A multiscale theoretical model for diffusive mass transfer in cellular biological media

Mathematical Biosciences

Volumn 210, Issue 1, 2007, Pages 177-237

  Kapellos, George E ^a   Alexiou, Terpsichori S ^a   Payatakes, Alkiviades C ^a  

^a UNIVERSITY OF PATRAS   (Greece)

Author keywords

Biofilm; Diffusion coefficient; Effective medium theory; Hierarchical multiscale modeling; Local spatial averaging; Tissue

Indexed keywords

BIOFILMS; CYTOLOGY; DIFFUSION; MATHEMATICAL MODELS; TISSUE;

EFFECTIVE MEDIUM THEORY; HIERARCHICAL MULTISCALE MODELING; LOCAL SPATIAL AVERAGING;

BIOLOGICAL SYSTEMS;

POLYMER;

BIOFILM; DIFFUSION; MAMMAL; MASS TRANSFER; MICROORGANISM; SOLUTE TRANSPORT; THEORETICAL STUDY;

ADSORPTION; ARTICLE; BIOFILM; CELL MEMBRANE; CULTURE MEDIUM; CYTOLOGY; DIFFUSION COEFFICIENT; HYDRODYNAMICS; MATHEMATICAL MODEL; MICROBIAL BIOMASS; MOLECULAR SIZE; PHYSICAL CHEMISTRY; POROSITY; THEORETICAL MODEL;

ANIMALS; BIOFILMS; BIOLOGICAL TRANSPORT, ACTIVE; BIOPOLYMERS; CELL MEMBRANE PERMEABILITY; DIFFUSION; EXTRACELLULAR FLUID; FLOCCULATION; HYDROGELS; MATHEMATICS; MICROBIOLOGY; MODELS, BIOLOGICAL;

MAMMALIA;

EID: 35248840117     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2007.04.008     Document Type: Article

References (134)

1. Adlercreutz P. Oxygen supply to immobilized cells: 5. Theoretical calculations and experimental data for the oxidation of glycerol by immobilized Gluconobacter oxydans cells with oxygen or p-benzoquinone as electron acceptor. Biotechnol. Bioeng. 28 (1986) 223
2. Agutter P.S., Malone P.C., and Wheatley D.N. Intra-cellular transport mechanisms: a critique of diffusion theory. J. Theor. Biol. 176 (1995) 261
3. Amsden B. Solute diffusion within hydrogels. Mechanisms and models. Macromolecules 31 (1998) 8382
4. Anderson T.B., and Jackson R. A fluid mechanical description of fluidized beds. Ind. Eng. Chem. Fundam. 6 4 (1967) 527
5. Aukland K., and Reed R.K. Interstitial-lymphatic mechanisms in the control of extra-cellular fluid volume. Physiol. Rev. 73 1 (1993) 1
6. Baveye P., and Sposito G. The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers. Water Resour. Res. 20 5 (1984) 521
7. Baveye P., and Sposito G. Macroscopic balance equations in soils and aquifers: the case of space- and time-dependent instrumental response. Water Resour. Res. 21 8 (1985) 1116
8. Beckie R. A comparison of methods to determine measurement support volumes. Water Resour. Res. 37 4 (2001) 925
9. Bentley T.B., and Pittman R.N. Influence of temperature on oxygen diffusion in hamster retractor muscle. Am. J. Physiol. Heart Circ. Physiol. 272 3 (1997) H1106
10. Berryman J.G. Comparison of upscaling methods in poroelasticity and its generalizations. J. Eng. Mech.-ASCE 131 9 (2005) 928
11. Betounes D.E. Nonclassical fields with singularities on a moving surface. J. Math. Phys. 23 12 (1982) 2304
12. Beyenal H., Şeker S., and Tanyolac A. Diffusion coefficients of phenol and oxygen in a biofilm of Pseudomonas putida. AIChE J. 43 1 (1997) 243
13. Bird R.B., Stewart W.E., and Lightfoot E.N. Transport Phenomena. second ed. (2002), Wiley
14. Blum J.J., Lawler G., Reed M., and Shin I. Effect of cytoskeletal geometry on intra-cellular diffusion. Biophys. J. 56 (1989) 995
15. Brenner H. Dispersion resulting from flow through spatially periodic porous media. Phil. Trans. R. Soc. Lond. A297 (1980) 81
16. Brenner H., and Adler P.M. Dispersion resulting from flow through spatially periodic porous media II. Surface and intraparticle transport. Phil. Trans. R. Soc. Lond. A307 (1982) 149
17. Brinkman H.C. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1 (1947) 27
18. Bruggeman D.A.G. Berechnung verschiedener physikalischer konstanten von heterogenen substanzen. I. Dielektrizitätskonstanten und leitfähigkeiten der mischkörper aus isotropen substanzen. Annalen der Physik 24 (1935) 636
19. Bryers J.D., and Drummond F. Local macromolecule diffusion coefficients in structurally non-uniform bacterial biofilms using fluorescence recovery after photobleaching (FRAP). Biotechnol. Bioeng. 60 4 (1998) 462
20. Cao H., Li X., Jiang B., Sun J., and Zhang Y. Evaluation of the influence of extra-cellular polymeric substances on the mass transport of substrate within multispecies biofilms. Chin. J. Chem. Eng. 12 4 (2004) 590
21. Chen K.C., and Nicholson C. Changes in brain cell shape create residual extra-cellular space volume and explain tortuosity behavior during osmotic challenge. PNAS USA 97 15 (2000) 8306
22. Choy T.C. Effective Medium Theory. Principles and Applications (1999), Oxford University Press, Great Britain
23. Cieslar J., Huang M.-T., and Dobson G.P. Tissue spaces in rat heart, liver, and skeletal muscle in vivo. Am. J. Physiol. Regul. Integr. Comp. Physiol. 275 (1998) R1530
24. Costerton J.W., Lewandowski Z., de Beer D., Caldwell D., Korber D., and James G. Biofilms: the customized microniche. J. Bacteriol. 176 8 (1994) 2137
25. Crapiste G.H., Rotstein E., and Whitaker S. A general closure scheme for the method of volume averaging. Chem. Eng. Sci. 41 2 (1986) 227
26. Cushman J.H. On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory. Water Resour. Res. 20 11 (1984) 1668
27. Cushman J.H. Stochastic filtering of multiphase transport phenomena. Transport Porous Media 2 (1987) 425
28. Deen W.M. Analysis of Transport Phenomena (1998), Oxford University Press, New York
29. Donlan R.M. Biofilms: microbial life on surfaces. Emerg. Infect. Dis. 8 9 (2002) 881
30. Dutta A., and Popel A.S. A theoretical analysis of intra-cellular oxygen diffusion. J. Theor. Biol. 176 (1995) 433
31. El-Kareh A.W., Braunstein S.L., and Secomb T.W. Effect of cell arrangement and interstitial volume fraction on the diffusivity of monoclonal antibodies in tissue. Biophys. J. 64 (1993) 1638
32. Ellis T.G., Eliosov B., Schmit C.G., Jahan K., and Park K.Y. Activated sludge and other aerobic suspended culture processes. Water Environ. Res. 74 4 (2002) 385
33. Estrada R., and Kanwal R.P. Applications of distributional derivatives to wave propagation. J. Inst. Math. Appl. 26 (1980) 19
34. Estrada R., and Kanwal R.P. Distributional analysis for discontinuous fields. J. Math. Anal. Appl. 105 (1985) 478
35. Evans N.T.S., Naylor P.F.D., and Quinton T.H. The diffusion coefficient of oxygen in respiring kidney and tumor tissue. Respir. Physiol. 43 (1981) 179
36. Fan L.-S., Leyva-Ramos R., Wisecarver K.D., and Zehner B.J. Diffusion of phenol through a biofilm grown on activated carbon particles in a draft-tube three-phase fluidized-bed bioreactor. Biotechnol. Bioeng. 35 (1990) 279
37. Fatin-Rouge N., Milon A., Buffle J., Goulet R.R., and Tessier A. Diffusion and partitioning of solutes in agarose hydrogels: the relative influence of electrostatic and specific interactions. J. Phys. Chem. B 107 (2003) 12126
38. Franzone P.C., Pavarino L.F., and Taccardi B. Simulating patterns of excitation, repolarization and action potential duration with cardiac Bidomain and Monodomain models. Math. Biosci. 197 (2005) 35
39. Grathwohl P. Diffusion in Natural Porous Media: Contaminant Transport, Sorption-Desorption and Dissolution Kinetics (1998), Kluwer Academic Publishers, USA
40. Gray W.G. A derivation of the equations for multi-phase transport. Chem. Eng. Sci. 30 (1975) 229
41. Gray W.G., and Miller C.T. Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview. Adv. Water Resour. 28 (2005) 161
42. Grote J. Die sauerstoffdiffusionskonstanten im lungengewebe und wasser und ihre temperaturabhängigkeit. Pflügers Archiv. 295 (1967) 245
43. Grote J., Süsskind R., and Vaupel P. Oxygen diffusivity in tumor tissue (DS-carcinosarcoma) under temperature conditions within the range of 20-40 °C. Pflügers Archiv. 372 (1977) 37
44. Grote J., and Thews G. Die bedingungen für die sauerstoffversorgung des herzmuskelgewebes. Pflügers Archiv. 276 (1962) 142
45. Hassanizadeh S.M. Derivation of basic equations of mass transport in porous media, Part 1. Macroscopic balance laws. Adv. Water Resour. 9 (1986) 196
46. Hassanizadeh S.M. Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws. Adv. Water Resour. 9 (1986) 207
47. Hershey D., and Karhan T. Diffusion coefficients for oxygen transport in whole blood. AIChE J. 14 6 (1968) 969
48. Hibiya K., Nagai J., Tsuneda S., and Hirata A. Simple prediction of oxygen penetration depth in biofilms for wastewater treatment. Biochem. Eng. J. 19 (2004) 61
49. Hinson R.K., and Kocher W.M. Model for effective diffusivities in aerobic biofilms. J. Environ. Eng.-ASCE 122 11 (1996) 1023
50. Ho C.S., and Ju L.-K. Effects of microorganisms on effective oxygen diffusion coefficients and solubilities in fermentation media. Biotechnol. Bioeng. 32 (1988) 313
51. Hoffman L.R., D'Argenio D.A., MacCoss M.J., Zhang Z., Jones R.A., and Miller S.I. Amino-glycoside antibiotics induce bacterial biofilm formation. Nature 436 (2005) 1171
52. Homer L.D., Shelton J.B., and Williams T.J. Diffusion of oxygen in slices of rat brain. Am. J. Physiol. Regul. Integr. Comp. Physiol. 244 (1983) R15
53. Horn H., and Morgenroth E. Transport of oxygen, sodium chloride, and sodium nitrate in biofilms. Chem. Eng. Sci. 61 (2006) 1347
54. Hunter R.C., and Beveridge T.J. High-resolution visualization of Pseudomonas aeruginosa PAO1 biofilms by freeze-substitution transmission electron microscopy. J. Bacteriol. 187 22 (2005) 7619
55. Hwang S.N., Chin C.-L., Wehrli F.W., and Hackney D.B. An image-based finite difference model for simulating restricted diffusion. Magnet. Reson. Med. 50 (2003) 373
56. Jackson T.L., and Byrne H.M. A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math. Biosci. 164 (2000) 17
57. Jacobson K., Sheets E.D., and Simson R. Revisiting the fluid mosaic model of membranes. Science 268 (1995) 1441
58. Jain R.K. Transport of molecules in the tumor interstitium - a review. Cancer Res. 47 12 (1987) 3039
59. Jang S.H., Wientjes M.G., Lu D., and Au J.L.-S. Drug delivery and transport in solid tumors. Pharm. Res. 20 9 (2003) 1337
60. Jones D.P., and Kennedy F.G. Analysis of intra-cellular oxygenation of isolated adult cardiac myocytes. Am. J. Physiol. Cell Physiol. 250 3 (1986) C384
61. Kanwal R.P. Generalized Functions: Theory and Applications. third ed. (2004), Birkhäuser, Boston
62. G.E. Kapellos, T.S. Alexiou, S. Pavlou, A.C. Payatakes, Hierarchical modeling approach for the prediction of effective hydraulic permeability and diffusion coefficient in biofilms, in: Proceedings of the International Conference: "Biofilms 2004: Biofilm Structure and Activity", Las Vegas, NV, USA, 2004, pp. 255-260.
63. Kapellos G.E., Alexiou T.S., and Payatakes A.C. Hierarchical simulator of biofilm growth and dynamics in granular porous materials. Adv Water Resour. 30 (2007) 1648
64. Karel S.F., Libicki S.B., and Robertson C.R. The immobilization of whole cells: engineering principles. Chem. Eng. Sci. 40 8 (1985) 1321
65. Kawashiro T., Nüsse W., and Scheid P. Determination of diffusivity of oxygen and carbon dioxide in respiring tissue: results in rat skeletal muscle. Pflügers Archiv. 359 (1975) 231
66. Kim Y.R., Savellano M.D., Savellano D.H., Weissleder R., and Bogdanov A. Measurement of tumor interstitial volume fraction: method and implication for drug delivery. Magnet. Reson. Med. 52 (2004) 485
67. Kurosawa H., Matsumura M., and Tanaka H. Oxygen diffusivity in gel beads containing viable cells. Biotechnol. Bioeng. 34 (1989) 926
68. Latour L.L., Svoboda K., Mitra P.P., and Sotak C.H. Time-dependent diffusion of water in a biological model system. PNAS USA 91 (1994) 1229
69. Lawrence J.R., Swerhone G.D.W., Leppard G.G., Araki T., Zhang X., West M.M., and Hitchcock A.P. Scanning transmission X-ray, laser scanning, and transmission electron microscopy mapping of the exopolymeric matrix of microbial biofilms. Appl. Environ. Microbiol. 69 9 (2003) 5543
70. Lawrence J.R., Wolfaardt G.M., and Korber D.R. Determination of diffusion coefficients in biofilms by confocal laser microscopy. Appl. Environ. Microbiol. 60 4 (1994) 1166
71. Levick J.R. Flow through the interstitium and other fibrous matrices. Q. J. Exp. Physiol. 72 (1987) 409
72. Libicki S.B., Salmon P.M., and Robertson C.R. The effective diffusive permeability of a nonreacting solute in microbial cell aggregates. Biotechnol. Bioeng. 32 (1988) 68
73. Ligeza A., Tikhonov A.N., Hyde J.S., and Subczynski W.K. Oxygen permeability of thylakoid membranes: electron paramagnetic resonance spin labeling study. BBA-Bioenergetics 1365 (1998) 453
74. Lipinski H.-G. Monte Carlo simulation of extra-cellular diffusion in brain tissues. Phys. Med. Biol. 35 3 (1990) 441
75. Mah T.-F.C., and O'Toole G.A. Mechanisms of biofilm resistance to antimicrobial agents. Trends Microbiol. 9 1 (2001) 34
76. Mahler M., Louy C., Homsher E., and Peskoff A. Reappraisal of diffusion, solubility, and consumption of oxygen in frog skeletal muscle, with applications to muscle energy balance. J. Gen. Physiol. 86 (1985) 105
77. Marcotte L., Therien-Aubin H., Sandt C., Barbeau J., and Lafleur M. Solute size effects on the diffusion in biofilms of Streptococcus mutans. Biofouling 20 4/5 (2004) 189
78. Marle C.M. Écoulements monophasiques en milieu poreux. Revue de l' Institut Français du Pétrole XXII 10 (1967) 1471
79. Marle C.M. On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. Int. J. Eng. Sci. 20 5 (1982) 643
80. Masaro L., and Zhu X.X. Physical models of diffusion for polymer solutions, gels and solids. Prog. Polym. Sci. 24 (1999) 731
81. Massol-Deyá A.A., Whallon J., Hickey R.F., and Tiedje J.M. Channel structures in aerobic biofilms of fixed-film reactors treating contaminated groundwater. Appl. Environ. Microbiol. 61 2 (1995) 769
82. Matheron G. Les variables régionalisées et léur estimation (1965), Masson et Cie, Paris
83. Maxwell J.C. A Treatise on Electricity and Magnetism vol. 1 (1954), Dover Publications
84. McCarty P.L., Reinhard M., and Rittmann B.E. Trace organics in groundwater. Environ. Sci. Technol. 15 1 (1981) 40
85. Muhr A.H., and Blanshard J.M.V. Diffusion in gels. Polymer 23 (1982) 1012
86. Neale G.H., and Nader W.K. Prediction of transport processes within porous media: diffusive flow processes within an homogeneous swarm of spherical particles. AIChE J. 19 1 (1973) 112
87. Neale G.H., and Nader W.K. Prediction of transport processes within porous media: creeping flow relative to a fixed swarm of spherical particles. AIChE J. 20 3 (1974) 530
88. Nicholson C. Diffusion and related transport mechanisms in brain tissue. Rep. Prog. Phys. 64 (2001) 815
89. Nicholson C., and Tao L. Hindered diffusion of high molecular weight compounds in brain extra-cellular microenvironment measured with integrative optical imaging. Biophys. J. 65 (1993) 2277
90. Ochoa-Tapia J.A., Stroeve P., and Whitaker S. Diffusive transport in two-phase media: spatially periodic models and Maxwell's theory for isotropic and anisotropic systems. Chem. Eng. Sci. 49 5 (1994) 709 Errata, Chem. Eng. Sci. 49 (13) (1994) 2205
91. Ogston A.G. The spaces in a uniform random suspension of fibres. Trans. Faraday Soc. 54 (1958) 1754
92. Parsek M.R., and Greenberg E.P. Sociomicrobiology: the connections between quorum sensing and biofilms. Trends Microbiol. 13 1 (2005) 27
93. Perry R.H., and Green D.W. Perry's Chemical Engineers' Handbook. seventh ed. (1997), McGraw-Hill
94. Phillips R.J. A hydrodynamic model for hindered diffusion of proteins and micelles in hydrogels. Biophys. J. 79 (2000) 3350
95. Popel A.S. Theory of oxygen transport to tissue. Crit. Rev. Biomed. Eng. 17 3 (1989) 257
96. Quintard M., and Whitaker S. Transport in ordered and disordered porous media. II: generalized volume averaging. Transport Porous Med. 14 (1994) 179
97. Reinoso R.F., Tefler B.A., and Rowland M. Tissue water content in rats measured by desiccation. J. Pharmacol. Toxicol. 38 (1997) 87
98. Riley M.R., Buettner H.M., Muzzio F.J., and Reyes S.C. Monte Carlo simulation of diffusion and reaction in two-dimensional cell structures. Biophys. J. 68 (1995) 1716
99. Riley M.R., Muzzio F.J., Buettner H.M., and Reyes S.C. Diffusion in heterogeneous media: application to immobilized cell systems. AIChE J. 41 3 (1995) 691
100. Riley M.R., Muzzio F.J., Buettner H.M., and Reyes S.C. A simple correlation for predicting effective diffusivities in immobilized cell systems. Biotechnol. Bioeng. 49 (1996) 223
101. Riley M.R., Muzzio F.J., and Reyes S.C. Experimental and modeling studies of diffusion in immobilized cell systems. A review of recent literature and patents. Appl. Biochem. Biotechnol. 80 2 (1999) 151
102. Roussel C.J., and Roussel M.R. Reaction-diffusion models of development with state-dependent chemical diffusion coefficients. Prog. Biophys. Mol Biol. 86 (2004) 113
103. Rumsey W.L., Schlosser C., Nuutinen E.M., Robiolio M., and Wilson D.F. Cellular energetics and the oxygen dependence of respiration in cardiac myocytes isolated from adult rat. J. Biol. Chem. 265 26 (1990) 15392
104. Saldeña T.A., Saraví F.D., Hwang H.-J., Cincunegui L.M., and Carra G.E. Oxygen diffusive barriers of rat distal colon: role of subepithelial tissue, mucosa, and mucus gel layer. Digest. Dis. Sci. 45 11 (2000) 2108
105. Sen P.N., and Basser P.J. A model for diffusion in white matter in the brain. Biophys. J. 89 5 (2005) 2927
106. Sen P.N., Scala C., and Cohen M.H. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics 46 5 (1981) 781
107. Sikavitsas V.I., Bancroft G.N., and Mikos A.G. Formation of three-dimensional cell/polymer constructs for bone tissue engineering in a spinner flask and a rotating wall vessel bioreactor. J. Biomed. Mater. Res. A 62 1 (2002) 136
108. Sokolnicki A.M., Fisher R.J., Harrah T.P., and Kaplan D.L. Permeability of bacterial cellulose membranes. J. Membr. Sci. 272 (2006) 15
109. Stein W.D. Transport and Diffusion Across Cell Membranes (1986), Academic Press, London
110. Stewart P.S., and Raquepas J.B. Implications of reaction-diffusion theory for the disinfection of microbial biofilms by reactive antimicrobial agents. Chem. Eng. Sci. 50 19 (1995) 3099
111. Stewart P.S. A review of experimental measurements of effective diffusive permeabilities and effective diffusion coefficients in biofilms. Biotechnol. Bioeng. 59 3 (1998) 261
112. Stoodley P., Yang S., Lappin-Scott H., and Lewandowski Z. Relationship between mass transfer coefficient and liquid flow velocity in heterogeneous biofilms using micro-electrodes and confocal microscopy. Biotechnol. Bioeng. 56 6 (1997) 681
113. Subczynski W.K., Hyde J.S., and Kusumi A. Oxygen permeability of phosphatidylcholine-cholesterol membranes. PNAS USA 86 (1989) 4474
114. Sun Y., Furusaki S., Yamauchi A., and Ichimura K. Diffusivity of oxygen into carriers entrapping whole cells. Biotechnol. Bioeng. 34 (1989) 55
115. Szafer A., Zhong J., and Gore J.C. Theoretical model for water diffusion in tissues. Magnet. Reson. Med. 33 (1995) 697
116. Tao L., and Nicholson C. Maximum geometrical hindrance to diffusion in brain extra-cellular space surrounding uniformly spaced convex cells. J. Theor. Biol. 229 (2004) 59
117. Tao A., Tao L., and Nicholson C. Cell cavities increase tortuosity in brain extra-cellular space. J. Theor. Biol. 234 (2005) 525
118. Thews G., and Niesel W. Zur theorie der sauerstoffdiffusion im erythrocyten. Pflügers Archiv. 268 (1959) 318
119. 2-löslichkeitskoeffizienten im gehirngewebe. Pflügers Archiv. 271 (1960) 227
120. Thurnheer T., Gmür R., Shapiro S., and Guggenheim B. Mass transport of macromolecules within an in vitro model of supragingival plaque. Appl. Environ. Microbiol. 69 3 (2003) 1702
121. Torquato S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties (2002), Springer-Verlag, New York
122. Tortora G.J., and Grabowski S.R. Principles of Anatomy and Physiology. 7th ed. (1993), HarperCollins College Publishers, New York
123. Turner M.G., Gardner R.H., and O'Neill R.V. Landscape Ecology in Theory and Practice: Pattern and Process (2001), Springer, New York
124. Vlachos D.G. A review of multiscale analysis: examples from systems biology, materials engineering, and other fluid-surface interacting systems. Adv. Chem. Eng. 30 (2005) 1
125. Westrin B.A., and Axelsson A. Diffusion in gels containing immobilized cells: a critical review. Biotechnol. Bioeng. 38 (1991) 439
126. Whitaker S. Multiphase transport phenomena: matching theory and experiment. In: Papanicolaou G. (Ed). Advances in Multiphase Flow and Related Problems (1986), SIAM, Philadelphia 273
127. Whitaker S. The Method of Volume Averaging (1999), Kluwer Academic Publishers, The Netherlands
128. Willaert R., and Baron G. Gel entrapment and micro-encapsulation: methods, applications and engineering principles. Rev. Chem. Eng. 12 (1996) 5
129. Wingender J., Neu T.R., and Flemming H.-C. What are bacterial extra-cellular polymeric substances?. In: Wingender J., Neu T.R., and Flemming H.-C. (Eds). Microbial extra-cellular Polymeric Substances: Characterization, Structure and Function (1999), Springer, Berlin 1
130. Wood B.D., Quintard M., and Whitaker S. Calculation of effective diffusivities for biofilms and tissues. Biotechnol. Bioeng. 77 5 (2002) 495
131. Wood B.D., and Whitaker S. Diffusion and reaction in biofilms. Chem. Eng. Sci. 53 3 (1998) 397 Erratum, Chem. Eng. Sci. 55 (2000) 2349
132. Wood B.D., and Whitaker S. Multi-species diffusion and reaction in biofilms and cellular media. Chem. Eng. Sci. 55 (2000) 3397
133. Xiang T.-X., and Anderson B.D. The relationship between permeant size and permeability in lipid bilayer membranes. J. Membr. Biol. 140 (1994) 111
134. Zhang T.C., and Bishop P.L. Density, porosity, and pore structure of biofilms. Water Res. 28 11 (1994) 2267