A new approach to the simulation of microbial biofilms by a theory of fluid-like pressure-restricted finite growth. (English) Zbl 1296.74061
Summary: In general, the term ‘growth’ characterises the process by which a living body increases in size by addition of mass. Living matter grows in various different ways, triggered by genetic and biological factors. In addition, the configuration of the grown body in space depends on its interaction with the environment at the boundaries. In this paper, we deal with mechanical constraints on growth at the boundary of the body. Particularly, we present a model for growth such that residual stresses resulting from an isotropic deposition of new material are continuously relieved and that depends on the hydrostatic pressure acting on the material. As an example for this pressure-restricted fluid-like type of growth, we consider microbial biofilms growing between rigid obstacles in geometrically confined environments. The presented concept unites two classical constitutive formulations of large strain viscoelasticity and finite growth. The model was implemented into a finite element framework to illustrate its performance in several benchmark problems.
MSC:
| 74K35 | Thin films |
| 74D05 | Linear constitutive equations for materials with memory |
| 92C42 | Systems biology, networks |
| 76Z05 | Physiological flows |
References:
| [1] | Abe, Y.; Skali-Lami, S.; Block, J. C.; Francius, G., Cohesiveness and hydrodynamic properties of young drinking water biofilms, Water. Res., 46, 4, 1155-1166 (2012) |
| [2] | Allison, D. G., The biofilm matrix, Biofouling, 19, 2, 139-150 (2003) |
| [3] | Alpkvist, E.; Picioreanu, C.; van Loosdrecht, M. C.M.; Heyden, A., Three-dimensional biofilm model with individual cells and continuum EPS matrix, Biotechnol. Bioeng., 94, 5, 961-979 (2006) |
| [4] | Ambrosi, D.; Mollica, F., On the mechanics of a growing tumor, Int. J. Eng. Sci., 40, 12, 1297-1316 (2002) · Zbl 1211.74161 |
| [5] | Andrews, J. S.; Rolfe, S. A.; Huang, W. E.; Scholes, J. D.; Banwart, S. A., Biofilm formation in environmental bacteria is influenced by different macromolecules depending on genus and species, Environ. Microbiol., 12, 9, 2496-2507 (2010) |
| [6] | Bader, F. G., Analysis of double-substrate limited growth, Biotechnol. Bioeng., 20, 2, 183-202 (1978) |
| [7] | Beeftink, H. H.; van der Heijden, R. T.; Heijnen, J. J., Maintenance requirements: energy supply from simultaneous endogenous respiration and substrate consumption, FEMS Microbiol. Lett., 73, 3, 203-209 (1990) |
| [8] | Ben-Jacob, E.; Schochet, O.; Tenenbaum, A.; Cohen, I.; Czirók, A.; Vicsek, T., Generic modelling of cooperative growth patterns in bacterial colonies, Nature, 368, 6466, 46-49 (1994) |
| [9] | Böl, M.; Bolea Albero, A., On a new model for inhomogeneous volume growth of elastic bodies, J. Mech. Behav. Biomed. Mater., 29, 582-593 (2014) |
| [10] | Böl, M.; Ehret, A. E.; Bolea Albero, A.; Hellriegel, J.; Krull, R., Recent advances in mechanical characterisation of biofilm and their significance for material modelling, Crit. Rev. Biotechnol., 33, 2, 145-171 (2013) |
| [11] | Bolea Albero, A.; Ehret, A. E.; Böl, M., A continuum model for free growth in living materials, Proc. Appl. Math. Mech., 12, 1, 123-124 (2012) |
| [12] | Buganza Tepole, A.; Gosain, A. K.; Kuhl, E., Stretching skin: the physiological limit and beyond, Int. J. Nonlinear Mech., 47, 8, 938-949 (2012) |
| [13] | Busscher, H. J.; Weerkamp, A. H., Specific and non-specific interactions in bacterial adhesion to solid substrata, FEMS Microbiol. Lett., 46, 2, 165-173 (1987) |
| [14] | Carlén, A.; Nikdel, K.; Wennerberg, A.; Holmberg, K.; Olsson, J., Surface characteristics and in vitro biofilm formation on glass ionomer and composite resin, Biomaterials, 22, 5, 481-487 (2001) |
| [15] | Chambless, J. D.; Hunt, S. M.; Stewart, P. S., A three-dimensional computer model of four hypothetical mechanisms protecting biofilms from antimicrobials, Appl. Environ. Microbiol., 72, 3, 2005-2013 (2006) |
| [16] | Characklis, W. G.; Marshall, K. C., Biofilms: a basis for an interdisciplinary approach, (Characklis, W. G.; Marshall, K. C., Biofilms (1990), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York), 3-15 |
| [17] | Chávez de Paz, L. E., Development of a multispecies biofilm community by four root canal bacteria, J. Endod., 38, 3, 318-323 (2012) |
| [18] | Chopp, D. L.; Kirisits, M. J.; Moran, B.; Parsek, M. R., A mathematical model of quorum sensing in a growing bacterial biofilm, J. Ind. Microbiol. Biotechnol., 29, 6, 339-346 (2002) |
| [19] | Chopp, D. L.; Kirisits, M. J.; Moran, B.; Parsek, M. R., The dependence of quorum sensing on the depth of a growing biofilm, Bull. Math. Biol., 65, 6, 1053-1079 (2003) · Zbl 1334.92254 |
| [20] | Cogan, N. G.; Keener, J. P., The role of the biofilm matrix in structural development, Math. Med. Biol., 21, 2, 147-166 (2004) · Zbl 1055.92034 |
| [21] | Cogan, N. G.; Keener, J. P., Channel formation in gels, SIAM J. Appl. Math., 65, 6, 1839-1854 (2005) · Zbl 1114.74011 |
| [22] | Colasanti, R. L., Cellular automata models of microbial colonies, Binary Comput. Microbiol., 4, 1, 191-193 (1992) |
| [23] | Costerton, J. W.; Lewandowski, Z.; Caldwell, D. E.; Korber, D. R.; Lappin-Scott, H. M., Microbial biofilms, Annu. Rev. Microbiol., 49, 1, 711-745 (1995) |
| [24] | Das, T.; Sharma, P. K.; Busscher, H. J.; van der Mei, H. C.; Krom, B. P., Role of extracellular DNA in initial bacterial adhesion and surface aggregation, Appl. Environ. Microbiol., 76, 10, 3405-3408 (2010) |
| [25] | DeAngelis, D. L.; Gross, L. J., Individual-based Models and Approaches in Ecology: Populations, Communities and Ecosystems (1992), Chapman and Hall/CRC: Chapman and Hall/CRC London |
| [26] | Dens, E. J.; Bernaerts, K.; Standaert, A. R.; Kreft, J. U.; van Impe, J. F., Cell division theory and individual-based modeling of microbial lag: Part II. Modeling lag phenomena induced by temperature shifts, Int. J. Food Microbiol., 101, 3, 319-332 (2005) |
| [27] | Dens, E. J.; Bernaerts, K.; Standaert, A. R.; van Impe, J. F., Cell division theory and individual-based modeling of microbial lag: Part I. The theory of cell division, Int. J. Food Microbiol., 101, 3, 303-318 (2005) |
| [28] | Dockery, J.; Klapper, I., Finger formation in biofilm layers, SIAM J. Appl. Math., 62, 3, 853-869 (2002) · Zbl 1012.92007 |
| [29] | Dominiak, D. M.; Nielsen, J. L.; Nielsen, P. H., Extracellular DNA is abundant and important for microcolony strength in mixed microbial biofilms, Environ. Microbiol., 13, 3, 710-721 (2011) |
| [30] | Donlan, R. M., Biofilm: microbial life on surfaces, Emerg. Infect. Dis., 8, 9, 881-890 (2002) |
| [31] | Duddu, R.; Bordas, S.; Chopp, D. L.; Moran, B., A combined extended finite element and level set method for biofilm growth, Int. J. Numer. Methods Eng., 74, 5, 848-870 (2008) · Zbl 1195.74169 |
| [32] | Duddu, R.; Chopp, D. L.; 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) |
| [33] | Dupin, H. J.; Kitanidis, P. K.; McCarthy, P. L., Pore-scale modeling of biological clogging due to aggregate expansion: a material mechanics approach, Water Resour. Res., 37, 12, 2965-2979 (2001) |
| [34] | Dupin, H. J.; Kitanidis, P. K.; McCarthy, P. L., Simulations of two-dimensional modeling of biomass aggregate growth in network models, Water Resour. Res., 37, 12, 2981-2994 (2001) |
| [35] | Eberl, H. J.; Parker, D. F.; van Loosdrecht, M. C.M., A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3, 3, 161-175 (2001) · Zbl 0985.92009 |
| [36] | Eberl, H. J.; Picioreanu, C.; Heijnen, J. J.; van Loosdrecht, M. C.M., A three-dimensional numerical study on the correlation of spatial structure, hydrodynamic conditions, and mass transfer and conversion in biofilms, Chem. Eng. Sci., 55, 24, 6209-6222 (2000) |
| [37] | Ehret, A. E.; Böl, M., Modelling mechanical characteristics of microbial biofilms by network theory, J. R. Soc. Interface, 10, 78 (2013) |
| [38] | Ehret, A. E.; Bolea Albero, A.; Böl, M., A network model for the EPS matrix of microbial biofilms, Proc. Appl. Math. Mech., 12, 1, 125-126 (2012) |
| [39] | Ferrer, J. D.; Prats, C.; López, D., Individual-based modelling: an essential tool for microbiology, J. Biol. Phys., 34, 1-2, 19-37 (2008) |
| [40] | Flemming, H. C.; Neu, T. R.; Wozniak, D. J., The EPS matrix: The house of biofilm cells, J. Bacteriol., 189, 22, 7945-7947 (2007) |
| [41] | Flemming, H. C.; Wimpenny, J. W., Biofilme - die bevorzugte Lebensform der Bakterien: Flocken, Filme und Schlämme, Biol. unserer Zeit, 31, 3, 169-180 (2001) |
| [42] | Flemming, H. C.; Wingender, J., Was Biofilme zusammenhält: Proteine, Polysaccharide Chem. unserer Zeit, 36, 1, 30-42 (2002) |
| [43] | Flemming, H. C.; Wingender, J., The biofilm matrix, Nat. Rev. Microbiol., 8, 9, 623-633 (2010) |
| [44] | Flory, P. J., Thermodynamics relation for high elastic materials, Trans. Faraday Soc., 57, 1, 829-838 (1961) |
| [45] | Fujikawa, H., Diversity of the growth patterns of Bacillus subtilis colonies on agar plates, FEMS Microbiol. Ecol., 13, 3, 159-168 (1994) |
| [46] | Garny, K.; Neu, T. R.; Horn, H., Sloughing and limited substrate conditions trigger filamentous growth in heterotrophic biofilms - Measurements in flow-through tube reactor, Chem. Eng. Sci., 64, 11, 2723-2732 (2009) |
| [47] | Gehrke, T.; Telegdi, J.; Thierry, D.; Sand, W., Importance of extracellular polymeric substances from Thiobacillus ferrooxidans for bioleaching, Appl. Environ. Microbiol., 64, 7, 2743-2747 (1998) |
| [48] | Göktepe, S.; Abilez, O. J.; Kuhl, E., A generic approach towards finite growth with examples of athlete’s heart, cardiac dilation, and cardiac wall thickening, J. Mech. Phys. Solids, 58, 10, 1661-1680 (2010) · Zbl 1200.74109 |
| [49] | Graf von der Schulenburg, D. A.; Pintelon, T. R.; Picioreanu, C.; van Loosdrecht, M. C.M.; Johns, M. L., Three-dimensional simulations of biofilm growth in porous media, AIChE J., 55, 2, 494-504 (2009) |
| [50] | Green, M. S.; Tobolsky, A. V., A new approach to the theory of relaxing polymeric media, J. Chem. Phys., 14, 2, 80-92 (1946) |
| [51] | Grimm, V., Ten years of individual-based modelling in ecology: what have we learned and what could we learn in the future?, Ecol. Model., 115, 2, 129-148 (1999) |
| [52] | Grimm, V.; Berger, U.; Bastiansen, F.; Eliassen, S.; Ginot, V.; Giske, J.; Goss-Custard, J.; Grand, T.; Heinz, S. K.; Huse, G.; Huth, A.; Jepsen, J. U.; Jørgensen, C.; Mooij, W. M.; Müller, B.; Pe’er, G.; Piou, C.; Railsback, S. F.; Robbins, A. M.; Robbins, M. M.; Rossmanith, E.; Rüger, N.; Strand, E.; Souissi, S.; Stillman, R. A.; Vabø, R.; Visser, U.; DeAngelis, D. L., A standard protocol for describing individual-based and agent-based models, Ecol. Model., 198, 1-2, 115-126 (2006) |
| [53] | Hermanowicz, S. W., A model of two-dimensional biofilm morphology, Water Sci. Technol., 37, 4-5, 219-222 (1998) |
| [54] | Hermanowicz, S. W., Two-dimensional simulations of biofilm development: effects of external environmental conditions, Water Sci. Technol., 39, 7, 107-114 (1999) |
| [55] | Hermanowicz, S. W., A simple 2D biofilm model yields a variety of morphological features, Math. Biosci., 169, 1, 1-14 (2001) · Zbl 0966.92003 |
| [56] | Holzapfel, G. A., Nonlinear Solid Mechanics: A Continuum Approach for Engineering (2000), Wiley: Wiley Chichester, New York · Zbl 0980.74001 |
| [58] | Huang, Z.; McLamore, E. S.; Chuang, H. S.; Zhang, W.; Wereley, S.; Leon, J. L.C.; Banks, M. K., Shear-induced detachment of biofilms from hollow fiber silicone membranes, Biotechnol. Bioeng., 110, 2, 525-534 (2013) |
| [59] | Jones, W. L.; Sutton, M. P.; McKittrick, L.; Stewart, P. S., Chemical and antimicrobial treatments change the viscoelastic properties of bacterial biofilms, Biofouling, 27, 2, 207-215 (2011) |
| [60] | Judson, O. P., The rise of the individual-based model in ecology, Trends Ecol. Evol., 9, 1, 9-14 (1994) |
| [61] | Khassehkhan, H.; Hillen, T.; Eberl, H. J., A nonlinear master equation for a degenerate diffusion model of biofilm growth, (Allen, G.; Nabrzyski, J.; Seidel, E.; van Albada, G. D.; Dongarra, J.; Sloot, P. M.A., Computational Science - ICCS 2009. Computational Science - ICCS 2009, Lecture Notes in Computer Science, vol. 5544 (2009), Springer), 735-744 |
| [62] | Khassehkhan, H.; Efendiev, M. A.; Eberl, H. J., A degenerate diffusion-reaction model of an amensalistic biofilm control system: existence and simulation of solutions, Discrete Contin. Dyn. Syst. Ser. B, 12, 2, 371-388 (2009) · Zbl 1183.35160 |
| [63] | Kishen, A., Advanced therapeutic options for endodontic biofilms, Endod. Topics, 22, 1, 99-123 (2010) |
| [64] | Klapper, I., Effect of heterogeneous structure in mechanically unstressed biofilms on overall growth, Bull. Math. Biol., 66, 4, 809-824 (2004) · Zbl 1334.92259 |
| [65] | Klapper, I.; Dockery, J., Role of cohesion in the material description of biofilms, Phys. Rev. E., 74, 3, 031902 (2006) |
| [66] | Klapper, I.; Rupp, C. J.; Cargo, R.; Purvedorj, B.; Stoodley, P., Viscoelastic fluid description of bacterial biofilm material properties, Biotechnol. Bioeng., 80, 3, 289-296 (2002) |
| [67] | Korber, D. R.; James, G. A.; Costerton, J. W., Evaluation of fleroxacin activity against established Pseudomonas fluorescens biofilms, Appl. Environ. Microbiol., 60, 5, 1663-1669 (1994) |
| [68] | Korber, D. R.; Lawrence, J. R.; Caldwell, D. E., Effect of motility on surface colonization and reproductive success of Pseudomonas fluorescens in dual-dilution continuous culture and batch culture systems, Appl. Environ. Microbiol., 60, 5, 1421-1429 (1994) |
| [70] | Körstgens, V.; Flemming, H. C.; Wingender, J.; Borchard, W., Influence of calcium ions on the mechanical properties of a model biofilm of mucoid Pseudomonas aeruginosa, Water Sci. Technol., 46, 6, 49-57 (2001) |
| [71] | Kreft, J. U.; Booth, G.; Wimpenny, J. W.T., BacSim, a simulator for individual-based modelling of bacterial colony growth, Microbiology, 144, 12, 3275-3287 (1998) |
| [72] | Kreft, J. U.; Picioreanu, C.; Wimpenny, J. W.T.; van Loosdrecht, M. C.M., Individual-based modelling of biofilms, Microbiology, 147, 11, 2897-2912 (2001) |
| [73] | Kreft, J. U.; Wimpenny, J. W.T., Effect of EPS on biofilm structure and function as revealed by an individual-based model of biofilm growth, Water Sci. Technol., 43, 6, 135-141 (2001) |
| [74] | Kuhl, E.; Steinmann, P., Mass- and volume-specific views on thermodynamics for open systems, Proc. R. Soc. A Math. Phys., 459, 2038, 2547-2568 (2003) · Zbl 1092.80500 |
| [75] | Kuhl, E.; Steinmann, P., On spatial and material settings of thermo-hyperelastodynamics for open systems, Acta Mech., 160, 3-4, 179-217 (2003) · Zbl 1064.74007 |
| [76] | Kuhl, E.; Steinmann, P., Theory and numerics of geometrically non-linear open system mechanics, Int. J. Numer. Methods Eng., 58, 11, 1593-1615 (2003) · Zbl 1032.74504 |
| [77] | Lappin-Scott, H. M.; Costerton, J. W., Microbial Biofilms (1995), Cambridge University Press: Cambridge University Press Cambridge, New York |
| [78] | Lardon, L. A.; Merkey, B. V.; Martins, S.; Dötsch, A.; Picioreanu, C.; Kreft, J. U.; Smets, B. F., iDynoMiCS: next-generation individual-based modelling of biofilms, Environ. Microbiol., 13, 9, 2416-2434 (2011) |
| [79] | Laspidou, C. S.; Kungolos, A.; Samaras, P., Cellular-automata and individual-based approaches for the modeling of biofilm structures: pros and cons, Desalination, 250, 1, 390-394 (2010) |
| [80] | Laspidou, C. S.; Liakopoulos, A.; Spiliotopoulos, M. G., A 2D cellular automaton biofilm detachment algorithm, (Sirakoulis, G. C.; Bandini, S., Cellular Automata. Cellular Automata, Lecture Notes in Computer Science, vol. 7495 (2012), Springer: Springer Wiesbaden), 415-424 |
| [81] | Laspidou, C. S.; Rittmann, B. E., Evaluating trends in biofilm density using the UMCCA model, Water Res., 38, 14-15, 3362-3372 (2004) |
| [82] | Laspidou, C. S.; Rittmann, B. E., Modeling the development of biofilm density including active bacteria, inert biomass, and extracellular polymeric substances, Water Res., 38, 14-15, 3349-3361 (2004) |
| [83] | Laspidou, C. S.; Rittmann, B. E.; Karamanos, S. A., Finite element modeling to expand the UMCCA model to describe biofilm mechanical behavior, Water Sci. Technol., 52, 7, 161-166 (2005) |
| [84] | Le Tallec, P.; Rahier, C.; Kaiss, A., Three-dimensional incompressible viscoelasticity in large strains: formulation and numerical approximation, Comput. Methods Appl. Math., 109, 3-4, 233-258 (1993) · Zbl 0845.73029 |
| [85] | Liao, Q.; Wang, Y. J.; Wang, Y. Z.; Chen, R.; Zhu, X.; Pu, Y. K.; Lee, D. J., Two-dimension mathematical modeling of photosynthetic bacterial biofilm growth and formation, Int. J. Hydrogen Energy, 37, 20, 15607-15615 (2012) |
| [86] | Linder, C.; Tkachuk, M.; Miehe, C., A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity, J. Mech. Phys. Solids, 59, 10, 2134-2156 (2011) · Zbl 1270.74035 |
| [87] | Lion, A., On the large deformation behaviour of reinforced rubber at different temperatures, J. Mech. Phys. Solids, 45, 11-12, 1805-1834 (1997) |
| [88] | Liu, Y.; Tay, J. H., The essential role of hydrodynamic shear force in the formation of biofilm and granular sludge, Water Res., 36, 7, 1653-1665 (2002) |
| [89] | Lubarda, V. A.; Hoger, A., On the mechanics of solids with a growing mass, Int. J. Solids Struct., 39, 18, 4627-4664 (2002) · Zbl 1045.74035 |
| [90] | Lubliner, J., A model of rubber viscoelasticity, Mech. Res. Commun., 12, 2, 93-99 (1985) |
| [91] | Mayer, C.; Moritz, R.; Kirschner, C.; Borchard, W.; Maibaum, R.; Wingender, J.; Flemming, H. C., The role of intermolecular interactions: studies on model systems for bacterial biofilms, Int. J. Biol. Macromol., 26, 1, 3-16 (1999) |
| [92] | McDougald, D.; Rice, S. A.; Barraud, N.; Steinberg, P. D.; Kjelleberg, S., Should we stay or should we go: Mechanisms and ecological consequences for biofilm dispersal, Nat. Rev. Microbiol., 10, 1, 39-50 (2012) |
| [93] | Menzel, A., Modelling of anisotropic growth in biological tissues. A new approach and computational aspects, Biomech. Model. Mechanobiol., 3, 3, 147-171 (2005) |
| [94] | Monod, J., The growth of bacterial cultures, Annu. Rev. Microbiol., 3, 1, 371-394 (1949) |
| [95] | Morgan, T. D.; Wilson, M., The effects of surface roughness and type of denture acrylic on biofilm formation by Streptococcus oralis in a constant depth film fermentor, J. Appl. Microbiol., 91, 1, 47-53 (2001) |
| [96] | Morikawa, M., Beneficial biofilm formation by industrial bacteria Bacillus subtilis and related species, J. Biosci. Bioeng., 101, 1, 1-8 (2006) |
| [97] | Nadell, C. D.; Xavier, J. B.; Foster, K. R., The sociobiology of biofilms, FEMS Microbiol. Rev., 33, 1, 206-224 (2009) |
| [98] | Neu, T. R., Significance of bacterial surface-active compounds in interaction of bacteria with interfaces, Microbiol. Mol. Biol. Rev., 60, 1, 151-166 (1996) |
| [99] | Nicolella, C.; van Loosdrecht, M. C.M.; Heijnen, J. J., Wastewater treatment with particulate biofilm reactors, J. Biotechnol., 80, 1, 1-33 (2000) |
| [100] | Noguera, D. R.; Pizarfo, G.; Stahl, D. A.; Rittmann, B. E., Simulation of multispecies biofilm development in three dimensions, Water Sci. Tech., 39, 7, 123-130 (1999) |
| [101] | O’Toole, G.; Kaplan, H. B.; Kolter, R., Biofilm formation as microbial development, Annu. Rev. Microbiol., 54, 1, 49-79 (2000) |
| [102] | Pan, J.; Sun, K.; Liang, Y.; Sun, P.; Yang, X.; Wang, J.; Zhang, J.; Zhu, W.; Fang, J.; Becker, K. H., Cold plasma therapy of a tooth root canal infected with Enterococcus faecalis biofilms in vitro, J. Endod., 39, 1, 105-110 (2013) |
| [103] | Patankar, S. V.; Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat. Mass. Trans., 15, 10, 1787-1806 (1972) · Zbl 0246.76080 |
| [104] | Percival, S. L.; Knapp, J. S.; Wales, D. S.; Edyvean, R. G., The effect of turbulent flow and surface roughness on biofilm formation in drinking water, J. Ind. Microbiol. Biotech., 22, 3, 152-159 (1999) |
| [105] | Picioreanu, C.; Head, I. M.; Katuri, K. P.; van Loosdrecht, M. C.M.; Scott, K., A computational model for biofilm-based microbial fuel cells, Water Res., 41, 13, 2921-2940 (2007) |
| [106] | Picioreanu, C.; Kreft, J. U.; van Loosdrecht, M. C.M., Particle-based multidimensional multispecies biofilm model, Appl. Environ. Microbiol., 70, 5, 3024-3040 (2004) |
| [107] | Picioreanu, C.; van Loosdrecht, M. C.M.; Heijnen, J. J., A new combined differential-discrete cellular automaton approach for biofilm modeling: application for growth in gel beads, Biotechnol. Bioeng., 57, 6, 718-731 (1998) |
| [108] | Picioreanu, C.; van Loosdrecht, M. C.M.; Heijnen, J. J., Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, Biotechnol. Bioeng., 58, 1, 101-116 (1998) |
| [109] | Picioreanu, C.; van Loosdrecht, M. C.M.; Heijnen, J. J., Two-dimensional model of biofilm detachment caused by internal stress from liquid flow, Biotechnol. Bioeng., 72, 2, 205-218 (2001) |
| [110] | Picioreanu, C.; Xavier, J. B.; van Loosdrecht, M. C.M., Advances in mathematical modeling of biofilm structure, Biofilms, 1, 4, 337-349 (2004) |
| [111] | Pirt, S. J., The energetics of microbes at slow growth rates: maintenance energies and dormant organisms, J. Ferment. Bioeng., 65, 2, 173-177 (1987) |
| [112] | Ramli, N. S.; Eng Guan, C.; Nathan, S.; Vadivelu, J., The effect of environmental conditions on biofilm formation of Burkholderia pseudomallei clinical isolates, PLoS One, 7, 9, e44104 (2012) |
| [113] | Rasmussen, B., Filamentous microfossils in a 3235-million-year-old volcanogenic massive sulphide deposit, Nature, 405, 6787, 676-679 (2000) |
| [114] | Reese, S.; Govindjee, S., Theoretical and numerical aspects in the thermo-viscoelastic material behaviour of rubber-like polymers, Mech. Time-Depend. Mater., 1, 4, 357-396 (1998) |
| [115] | Ricucci, D.; Siqueira, J. F., Biofilms and apical periodontitis: study of prevalence and association with clinical and histopathologic findings, J. Endod., 36, 8, 1277-1288 (2010) |
| [116] | Rittmann, B. E.; McCarthy, P. L., Model of steady-state-biofilm kinetics, Biotechnol. Bioeng., 22, 11, 2343-2357 (1980) |
| [117] | Rodriguez, E. K.; Hoger, A.; McCulloch, A. D., Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27, 4, 455-467 (1994) |
| [118] | Rupp, C. J.; Fux, C. A.; Stoodley, P., Viscoelasticity of Staphylococcus aureus biofilms in response to fluid shear allows resistance to detachment and facilitates rolling migration, Appl. Environ. Microbiol., 71, 4, 2175-2178 (2005) |
| [119] | Sand, W.; Gehrke, T., Extracellular polymeric substances mediate bioleaching/biocorrosion via interfacial processes involving iron(III) ions and acidophilic bacteria, Res. Microbiol., 157, 1, 49-56 (2006) |
| [120] | Sauer, K.; Camper, A. K.; Ehrlich, G. D.; Costerton, J. W.; Davies, D. G., Pseudomonas aeruginosa displays multiple phenotypes during development as a biofilm, J. Bacteriol., 184, 4, 1140-1154 (2002) |
| [121] | Schiff, J. L., Cellular Automata: A Discrete View of the World (2008), John Wiley & Sons: John Wiley & Sons New York · Zbl 1142.68052 |
| [122] | Schindler, J.; Rataj, T., Fractal geometry and growth models of a Bacillus subtilis colony, Binary Comput. Microbiol., 4, 1, 66-72 (1992) |
| [123] | Schindler, J.; Rovensky, L., A model of intrinsic growth of a Bacillus colony, Binary Comput. Microbiol., 6, 3, 105-108 (1994) |
| [124] | Schopf, J. W., Microfossils of the early archean apex chert: new evidence of the antiquity of life, Science, 260, 5108, 640-646 (1993) |
| [125] | Shaw, T.; Winston, M.; Rupp, C. J.; Klapper, I.; Stoodley, P., Commonality of elastic relaxation times in biofilms, Phys. Rev. Lett., 93, 9, 0981021-0981024 (2004) |
| [126] | Sidoroff, F., Nonlinear viscoelastic model with an intermediate configuration: Un modèle viscoélastique non linéaire avec configuration intermédiaire, J. Mech., 13, 4, 679-713 (1974) · Zbl 0321.73029 |
| [127] | Simo, J. C.; Ortiz, M., A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comput. Methods Appl. Math., 49, 2, 221-245 (1985) · Zbl 0566.73035 |
| [128] | Siqueira, J. F.; Rôças, I. N.; Ricucci, D., Biofilms in endodontic infection, Endod. Topics, 22, 1, 33-49 (2012) |
| [129] | Stoodley, P.; Cargo, R.; Rupp, C. J.; Wilson, S.; Klapper, I., Biofilm material properties as related to shear-induced deformation and detachment phenomena, J. Ind. Microbiol. Biotechnol., 29, 6, 361-367 (2002) |
| [130] | Stoodley, P.; Lewandowski, Z.; Boyle, J. D.; Lappin-Scott, H. M., Structural deformation of bacterial biofilms caused by short-term fluctuations in fluid shear: An in situ investigation of biofilm rheology, Biotechnol. Bioeng., 65, 1, 83-92 (1999) |
| [131] | Stoodley, P.; Sauer, K.; Davies, D. G.; Costerton, J. W., Biofilms as complex differentiated communities, Annu. Rev. Microbiol., 56, 1, 187-209 (2002) |
| [132] | Sutherland, I. W., Biofilm exopolysaccharides: a strong and sticky framework, Microbiology, 147, 1, 3-9 (2001) |
| [133] | Telgmann, U.; Horn, H.; Morgenroth, E., Influence of growth history on sloughing and erosion from biofilms, Water Res., 38, 17, 3671-3684 (2004) |
| [134] | Toffoli, T.; Marqolus, N., Cellular Automata Machines: A New Environment for Modeling (1987), MIT Press: MIT Press Cambridge, USA |
| [135] | Towler, B. W.; Rupp, C. J.; Cunningham, A. B.L.; Stoodley, P., Viscoelastic properties of a mixed culture biofilm from rheometer creep analysis, Biofouling, 19, 5, 279-285 (2003) |
| [136] | Vilain, S.; Pretorius, J. M.; Theron, J.; Brözel, V. S., DNA as an adhesin: Bacillus cereus requires extracellular DNA to form biofilms, Appl. Environ. Microbiol., 75, 9, 2861-2868 (2009) |
| [137] | Vinogradov, A. M.; Winston, M.; Rupp, C. J.; Stoodley, P., Rheology of biofilms formed from the dental plaque pathogen Streptococcus mutans, Biofilms, 1, 01, 49-56 (2004) |
| [138] | Wang, Q.; Zhang, T., Review of mathematical models for biofilms, Solid State Commun., 150, 21-22, 1009-1022 (2010) |
| [139] | Wanner, O.; Gujer, W., A multispecies biofilm model, Biotechnol. Bioeng., 28, 3, 314-328 (1986) |
| [140] | Wanner, O.; Reichert, P., Mathematical modeling of mixed-culture biofilms, Biotechnol. Bioeng., 49, 2, 172-184 (1996) |
| [141] | Watnick, P.; Kolter, R., Biofilm, city of microbes, J. Bacteriol., 182, 10, 2675-2679 (2000) |
| [142] | Westall, F.; de Wit, M. J.; Dann, J.; van der Gaast, S.; de Ronde, C. E.; Gerneke, D., Early Archean fossil bacteria and biofilms in hydrothermally-influenced sediments from the Barberton greenstone belt, South Africa, Precambrian Res., 106, 1-2, 93-116 (2001) |
| [143] | Wilson, M., Bacterial biofilms and human disease, Sci. Prog., 84, 3, 235-254 (2001) |
| [144] | Wimpenny, J. W.T., Community structure and co-operation in biofilms, (Allison, D. G.; Gilbert, P.; Lappin-Scott, H. M.; Wilson, M., Community Structure and Co-operation in Biofilms (2000), Cambridge University Press: Cambridge University Press Cambridge, UK, New York), 1-24 |
| [145] | Wimpenny, J. W.T.; Colasanti, R. L., A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models, FEMS Microbiol. Ecol., 22, 1, 1-16 (1997) |
| [146] | Wingender, J.; Neu, T. R.; Flemming, H. C., What are bacterial extracellular polymeric substances?, (Wingender, J.; Neu, T. R.; Flemming, H. C., Microbial Extracellular Polymeric Substances (1999), Springer: Springer Berlin, Heidelberg), 1-19 |
| [147] | Winstanley, H. F.; Chapwanya, M.; McGuinness, M. J.; Fowler, A. C., A polymer-solvent model of biofilm growth, Proc. R. Soc. A Math. Phys., 467, 2129, 1449-1467 (2011) · Zbl 1219.35338 |
| [148] | Witten, T. A.; Sander, L. M., Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. E, 47, 19, 1400-1403 (1981) |
| [149] | Wloka, M.; Rehage, H.; Flemming, H. C.; Wingender, J., Structure and rheological behaviour of the extracellular polymeric substance network of mucoid Pseudomonas aeruginosa biofilms, Biofilms, 2, 4, 275-283 (2005) |
| [150] | Wolfram, S., Cellular Automata and Complexity: Collected Papers (1994), Westview Press: Westview Press Boulder, USA · Zbl 0823.68003 |
| [151] | Xavier, J. B.; Picioreanu, C.; Rani, S. A.; van Loosdrecht, M. C.M.; Stewart, P. S., Biofilm-control strategies based on enzymic disruption of the extracellular polymeric substance matrix - A modelling study, Microbiology, 151, 12, 3817-3832 (2005) |
| [152] | Xavier, J. B.; Picioreanu, C.; van Loosdrecht, M. C.M., Assessment of three-dimensional biofilm models through direct comparison with confocal microscopy imaging, Water Sci. Technol., 49, 11-12, 177-185 (2004) |
| [153] | Xavier, J. B.; Picioreanu, C.; van Loosdrecht, M. C., A framework for multidimensional modelling of activity and structure of multispecies biofilms, Environ. Microbiol., 7, 8, 1085-1103 (2005) |
| [154] | Xavier, J. B.; Picioreanu, C.; van Loosdrecht, M. C.M., A general description of detachment for multidimensional modelling of biofilms, Biotechnol. Bioeng., 91, 6, 651-669 (2005) |
| [155] | Yürüsoy, M.; Yilbaş, B. S.; Pakdemirli, M., Non-Newtonian fluid flow in annular pipes and entropy generation: temperature-dependent viscosity, Sadhana, 31, 6, 683-695 (2006) · Zbl 1149.76003 |
| [156] | Zhang, T.; Cogan, N. G.; Wang, Q., Phase-field models for biofilms. I. Theory and one-dimensional simulations, SIAM J. Appl. Math., 69, 3, 641-669 (2008) · Zbl 1186.92015 |
| [157] | Zhang, T.; Cogan, N. G.; Wang, Q., Phase-field models for biofilms II. 2-D numerical simulations of biofilm-flow interaction, Commun. Comput. Phys., 4, 1, 72-101 (2008) · Zbl 1365.92024 |
| [158] | Zöllner, A. M.; Abilez, O. J.; Böl, M.; Kuhl, E., Stretching skeletal muscle: chronic muscle lengthening through sarcomerogenesis, PLoS One, 7, 10, e45661 (2012) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
