Cellular Growth in Biofilms Brian D. Wood, 1 Stephen Whitaker 2 1 Pacific Northwest National Laboratory, Richland, Washington 99352 2 Department of Chemical Engineering and Material Science, University of California at Davis, Davis, California, 95616 Received 12 November 1998, accepted 16 February 1999 Abstract: In this paper we develop a macroscopic evolu- tionary equation for the growth of the cellular phase starting from a microscopic description of mass trans- port and a simple structured model for product forma- tion. The methods of continuum mechanics and volume averaging are used to develop the macroscopic repre- sentation by carefully considering the fluxes of chemical species that pertain to cell growth. The concept of struc- tured models is extended to include the transport of re- acting chemical species at the microscopic scale. The re- sulting macroscopic growth model is similar in form to previously published models for the transport of a single substrate and electron donor and for the production of cellular mass and exopolymer. The method of volume averaging indicates under what conditions the devel- oped growth model is valid and provides an explicit con- nection between the relevant microscopic model param- eters and their corresponding macroscopic counter- parts. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 64: 656—670, 1999. Keywords: biofilms; cell growth; volume averaging; structured models INTRODUCTION The analysis and modeling of mass transport and reaction in biofilms has undergone substantial evolution from early re- search efforts (e.g., Aiba et al., 1968; Young and McCarty, 1969). Numerous studies exist in which conservation equa- tions for solutes and microbial biomass are developed, and recent examples include the work of Wanner and Gujer (1986), Hsieh et al., (1994), Wanner et al. (1995), and Stew- art et al. (1996). One characteristic of these studies is that the biofilm is treated as a continuum (Wanner and Gujer, 1986) rather than as a complex multiphase system. Wanner et al. (1995) have identified this simplification in the fol- lowing statement: The accumulation and activity of biofilms varies from point to point . . . and thus are considered to be microscale phenomena. Ideally, mechanistic modeling should likewise be carried out at the microscale in order to completely describe these processes. However [this] represents a major computational challenge, and the development and evaluation of such models is just beginning. Few studies thus far have examined transport and reactions in biofilms starting from a microscopic point of view. In this study, we use the word microscopic to mean a description given in terms of the point equations for the - and -phases illustrated in Fig. 1, and we use the word macro- scopic to refer to equations that are valid at the scale of the averaging volume, , also illustrated in Fig. 1. There are a small number of previous studies available that have con- sidered the influence of microscopic information on the de- scription of macroscopic biofilm behavior. Ochoa et al. (1986) used volume averaging to develop the macroscopic solute conservation equations in membrane-bound cellular films. They applied their work to solute transport in tissues (Ochoa et al., 1987) and obtained good agreement with experimental data. Libicki et al. (1988) suggested that bio- films should be treated as multiphase systems and postu- lated that volume averaging should be used to develop the macroscopic equations for solute transport in biofilms under the conditions of local mass equilibrium. However, the de- velopment of volume-averaged transport equations was not presented in their study. Recently, Wood and Whitaker (1998, 1999) have derived volume-averaged expressions for solute transport in biofilms under conditions of local mass equilibrium. In their work nonlinear intercellular reactions are considered, and transport across the cell membrane is represented by a simple carrier model. Although the result- ing macroscopic equations are of the form that is conven- tionally used for describing diffusion and reactions in bio- films, the process of volume averaging provides additional information illustrating how the effective (macroscopic) pa- rameters are defined in terms of the microscale parameters and the structure of the biofilm. In the studies described above, information about mass transport and reactions is treated at the microscopic scale. The use of structured models is one mechanism that has been employed to add microscopic detail to reaction pro- cesses in biofilms. For example, in the work of Hsieh et al. (1994) the multiphase nature of biofilms was accounted for by conceptualizing the biofilm to consist of a cellular com- ponent bound by the cell membrane and wall and an asso- ciated extracellular biopolymer that forms the second phase. Although structured models have been used in the descrip- tion of batch kinetics for some time (e.g., Ramkrishna et al., 1967; Domach et al., 1984; Steinmeyer and Shuler, 1989; Schaff et al., 1997), these models generally do not account Correspondence to: B. D. Wood © 1999 John Wiley & Sons, Inc. CCC 0006-3592/99/060656-15