng., 68(5):536–547, 2000.Google Scholar
[14]Dillon, R., Fauci, L., Fogelson, A., and D. Gaver III, . Modeling biofilm processes using the immersed boundary method. J. Comput. Phys., 129(1):57–73, 1996.Google Scholar
[15]Donlan, R. M. and Costerton, W. J.. Biofilms: survival mechanisms of clinically relevant microorganisms. Clin. Microbiol. Rev., 15(2):167–193, 2002.CrossRefGoogle ScholarPubMed
[16]Duddu, R., Chopp, D. L., and Moran, B.. A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment. Biotechnol. Bioeng., 103(1):92–104, 2009.CrossRefGoogle ScholarPubMed
[17]Eberl, H. J., Morgenroth, E., Noguera, D., Picioreanu, C., Rittmann, B., van Loosdrecht, M., and Wanner, O.. Mathematical Modeling of Biofilms, volume 18 of Scientific and Technical Report Series. International Water Association, London, 2006.Google Scholar
[18]Eberl, H. J., Parker, D. F., and van Loosdrecht, M. C. M.. A new deterministic spatio-temporal continuum model for biofilm development. Theor. Med., 3(3):161–175, 2001.Google Scholar
[19]Eberl, H. J. and Sudarsan, R.. Exposure of biofilms to slow flow fields: The convective contribution to growth and disinfection. J. Theor. Biol., 253(4):788–807, 2008.CrossRefGoogle ScholarPubMed
[20]Elias, R. N., Martins, Marcos A. D., and Alvaro, L. G. A.Coutinho. Simple finite element-based computation of distance functions in unstructured grids. Int. J. Numer. Meth. Eng., 72(9):1095–1110, 2007.Google Scholar
[21]Fauci, L. J. and McDonald, A.. Sperm motility in the presence of boundaries. Bull.Math. Biol., 57(5):679–699, 1995.Google Scholar
[22]Fogelson, A. L. and Guy, R. D.. Platelet–wall interactions in continuum models of platelet thrombosis: formulation and numerical solution. Math. Med. Biol., 21(4):293–334, 2004.Google Scholar
[23]Fortin, J., Millet, O., and De Saxcé, G.. Construction of an averaged stress tensor for a granular medium. Euro. J. Mech. A Solids, 22(4):567–582, 2003.Google Scholar
[24]Gaboriaud, F., Gee, M.L., Strugnell, R., and Duval, J.F.. Coupled electrostatic, hydrodynamic, and mechanical properties of bacterial interfaces in aqueous media. Langmuir, 24(19):10988– 10995, 2008.Google Scholar
[25]Gaver III, D. P. and Kute, S.M.. A theoretical model study of the influence of fluid stresses on a cell adhering to a microchannel wall. Biophys. J., 75(2):721–733, 1998.Google Scholar
[26]Ghosh, S. and Stockie, J. M.. Numerical simulations of particle sedimentation using the immersed boundary method. Commun. Comput. Phys., 18(2):380–416, 2015.Google Scholar
[27]Griffith, B. E.. On the volume conservation of the immersed boundary method. Commun. Comput. Phys., 12(2):401–432, 2012.Google Scholar
[28]Griffith, B. E., Hornung, R. D., McQueen, D. M., and Peskin, C. S.. An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys., 223(1):10–49, 2007.Google Scholar
[29]Guélon, T. H., Mathias, J. D., and Stoodley, P.. Advances in biofilm mechanics. In Flemming, H.-C., Wingender, J., and Szewzyk, U., editors, Biofilm Highlights, volume 5 of Springer Series on Biofilms, pages 111–139. Springer, 2011.CrossRefGoogle Scholar
[30]Hammond, J. F., Stewart, E. J., Younger, J. G., M. J. Solomon, , and Bortz, D. M.. Spatially heterogeneous biofilm simulations using an immersed boundary method with lagrangian nodes defined by bacterial locations. arXiv preprint arXiv:1302.3663, 2013.Google Scholar
[31]Hammond, J. F., Stewart, E. J., Younger, J.G., Solomon, M. J., and Bortz, D.M.. Variable viscosity and density biofilm simulations using an immersed boundary method, part I: Numerical ccheme and convergence results. Comput. Model. Eng. Sci, 98(3):295–340, 2014.Google Scholar
[32]Hohne, D. N., Younger, J. G., and Solomon, M. J.. Flexible microfluidic device for mechanical property characterization of soft viscoelastic solids such as bacterial biofilms. Langmuir, 25(13):7743–7751, 2009.Google Scholar
[33]Jagota, A. and Bennison, S. J.. Spring-network and finite-element models for elasticity and fracture. In Non-linearity and Breakdown in Soft Condensed Matter, pages 186–201. Springer, 1994.Google Scholar
[34]Jagota, A. and Bennison, S.J.. Element breaking rules in computational models for brittle fracture. Modell. Simul. Mater. Sci. Eng., 3(4):485, 1995.Google Scholar
[35]Bolander, J.E. Jr, Hong, G. S., and Yoshitake, K.. Structural concrete analysis using rigid-body-spring networks. Comput. Aided Civ. Infrastruct. Eng., 15(2):120–133, 2000.Google Scholar
[36]Kissel, J. C., McCarty, P. L., and Street, R. L.. Numerical simulation of mixed-culture biofilm. J. Environ. Eng., 110(2):393–411, 1984.Google Scholar
[37]Klapper, I. and Dockery, J.. Finger formation in biofilm layers. SIAMJ. Appl.Math., 62(3):853–869, 2002.Google Scholar
[38]Klausen, M., Aaes-Jørgensen, A., Molin, S., and Tolker-Nielsen, T.. Involvement of bacterial migration in the development of complex multicellular structures in Pseudomonas aeruginosa biofilms. Mol. Microbiol., 50(1):61–68, 2003.Google Scholar
[39]Kreft, J. U., Picioreanu, C., Wimpenny, J.W. T., and van Loosdrecht, M. C.M.. Individual-based modelling of biofilms. Microbiology+, 147(11):2897–2912, 2001.Google Scholar
[40]Lai, M.-C. and Peskin, C. S.. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys., 160(2):705–719, 2000.CrossRefGoogle Scholar
[41]Lewandowski, Z. and Beyenal, H.. Fundamentals of Biofilm Research. CRC press, 2013.CrossRefGoogle Scholar
[42]Lindley, B., Wang, Q., and Zhang, T.. Multicomponent hydrodynamic model for heterogeneous biofilms: Two-dimensional numerical simulations of growth and interaction with flows. Phys. Rev. E, 85(3):031908, 2012.Google Scholar
[43]Liu, D., Li, H., and Liu, Y.. Numerical simulation of creep damage and life prediction of superalloy turbine blade. Math Probl. Eng., 2015:732502, 2015.Google Scholar
[44]Lloyd, B. A., G. Székely, , and Harders, M.. Identification of spring parameters for deformable object simulation. IEEE Trans. Vis. Comput. Graphics, 13(5):1081–1094, 2007.Google Scholar
[45]Van Loosdrecht, M.C.M., Heijnen, J. J., Eberl, H., Kreft, J., and Picioreanu, C.. Mathematical modelling of biofilm structures. ANTON LEEUW INT J G, 81(1–4):245–256, 2002.Google Scholar
[46]Manz, B., Volke, F., Goll, D., and Horn, H.. Measuring local flow velocities and biofilm structure in biofilm systems with magnetic resonance imaging(mri). Biotechnol. Bioeng., 84(4):424–432, 2003.Google Scholar
[47]Merkey, B. V., Rittmann, B. E., and Chopp, D. L.. Modeling how soluble microbial products (SMP) support heterotrophic bacteria in autotroph-based biofilms. J. Theor. Biol., 259(4):670–683, 2009.Google Scholar
[48]Mori, Y.. Convergence proof of the velocity field for a Stokes flow immersed boundary method. Commun. Pure Appl. Math., 61:1213–1263, 2008.Google Scholar
[49]Mori, Y. and Peskin, C. S.. Implicit second-order immersed boundary methods with boundary mass. Comput.Method. Appl. Mech. Eng., 197(25):2049–2067, 2008.CrossRefGoogle Scholar
[50]Nicolella, C., van Loosdrecht, M. C. M., and Heijnen, J. J.. Wastewater treatment with particulate biofilm reactors. J. Biotechnol., 80(1):1–33, 2000.CrossRefGoogle ScholarPubMed
[51]Ohashi, A. and Harada, H.. A novel concept for evaluation of biofilm adhesion strength by applying tensile force and shear force. Water Sci. Technol., 34(5-6):201–211, 1996.CrossRefGoogle Scholar
[52]Ostoja-Starzewski, M.. Latticemodels in micromechanics. Appl.Mech. Rev., 55(1):35–60, 2002.Google Scholar
[53]Ostoja-Starzewski, M., Sheng, P. Y., and Alzebdeh, K.. Spring network models in elasticity and fracture of composites and polycrystals. Comp. Mater. Sci., 7(1):82–93, 1996.Google Scholar
[54]Persson, P.-O. and Strang, G.. A simple mesh generator in MATLAB. SIAM Review, 46(2):329–345, 2004.CrossRefGoogle Scholar
[55]Peskin, C. S.. Numerical analysis of blood flow in the heart. J. Comput. Phys., 25(3):220–252, 1977.Google Scholar
[56]Peskin, C. S.. The immersed boundary method. Acta Numerica, 11:479–517, 2003.Google Scholar
[57]Picioreanu, C.,van Loosdrecht, M.C.M., and Heijnen, J. J.. Two-dimensionalmodel of biofilm detachment caused by internal stress from liquid flow. Biotechnol. Bioeng., 72(2):205–218, 2001.Google Scholar
[58]Press, W. H.. Numerical Recipes in Fortran 77: The Art of Scientific Computing, volume 1. Cambridge University Press, 1992.Google Scholar
[59]Rusconi, R., Lecuyer, S., Autrusson, N., Guglielmini, L., and Stone, H. A.. Secondary flow as a mechanism for the formation of biofilm streamers. Biophys. J., 100(6):1392–1399, 2011.Google Scholar
[60]Schlangen, E. and Garboczi, E. J.. New method for simulating fracture using an elastically uniform random geometry lattice. Int. J. Eng. Sci., 34(10):1131–1144, 1996.Google Scholar
[61]Schlangen, E. and Garboczi, E. J.. Fracture simulations of concrete using latticemodels: Computational aspects. Eng. Fract. Mech., 57(2):319–332, 1997.Google Scholar
[62]Seeluangsawat, P.. 3-D computational investigation of viscoelastic biofilms using GPUs. PhD thesis, Department of Mathematics, University of South Carolina, Columbia, SC, 2011.Google Scholar
[63]Shi, X. and Zhu, X.. Biofilm formation and food safety in food industries. Trends Food Sci. Technol., 20(9):407–413, 2009.Google Scholar
[64]Smith, B., B. Vaughan, , and Chopp, D.. The extended finite element method for boundary layer problems in biofilm growth. Commun. Appl. Math. Comput. Sci., 2(1):35–56, 2007.Google Scholar
[65]Stockie, J. M.. Analysis and computation of immersed boundaries, with application to pulp fibers. PhD thesis, Department of Mathematics, University of British Columbia, 1997.Google Scholar
[66]Stockie, J. M.. Modelling and simulation of porous immersed boundaries. Comput. Struct., 87(11-12):701–709, 2009.Google Scholar
[67]Stotsky, J. A., Hammond, J. F., Pavlovsky, L., Stewart, E. J., Younger, J. G., Solomon, M. J., and Bortz, D.M.. Variable viscosity and density biofilm simulations using an immersed boundary method, part II: Experimental validation and the heterogenous rheology-ibm. arXiv preprint arXiv:1504.07326, 2015.Google Scholar
[68]Sudarsan, R., Milferstedt, K., Morgenroth, E., and Eberl, H. J.. Quantification of detachment forces on rigid biofilm colonies in a roto-torque reactor using computational fluid dynamics tools. Water Sci., 52(7):149–154, 2005.Google Scholar
[69]Taherzadeh, D., Picioreanu, C., and Horn, H.. Mass transfer enhancement in moving biofilm structures. Biophys. J., 102(7):1483–1492, 2012.CrossRefGoogle ScholarPubMed
[70]Towler, B. W., Cunningham, A., Stoodley, P., and McKittrick, L.. A model of fluid-biofilm interaction using a Burger material law. Biotechnol. Bioeng., 96(2):259–271, 2007.Google Scholar
[71]Towler, B. W., Rupp, C. J., Cunningham, A. B., and Stoodley, P.. Viscoelastic properties of a mixed culture biofilm from rheometer creep analysis. Biofouling, 19(5):279–285, 2003.Google Scholar
[72]Valiei, A., Kumar, A., Mukherjee, P. P., Liu, Y., and Thundat, T.. A web of streamers: Biofilm formation in a porous microfluidic device. Lab on a Chip, 12(24):5133–5137, 2012.Google Scholar
[73]Vo, G. D., Brindle, E., and Heys, J.. An experimentally validated immersed boundary model of fluid-biofilm interaction. Water Sci. Technol., 61(12):3033–3040, 2010.Google Scholar
[74]Vo, G. D. and Heys, J.. Biofilm deformation in response to fluid flow in capillaries. Biotechnol. Bioeng., 108(8):1893–1899, 2011.Google Scholar
[75]Voller, V. R.. Basic Control Volume Finite Element Methods for Fluids and Solids. World Scientific, Singapore, 2009.Google Scholar
[76]Wang, Q. and Zhang, T.. Review of mathematical models for biofilms. Solid State Commun., 150(21-22):1009–1022, 2010.Google Scholar
[77]Wanner, O. and Gujer, W.. A multispecies biofilm model. Biotechnol. Bioeng., 28(3):314–328, 1986.Google Scholar
[78]Wiens, J. K. and Stockie, J. M.. An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver. J. Comput. Phys., 281:917–941, 2015.Google Scholar
[79]Williams, H. A. R., Fauci, L. J., and Gaver, D. P.. Evaluation of interfacial fluid dynamical stresses using the immersed boundary method. Disc. Contin. Dyn. Syst. B, 11(2):519–540, 2009.Google Scholar
[80]Wimpenny, J. W. T. and Colasanti, R.. A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models. FEMS Microbiol. Ecol., 22(1):1–16, 1997.Google Scholar
[81]Xavier, J. D. B., Picioreanu, C., and van Loosdrecht, M. C.M.. A general description of detachment for multidimensional modelling of biofilms. Biotechnol. Bioeng., 91(6):651–669, 2005.Google Scholar
[82]Xu, J., Sudarsan, R., Darlington, G. A., and Eberl, H. J.. A computational study of external shear forces in biofilm clusters. In 22nd International Symposium on High Performance Computing Systems and Applications, pages 139–145, Québec, Canada, 2008. IEEE.Google Scholar
[83]Yatomi, M., Bettinson, A. D., O’dowd, N. P., and K. M. Nikbin, . Modelling of damage development and failure in notched-bar multiaxial creep tests. Fatigue Fract. Eng. M., 27(4):283–295, 2004.CrossRefGoogle Scholar
[84]Zhang, T., Cogan, N., and Wang, Q.. Phase-field models for biofilms. II. 2-D numerical simulations of biofilm-flow interaction. Commun. Comput. Phys., 4(1):72–101, 2008.Google Scholar
[85]Zhao, P. and Heinrich, J. C.. Front-tracking finite element method for dendritic solidification. J. Comput. Phys., 173(2):765–796, 2001.CrossRefGoogle Scholar