Simple model of attachment and detachment of pathogens in water distribution system biofilms Schrottenbaum, I.1 , Uber, J.1 , Ashbolt, N.2 , Murray, R.2 , Janke, R.2 , Szabo, J.2 , Boccelli, D. 1 1
Department of Civil and Environmental Engineering, University of Cincinnati,Cincinnati, OH, 45220; email:
[email protected],
[email protected]. 2 Research Scientist, U.S. Environmental Protection Agency, Cincinnati OH 45268 Abstract The influence of wall biofilms on the transport of pathogens through a water pipe is mathematically described using an attachment - detachment model. The process coefficients of attachment and detachment are estimated from experimental data. A biofilm annular reactor model is presented and model predictions are compared to experimentally derived bulk phase and biofilm-sorbed pathogen data. The resulting pair of parameters, through minimizing the weighted root mean square error, are used in a single pipe model, and the results are discussed.
INTRODUCTION Biofilms (BF) are ubiquitous and a common matrix for microbial growth and environmental persistence. While the majority of a biofilm is water, its other major constituents are microorganisms and various extracellular polymeric substances (carbohydrates, proteins, nucleic acids) interspersed with colloidal detritus (e.g. iron oxides) and water channels. Biofilms are beneficial in nature and in engineered systems where conversion of organic and inorganic substrates is desirable (i.e. wastewater treatment). However, in water distribution systems (WDS), microbial activity within the biofilm can deteriorate water quality in terms of aesthetics and public health (LeChevallier et al. (1987); Characklis and Marshall (1990)). Of particular concern for this paper is the potential role biofilm may play in modifying the temporal signature of human exposure to pathogens, and, as a result, potential disease risk. Pathogens introduced into the WDS may attach and accumulate in the biofilm matrix, then subsequently detach in higher densities either randomly or because of the changes in hydraulic conditions. Early biofilm models were limited to linking substrate flux to the mechanisms of mass transport and substrate utilization (Rittmann and McCarrty (1980); LaMotta (1976)).
1
These models were further extended to multispecies models for 1-D biofilms by Rittmann (1992) and Wanner and Gujer (1986). Subsequent modeling attempts were expanded in two directions, a discrete versus a continuum approach. Examples of discrete descriptions of a heterogeneous multidimensional biofilm morphology are grid based Cellular Automata (CA) models (e.g. Noguera et al. (2004); Pizarro et al. (2001); Hermanowicz (2001)) and individual particle based models (IbM) (e.g. Kreft et al. (2001); Kreft and Wimpenny (2001); Piciorenau et al. (2004); Xavier et al. (2005)). Continuum models describe transport phenomena assuming that concentration is expressed by a density field that follows conservation laws, (i.e.conservation of mass and momentum), resulting in a system of differential equations. Newer continuum models describing multispecies interactions often are based on the model developed by Wanner and Gujer (1986), representing dissolved components in a diffusion reaction equation. Eberl et al. (2001), for example, describes a system of advection diffusion equations for the liquid phase and diffusion reaction equations for the biofilm, while others have focused on redistribution mechanisms using viscous fluid models (Dockery and Klapper (2001) and Cogan and Keener (2004)). Due to computational demands, models of biofilm processes appropriate for WDS scale would require a continuum approach. Early models by Dukan et al. (1996) and Servais and Prvost (1994) describing biofilm growth processes, were modified by Munavalli and Kumar (2004) and Zhang et al. (2004) to reduce complexity and allow application to hydraulic networks. Whereas these and other WDS scale biofilm models have focused on biomass processes such as growth and accumulation on pipe surfaces, we are interested in the effect of established biofilm on pathogen transport in water distribution pipes. Experimental studies in biofilm reactors (e.g.Storey (2002)) and pilot scale systems (e.g.Quignon et al. (1997)) indicate that microorganisms transported in bulk flow can be incorporated into biofilm. Different surrogates have been investigated on different surfaces. Szabo (2006) for instance observed accumulation and persistence of Klebsiella pneumoniae (KP) on corroding biofilm coupons in annular reactors (AR). In this paper we will present a modeling approach that includes the influence of the biofilm interactions on the transport of pathogens. Measurements of KP concentrations in the bulk and on the coupons in the dechlorinated AR system from Szabo (2006) will be used for the estimation of attachment and detachment model coefficients. First, pathogen pipe wall interaction (attachment - detachment) will be presented. Then, a comparable AR model will be established to support the estimation of the model parameters (attachment and detachment coefficient). Finally the model and estimated parameters will be applied to a simple WDS network of a single pipe. MODEL DESCRIPTION The modeling domain is divided into three compartments: the bulk phase, the concentration boundary layer, and the biofilm itself (see Figure.1). The pathogen concentration in the bulk (Pb ), at the biofilm surface (Ps ), in the biofilm (Pf ) and in the detached
biofilm material (Pd ) are predicted within their compartment. Note that the bulk phase concentrations are stated per volume, the biofilm concentration has units per area. Model Assumptions The one dimensional model describes the change in pathogen concentration over time. Advectional transport is solved with a Lagrangian scheme within the WDS network application used to interpret the temporal patterns of pathogen exposure. Biofilm is assumed to be a homogenous fixed layer at steady state, covering the inner pipe wall entirely. The nonmotile pathogenic matter attaches to the biofilm via transport to the biofilm and adsorption to the biofilm surface. It is assumed that there are unlimited sorption sites, and particulates, once attached, are assumed to be uniformly distributed within the biofilm. The pathogens colonizing the biofilm do not grow and natural mortality and inactivation are not included. This is also true for the bulk phase. It is assumed that the detached biofilm with incorporated pathogens does not reattach. Attachment Process Description Attachment processes are described as a combination of particle transport from the bulk liquid to the wall and the adsorption of those particles to the surface. Convectional transfer is due to superposition of diffusion and advection, and occurs between a fluid in motion and a surface. If the concentration at the surface (Cs ) differs from that in the bulk (Cb ) (cross sectional mean) a concentration gradient is developed by establishing an equilibrium beFigure 1: Concentration profile tween adjacent layers of the bulk phase. Figure 1 shows the concentration profile for turbulent flow as a dashed line. Although the transition is gradual, the flux over the concentration boundary layer, perpendicular to the surface, is approximately linear as illustrated in Figure 1 with a blue line. The change of pathogen concentration in the bulk due to transport of matter to the wall in order to achieve equilibrium, yields: Vb
dPb = −hm A(Pb − Ps ) dt
(1)
where Vb is the bulk volume, A is the surface of the pipe and hm is the mass transfer coefficient. The boundary layer theory states that the transfer coefficient (hm = Sh Dd ) is proportional to the Sherwood number (Sh) and the diffusion coefficient (D) over the pipe diameter (d) derived after normalization of transfer equations. Chilton-Colburn’s anal-
ogy, which is most accurate in the turbulent region, relates the Sherwood number to the D’Arcy Weisbach friction factor (f ), the Reynolds number (Re) and the Schmidt number (Sc = Dν ), thus providing a tool to estimate Sh = f8 ReSc1/3 , and the mass transfer coefficient. Adsorption depends on particle surface properties and biofilm sorption sites. The rate of incorporation of particles that are situated close to the wall in the boundary layer into the biofilm is assumed to be first order. Therefore the change of wall pathogen concentration due to transport to the wall and adsorption to the biofilm is expressed as Vb
dPs = hm A(Pb − Ps ) − Af ka Ps dt
(2)
where Af is the biofilm surface area and ka is the attachment coefficient. Detachment Process Description The dominant detachment mechanism under the given hydrodynamic condition is assumed to be erosion, however, biofilm sloughing is the more important event. Continuous loss of biofilm particles will be a function of biofilm density and shear stress on the biofilm surface (τ ). The detachment rate expression (rd = kd Af τ Pf ) proposed in similar form by Bakke et al. (1990) depends on the concentration of pathogens within the biofilm. Thus, the pathogen concentration change in the biofilm, increasing with attachment of particles and decreasing due to detachment, can be written as: Af
dPf = ka Af Ps − kd Af τ Pf dt
(3)
where kd is the detachment coefficient. The concentration of detached pathogens is tracked separately from the bulk matter in order to ensure no reattachment in the model. The change in detached concentration is expressed through the detachment rate. Vb
dPd = kd τ Af Pf dt
(4)
Model Summary Equations (1), (2), (3) and (4) simplify to the system of differential equations of first order: dPb dt dPs dt dPf dt dPd dt
= − =
1 hm (Pb − Ps ) rh
1 Af hm (Pb − Ps ) − ka Ps rh Vb
= ka Ps − kd τ Pf =
Af kd τ Pf Vb
where we introduce the hydraulic radius (rh ) as the cross sectional area of flow per wetted perimeter. PARAMETER ESTIMATION Pathogen persistence experiments conducted by Szabo (2006) simulated wall shear stress in a pipe with a diameter d = 0.102 m and a flow velocity of v = 0.305 m/s, using an annular reactor (AR) with a motor speed of 100 rpm, a volume of one liter, and a residence time (θ) of two hours. Three iron foil coupons per slide were inserted in the AR and biofilm and corrosion were established through continuous contact with dechlorinated Cincinnati tap water for three to four weeks before KP was spiked into the system. Five different experiment with different initial KP concentrations and cultivating broth strength were performed in two reactor duplicates. Bulk and coupon concentrations were measured 30 minutes after inoculation and each day thereafter until KP was not detected in either phase. Annular Reactor Model for Parameter Estimation To allow estimation of detachment and attachment parameters, the pathogen transport model was adjusted to the geometry of an annular reactor. We assume that the shear stress at the AR wall imitates the shear stress at a pipe wall, and therefore that the velocity gradient over the boundary layer at the AR wall is equal to that at the pipe wall. Then the shear stress (τ ) and the mass transfer coefficient (hm ) describing the transport of particles towards the biofilm surface are calculated for pipe flow, and used in the AR model to estimate the attachment and detachment coefficient ka and kd , respectively. The bulk concentration (Pb ) in the AR case includes detached matter (Pd ). Therefore the AR model is given as, 1 AAR AAR dPb = − Pb − hm (Pb − Ps ) + kd τ Pf dt θ VAR VAR dPs AAR AAR = hm (Pb − Ps ) − ka Ps dt VAR VAR dPf = ka Ps − kd τ Pf dt where VAR is the AR volume, AAR is the AR biofilm surface area and θ is the mean residence time. Note this is a linear system as θ, hm , Af AR , VAR , kd , ka and τ are constant. With the characteristic quantities Pb∗ = Pb0 , Ps∗ = Pb0 , Pf∗ = rPb0 , t∗ = θ, where 1/r = AAR /VAR , the AR biofilm model is nondimensionalized by substituting Pb0 = Pb /Pb∗ , Ps0 = Ps /Ps∗ , Pf0 = Pf /Pf∗ and t0 = t/t∗ .
dPb0 = −Pb0 − α(Pb0 − Ps0 ) + βPf0 dt0 dPs0 = α(Pb0 − Ps0 ) − γPs0 dt0 dPf0 = γPs0 − βPf0 dt0 With the dimensionless parameter groups α = θhrm , β = kd τ θ, and γ = kar θ . α, β and γ can be regarded as the dimensionless mass transfer, detachment and attachment coefficients, respectively. By nondimensionalizing the AR experimental data in the same way we have previously, a consistent basis is created for combining field data to model predictions for estimation of β and γ (note that α is calculated). This nondimensionalization also imposes the natural scaling of the problem with benefits for numerical computation. Estimation of Attachment and Detachment Coefficients The AR measurements of bulk and biofilm KP concentrations by Szabo (2006) are used to estimate the attachment and detachment coefficients γ and β for the non-dimensional AR biofilm model. A grid search is implemented over a plausible parameter space (γ, β) and the minimum weighted root mean squared error is r X X 1 0 2 0 2 b b δ× (Pb − Pb ) + (1 − δ) × ( Pf − Pf ) (5) J= N where δ = 0.4 was selected using judgment, N is the number of measurements, Pb0 and Pf0 are the nondimensional concentrations predicted by the model and Pbb and Pbf are the nondimensionalized measured bulk and coupon biofilm concentrations of KP measures from the five experiments. RESULTS AND CONCLUSION Parameter Estimation Calculated bulk and biofilm KP concentrations from the nondimensional AR model are compared to the measured data over the parameter space (γ, β). The response surface (J), defined by Eq. 5, is presented in Figure 2, using a 10 × 10 grid. The minimum of J occurs at the grid point (γ, β) = (32080, 1.32) where J = 0.86694. The parameter pair at this minimum is assumed to best represent the experimental data. The results clearly show, however, that the estimate of γ is highly uncertain due to the experimental conditions and sampling intervals used, which were not designed for model parameter estimation.
Figure 2: Response surface (left) and contours (right) of the weighted root mean squared error over the parameter space The complete set of parameters for the AR model application is listed in Table 1. The nondimensional mass transfer coefficient used in the nondimesional AR model is determined to be α = 7.1715. The estimated coefficients γ and β in dimensional form give, = 0.022057, and the detachment coefficient for the attachment coefficient ka = γr θ β −7 kd = τ θ = 6.6179 × 10 . The AR model with nondimensional model coefficients computes bulk and biofilm KP concentrations over time as presented in Figure 3 and Figure 4 respectively (black line). The two figures also show the nondimensional ex-
Figure 3: Dimensionless logarithmic KP concentration in the AR bulk phase versus dimensionless log time (where (log(t) = log(θ)). The nondimensional experimental data are shown for duplicated reactors and five different experiments (the legend indicate the initial concentration of each experiment). Concentration over time estimated by the AR model is indicated by the solid black line. perimental data for five different experiments (each conducted in duplicate (R1 and R2)) with varying initial KP concentration. The model prediction underestimates the experimental biofilm concentrations. This results from mass transfer limitations, as the
larger attachment coefficient γ does not decrease the error . This phenomena could be due to a variety of reasons. The calculated mass transfer coefficient could be too small, due to the friction factor (f ) calculation using the Blasius formula for pipe flow. The mass transfer coefficient also depends on the diffusion coefficient (D), and the diffusion coefficient of KP could be too small (it is assumed equal to the value for Escherichia coli reported by Berg (1993)). Finally, the assumption of a similarity between the concentration gradient adjacent to the wall in pipe flow, and in an annular reactor could be inaccurate. Experiments, using smaller sampling steps, would be more suitable for attachment investigation and varying rotational AR speed, would help to understand the influence of the shear stress induced detachment.
Figure 4: Dimensionless logarithmic KP concentration in the AR biofilm phase versus dimensionless log time (where log(t) = log(θ)). The nondimensional experimental data are shown for duplicated reactors and five different experiments (the legend indicate the initial concentration of each experiment). Concentration over time estimated by the AR model is indicated by the solid black line.
WDS System Application of the Pathogen Transport Model The set of parameters (ka , kd ) = (0.022057, 6.6179 × 10−7 ) estimated by minimizing the weighted root mean squared error over an appropriate parameter space were applied to the attachment-detachment model for a pipe. The predictions are presented for a single pipe advection model application. A single pipe of 21km length with a residence time of 20 hours for given flow (v) and pipe diameter (d) describes the hydraulics for the model. The model was executed with a initial pathogen concentration (Pb0 = 4.8 × 1010 M P N/m3 ) introduced in the system for the initial 20 hours, and run for a simulation time of 200 hours. The resulting pathogen concentration pattern for
bulk, surface, biofilm and detached material can be observed in Figure 5. The bulk concentration and the concentration of detached material at the end of the pipe are shown in Figure 6. The results predict a fast washout of the system, so the system is nearly cleaned out after twice the residence time. This behavior is likely a result of the discussed shortcoming of the parameter prediction. The model is under-predicting biofilm concentration in the AR, thus the washout rate would also be higher than expected.
Figure 5: Average concentration for bulk (upper left), the surface (upper right), in the biofilm (lower left) and detached material (lower right) over time.
Conclusion A simple linear pathogen attachment-detachment model was developed for applications to pipe flow, where advection would be incorporated by a Lagrangian scheme. The unknown coefficients of attachment and detachment are estimated with a least square search over the plausible parameter space using experimental data collected in annular reactors. The results suggest limitations from mass transfer assumptions, which require further investigation. Nevertheless, a first application of the transport model is presented, for a hydraulic pipe model with the set of fitted parameters. This results indicate that future work should be directed towards a more sound prediction of the unknown model parameters. The form of the mass transfer model should also be investigated. In addition to improvements in pathogen measurement frequency, investigation of the
influence of the AR mass transfer and the impact of the shear stress on detachment will improve the pathogen transport model predictions.
Figure 6: Logarithmic concentration at the effluent of the pipe of bulk material (left) and detached material (right) over time.
Acknowledgements The U.S. Environmental Protection Agency through the Office of Research and Development funded, managed, and participated in the research described here under a contract with the University of Cincinnati. The views expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the USEPA. Mention of trade names or commercial products does not constitute endorsement or recommendation for use. Work at the University of Cincinnati was sponsored by the USEPA under the Pegasus Contract, Contract Number EP-C-05-056, Work Assignment WA 3-06.
α β γ kd ka D v d θ 1/r rh Af f Re τ Sc Sh hm ν ρ
7.1715 1.32 32080 6.6179 × 10−7 0.022057 4.000000 × 10−10 0.305 0.102 2 202 2.550000 × 10−2 351.8458 2.382382 × 10−2 31110 2.770264 × 10+2 2500 1.257385 × 10+3 4.930920 × 10−6 1.0 × 10−6 1.0 × 10+6
non-dimensional mass transfer coefficient non-dimensional detachment coefficient non-dimensional adsorption coefficient [ms/g] detachment coefficient [m/s] adsorption coefficient [m2 /s] diffusion coefficient [m/s] mean velocity [m] diameter [hr] mean residence time of AR [1/m] surface area per volume in the AR [m] hydraulic radius for fully filled smooth pipe [m2 ] biofilm surface area in pipe equals pipe surface area friction factor Reynolds number pipe [g/ms2 ] wall shear stress Schmidt number Sherwood number [m/s] mass transfer coefficient [m2 /s] kinematic viscosity of water [g/m3 ] density of water Table 1: Model Parameters
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