Statistical Evaluation of Biochemical Kinetic Models for BTX

Article pubs.acs.org/IECR

Statistical Evaluation of Biochemical Kinetic Models for BTX Degradation Heveline Enzweiler,† Luiz Jardel Visioli,† Josiane Maria Muneron de Mello,‡,§ Selene Maria de Arruda Guelli Ulson de Souza,‡ Antônio Augusto Ulson de Souza,‡ Adriano da Silva,∥ Daniela Estelita Goes Trigueros,⊥ and Marcio Schwaab*,† †

Departamento de Engenharia Química, Universidade Federal de Santa Maria, Av. Roraima, 1000, Cidade Universitária, Santa Maria, RS 97105-900, Brazil ‡ Departamento de Engenharia Química e Engenharia de Alimentos, Universidade Federal de Santa Catarina, Caixa Postal 476, Florianópolis, SC 88040-900, Brazil § Programa de Pós-Graduaçaõ em Ciência Ambientais, Universidade Comunitária da Região de Chapecó, Av. Senador Atílio Fontana, 591-E, Caixa Postal 1141, Chapecó, SC 89809-000, Brazil ∥ Escola de Química e Alimentos, Universidade Federal do Rio Grande, Rua Barão do Caí, 125, Santo Antônio da Patrulha, RS 95.500-000, Brazil ⊥ Universidade Estadual do Oeste do Paraná, Campus de Toledo, Rua Da Faculdade 645, Jardim Santa Maria, Toledo, PR 85903-000, Brazil ABSTRACT: Microbial degradation of toxic compounds like benzene, toluene, and xylene in contaminated water is an important alternative for wastewater treatment processes since it is able to eliminate these toxic compounds even when they are present at low concentrations. Besides, when one is interested in the development of a large scale process, kinetic models of the substrate degradation rates are necessary in order to allow design and sizing of the reactors and whole process operation. During kinetic model development, the parameter estimation step is of fundamental importance, since it allows determination of unknown parameter values that will enable model use for design purposes. In this work, experimental data for benzene, toluene, and xylene biodegradation is presented and used for the estimation of parameter values of eight different biokinetic models. The Monod model was shown to be suitable for representation of the experimental data. Besides, statistical analysis of parameter estimation results, in particular with respect to the model parameter uncertainties, was performed using two different approaches. It was shown that the elliptical approach for the determination of confidence regions for parameter estimates leads to a poor evaluation of parameter confidence. On the other hand, the likelihood method preserves the shape of the confidence regions for parameter estimates and provides a more rigorous statistical analysis for the parameter estimation procedure. It is important to notice that this rigorous analysis is very helpful for researchers in selection of the most appropriate biokinetic model and can be directly extended to any filed of expertise.

1. INTRODUCTION Benzene, toluene, and xylene (BTX compounds) are very toxic substances present in petroleum and in contaminated soil. Consequently, they are present in the water originated from petroleum processing or in the water contacted with contaminated soil.1 It must be observed that these compounds present a low solubility in water, but even in small quantities they are very dangerous to the environment due to their toxicity and genotoxicity.2,3 The BTX compounds have high toxicity potential, representing a serious risk to environment and to human health. These compounds are listed in the Environmental Protection Agency (EPA) as chemical priorities due to their toxic properties.4,5 Since these compounds are present in aqueous effluents that are produced during petroleum processing, the use of some efficient and environmental friendly process for elimination of these toxic compounds is necessary, allowing for an appropriate discard of this effluent. Since these compounds are present in very low concentrations, traditional wastewater treatment processes are not suitable to eliminate the BTX compounds. In this scenario, microbial © 2014 American Chemical Society

degradation of benzene, toluene, and xylene must be considered, since it is a process able to remove even small amounts of these compounds.2,3 There are a lot of advantages of microbial degradation, since it can be used to remove contaminants of water, soil, and air. It is effective in the removal of the BTX compounds even in low concentrations, and the cost of this bioprocess is usually lower than the traditional chemical process.6 Microbial degradation processes are currently well developed, and one of the largest contributions to this evolution was the development of processes using biofilm supported on inert particulates.7 The method for microorganism growth in a biofilm presents several benefits, as protection against aggressive environmental factors, for example, fluctuations of pH, concentrations of salts and heavy metals, dehydration, tensional forces, aggressive Received: Revised: Accepted: Published: 19416

August 28, 2014 November 20, 2014 November 25, 2014 November 25, 2014 dx.doi.org/10.1021/ie503408g | Ind. Eng. Chem. Res. 2014, 53, 19416−19425

Industrial & Engineering Chemistry Research

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chemicals, bactericides, antibiotics, and predators.8−10 It also allows genetic material exchange due to long retention times of microorganisms and facilitates the development of a microconsortia that allows the establishment of symbiotic relationships, as well as the use of substrates with difficult degradation. Biofilm bioreactors have been proposed owing to some advantages,8−12 such as great biomass concentration and metabolic activity, small hydraulic retention time, the sludge does not need to be returned to the biological reactor, coexistence between aerobic and anaerobic metabolic activity, great reduction of biochemical oxygen demand (BOD), and a higher tolerance to toxic pollutants when compared to other conventional processes. To evaluate the BTX microbial degradation process and also allow design of a real process, the determination of biochemical kinetic models comprises a very important stage in the development of this technology, since it leads to an appropriate comprehension of the chemical phenomena and the developed model can be used for design, control, and optimization of biochemical reactors. In many chemical and biochemical reactions the number of kinetic models that can be used for data interpretation and representation is usually very high. In biochemical systems, the Monod model is the most reported model.3,13,14 However, in some cases the Monod model was not suitable to provide a good representation of the experimental data and other biochemical models were proposed, such as the Tessier model, Moser model, Contois model, modified Monod model, Andrews or Haldane model, Webb model, Edwards model, and Wu model.3,13−18 Besides, some authors also proposed the use of specific empirical models to particular problems. With this variety of biochemical models that can be used, researches must perform a proper statistical analysis in order to define the most appropriate biochemical model among proposed models. There are several methods proposed in the literature for discrimination among two or more kinetic models. Ahmad13 et al. propose the use of the determination coefficient R2 and model variance to compare four kinetic models of ethylic fermentation, and the best model was determined by a high R2 value and a low model variance value. On the other side, the objective function used in the parameter estimation procedure can be directly used for model comparison.14,19 Although a lower objective function value indicates a better model performance, it is also necessary to use a statistical test to ensure the significance of the parameter estimation results. It is important to notice that when an objective function based on the Maximum Likelihood Principle is used, the objective function value can be directly used for statistical evaluation of model performance. In particular, when the Normal probability function is used, the derived objective function consists of a χ2 (chi-square) probability distribution value and can be used directly in a χ2-test for adequacy or for model probability computation.20 When model discrimination cannot be achieved with the experimental data available, one can utilize experimental design techniques for model discrimination19,21,22 or one can select one model among the suitable ones, which is usually the simplest model that is able to provide a good representation of the experimental data. Model quality may also be evaluated through analysis of estimated parameter values and its uncertainties. An appropriate model must provide a good experimental data representation and also estimated parameter values with physical meaning and low uncertainties, what is usually done through analysis of estimated parameter confidence intervals based on a t-Student probability

distribution.19,23 However, as pointed out by Draper and Guttman,24 analysis of elliptic confidence regions of parameter pairs gives more information about the parameter uncertainties than analysis of individual confidence intervals of each parameter. Furthermore, elliptic confidence regions for parameter estimates cannot represent adequately the real shape of confidence regions when nonlinear models are considered, and the likelihood ratio method must be used in order to provide confidence regions with shapes close to the real ones.25,26 The objective of this paper is to present experimental data of BTX degradation and evaluate some biochemical kinetic models through statistical analysis. Moreover the significance and uncertainty of the estimated parameter values were evaluated through their confidence intervals and confidence regions, and determined through elliptic and nonelliptic procedures, providing a rigorous statistical evaluation of biochemical models and their estimated parameter values.

2. METHODOLOGY 2.1. Experimental Methodology. Initially, microorganisms were obtained from a sewage treatment plant from the city of Florianópolis-SC in Brazil and adapted to consume BTX compounds. Adaptation and immobilization onto a support of biomass was done in three separate vessels for benzene, toluene, and xylene compounds. The culture medium was composed of 150 mL of a mineral medium solution; 100 mL of activated sludge, with volatile suspended solids concentration of 85 g·L−1; 5 g of activated charcoal particles used as biofilm support; 150 mg·L−1 of glucose as carbon source; and 30 μL of hydrogen peroxide as oxygen source. The mineral medium solution (MMS) is composed by MgSO4·6H2O, 464 mg·L−1; K2HPO4, 500 mg·L−1; KH2PO4, 500 mg·L−1; (NH4)2SO4, 500 mg·L−1; CaCl2·2H2O, 9.8 mg·L−1; MnSO4·H2O, 10 mg·L−1; Fe(NH4)2(SO4)·6H2O, 8 mg·L−1; ZnSO4·7H2O, 2 mg·L−1; H3BO3, 1 mg·L−1; and CuSO4·5H2O, 0.5 mg·L−1. These minerals were weighted in an analytical balance (Shimadzu AW 220 model) and then dissolved in distillated water. Over a period of 20 days a solution containing MMS, hydrogen peroxide, and glucose was added daily to this mixture. This period was necessary to fix the microorganisms at the support. After this time, each of the BTX compounds were gradually added to the medium in three separate vessels, gradually replacing glucose as the carbon source until only the BTX compounds were added. This replacement period was about three months. After this replacement time, the liquid solution in each bioreactor was daily removed and replaced by a solution that contained MMS, hydrogen peroxide, and one of the BTX compounds. These vessels were maintained at 25 °C and stirred at 120 rpm in an incubator shaker (Logen Scientific, model LS 4500). The dissolved oxygen was monitored by digital oxymeter (WTW, model OXI 340i/SET). Kinetic experiments of BTX biodegradation were performed in Erlenmeyer flasks placed on an incubator shaker, at 25 °C and 150 rpm. The flasks were completely closed with a liquid thread seal and samples were collected along the reaction time with the help of a syringe, avoiding physical losses of solvents. To quantify eventual abiotic losses of BTX compounds, one bioreactor without microorganisms was used as a control reactor. This reactor had the same configuration as the bioreactors, but it was not inoculated with bioparticles. Abiotic loss of BTX in the control reactor was found to be negligible (a maximum value about 3% for higher concentrations). 19417

dx.doi.org/10.1021/ie503408g | Ind. Eng. Chem. Res. 2014, 53, 19416−19425


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For microbial degradation experiments, support with adhered biofilm was collected from the vessels used for adaptation and immobilization and the sludge that was not adhered to the charcoal was washed out. Then, 20 g of support with adhered biofilm were placed in each Erlenmeyer, also containing MMS, hydrogen peroxide, and one of the BTX compounds. The initial concentration of each BTX compound was defined as 20, 40, and 60 mg·L−1, but due to the experimental complexity of this system it was possible to fix these values exactly. BTX compounds concentration was measured through high performance liquid chromatography (HPLC) with plumb model CG 480-E, an ultraviolet−visible spectroscopy detector model CG 437-B, and a Nucleosil C18 column (Macherey-Nagel Model EC250/4.6). The eluent was methanol and Mili-Q water (80:20); the flow was maintained at 1.0 mL·min−1. The BTX compounds elution was detected at a wavelength of 254 nm. 2.2. Mathematical Methodology. The mathematical model of the process consists of a differential equation that describes the substrate concentration variation along the reaction time in a constant volume batch reactor with a mass balance for a bath reactor (eq 1).

dS = −μS X dt

Table 1. Kinetic Models Purposed for the BTX Degradation identification

(2)

Equation 2 is the differential equation that has been used in this process modeling. Different biokinetic models were considered for the apparent specific rate of substrate consumption, as presented in Table 1. These biokinetic models3,13−18 were selected in order to evaluate possible behavior that was reported before, that is, with and without inhibition by the substrate. Models 4 to 8 are proposed to represent biodegradation process with inhibition by a substrate. In the models presented in Table 1, μmax ′ is the apparent maximum specific rate of substrate consumption, KS describes the substrate overall affinity of the microorganism27 and KI is the inhibition constant. The parameters from the models presented in Table 1 were estimated assuming that experimental fluctuations follow the normal probability distribution and experimental measurements of dependent variables are uncorrelated.26 Furthermore, assuming that measurement errors in all experimental conditions are constant, the objective function that must be minimized is the least-squares function, presented in eq 3. NE NT

Fobj =

mod 2 ∑ ∑ (Siexp , j − Si , j ) i=1 j=1

(1) Monod

′ μS′ = μmax

S KS + S

(2) Moser

′ μS′ = μmax

Sn KS′ + Sn

(3) Tessier

′ [1 − exp(−S /KS)] μS′ = μmax

(4) Andrews or Haldane

′ μS′ = μmax

S KS + S + KIS2

(5) Wu

′ μS′ = μmax

1 1 + KS/S + KI′Sn

(6) modified Monod

′ μS′ = μmax

S (KS + S)(1 + KIS)

(7) Webb

′ μS′ = μmax

S(1 + KIS) KS + S + KIS2

(8) Eduards

′ μS′ = μmax

S exp(− KIS) KS + S

Si,jmod of the substrate concentration (dependent variable), measurement at time j in the experiment i. NE and NT are the number of experiments and the number of measurement in each experiment, respectively. Model predicted values of substrate concentration were obtained through the numerical solution of the differential model presented in eq 2, with each one of the biokinetic models presented in Table 1. Numerical integration was performed with the help of the DASSL routine.28 It is important to notice that the initial concentration of each BTX compound was not considered in the parameter estimation procedure, since these values are only badly known because of the experimental complexity of this system, and in order to avoid disturbances of the reaction system at the beginning of the process, the initial concentration was not experimentally measured. Consequently, the BTX concentrations at time zero are assumed to be unknown and are simultaneously estimated with kinetic parameters, avoiding the use of poor initial BTX concentration values. This procedure resembles a data reconciliation method, where unmeasured experimental values (in our case, initial concentrations) are estimated simultaneously with model parameters.29,30 For estimation of model parameters, an appropriate numerical method must be used for minimization of the objective function with respect to the p model parameters. In this work a hybrid optimization algorithm that combines a particle swarm optimization (PSO) method and a Gauss−Newton method was used. The optimization procedure begins with a PSO method,31 a stochastic optimization method that performs a large number of objective function evaluations, which ensures a higher probability to find the global minimum. The PSO method also does not require initial guesses for parameter values and computation of objective function derivatives is also not necessary. It performs a direct search using many objective function values evaluated at points distributed over all the search space, which must be provided to the PSO method. Besides, this large number of objective function evaluations can be readily used for the construction of nonelliptic confidence regions for parameter estimates.26 The PSO algorithm described in detail by Schwaab and co-workers26 was used in this work. The performance of the PSO algorithm and other stochastic optimization methods in parameter estimation of biochemical

(1)

where S is the concentration of substrate (benzene, toluene, or xylene), t is the reaction time, μS is the specific rate of substrate consumption, and X is the microorganism concentration. In the experimental setup used in this work, the microorganisms are immobilized inside porous charcoal, and due to spatial restrictions, microbial growth can be neglected and microorganism concentration can be considered constant. Since the mass of immobilized microorganisms used in all experiments are the same, eq 1 can be rewritten, leading to eq 2, where μS′ is the apparent specific rate of substrate consumption and is equal to μSX. dS = −μS′ dt

model equation

(3)

Equation 3 consists in the sum of the squared differences between experimental values Si,jexp and model predicted values 19418



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models was recently evaluated by Da Ros and co-workers,32 and PSO was considered one of the best algorithms. The optimization procedure is ended by a Gauss−Newton method,33 a local search method that uses the best point found by the PSO as a starting point, assuring that the final solution found is a minimum value. After minimization procedure, comparison among models is performed with respect to the minimum value of the objective function. Estimated parameter values were evaluated according to their physical meaning and their uncertainty. Confidence intervals of estimated parameter values were constructed using the t-Student distribution,23 according to eq 4, where θi is the parameter i and ̂ indicates that this is the estimated parameter value, t(1+α)/2 is the t-Student distribution value with a confidence n−p level equal to α and n − p degrees of freedom (n is equal to the total number of experimental measurements and p is the number of estimated parameters), and vii is the diagonal element of the covariance matrix of parameter estimates and corresponds to the variance of the ith parameter. θî − tn(1−+p α)/2 vii ≤ θi ≤ θî + tn(1−+p α)/2 vii

an equation where each side follows a F distribution with p and n − p degrees of freedom, as indicates in eq 9. ̂ p [Fobj(θ) − Fobj(θ)]/ ̂ n − p) F (θ)/( obj

=

̂ p [(θ − θ)̂ T V −θ 1(θ − θ)]/ ≡ Fp , n − p ̂ n − p) F (θ)/( obj

(9)

Considering the left side of eq 9, one can obtain eq 6, and considering the left side of eq 9, one can obtain eq 5. It was clearly demonstrated that both equations for the determination of the confidence regions are derived with same assumption. In fact, for linear models both eqs 5 and 6 lead to identical elliptical confidence regions. However, for nonlinear models, eq 6 does not constrain the shape of the confidence regions as elliptical ones and can provide confidence regions with a shape very close to that of the real ones.

3. RESULTS Degradation of BTX compounds were evaluated at three different initial concentrations, which were used simultaneously to estimate the parameter values of models presented in Table 1. In Figure 1, lines correspond to predictions of the Monod model for each of the batch experiments. The quality of the model adjustment to experimental data can be considered adequate,

(4)

Confidence regions for the estimated parameter values were constructed according to the elliptic and likelihood methods. The elliptic method is based on assuming that the estimated parameter values follow a normal distribution and consequently, an elliptical shape for the confidence region is obtained, which is constructed according to eq 5, where Vθ is the covariance matrix of parameter estimates and Fαp,n−p is the Fisher probability distribution value with p and n − p degrees of freedom and a confidence level equal to α. p (θ − θ)̂ T V −θ 1(θ − θ)̂ ≤ Fobj(θ)̂ F pα, n − p n−p (5) On the other side, nonelliptical confidence regions were constructed through the likelihood method25 with the use of the points evaluated by the PSO method during the minimization procedure that satisfies eq 6.26 ⎛ ⎞ p Fobj(θ) ≤ Fobj(θ)̂ ⎜1 + F pα, n − p⎟ n−p ⎝ ⎠

(6)

Equations 5 and 6 can be obtained through a quadratic approximation of the objective function through a second order truncated Taylor series around the optimum parameter estimated values, as shown in eq 7. ̂ T (θ − θ)̂ Fobj(θ) = Fobj(θ)̂ + [∇Fobj(θ)] +

1 ̂ θ − θ)̂ (θ − θ)̂ T H(θ)( 2

(7)

where ∇Fobj(θ̂) is the gradient vector of objective function and H(θ̂) is the Hessian Matrix that is the matrix of second derivatives of the objective function. Hessian matrix multiplied by one-half is equal to the inverse of the covariance matrix of 19,26 parameter estimates V−1 θ , as presented in the literature. Besides, since at minimum the gradient vector of the objective function must be null, eq 7 can be rewritten as Fobj(θ) − Fobj(θ)̂ = (θ − θ)̂ T V −θ 1(θ − θ)

(8)

Each side of eq 8 follows a chi-square distribution with p degrees of freedom, when the mathematical model is linear. Dividing each side by p, and dividing again by Fobj(θ̂)/(n − p), the result is

Figure 1. Experimental data and Monod fitted model for (a) benzene, (b) toluene, and (c) xylene substrates. 19419



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Table 2. Parameter Estimation Results of Benzene Biodegradationa Fobj

parameter

θ

θmin

θmax

standard deviation

(1) Monod

215.95

(2) Moser

213.56

(3) Tessier

217.86

(4) Andrews

216.62

(5) Wu

224.22

(6) modified Monod

215.95

(7) Webb

215.95

(8) Eduards

215.95

μmax ′ KS μ′max K′S n μ′max KS μ′max KS KI μ′max KS K′I n μ′max KS KI μ′max KS KI μ′max KS KI

7.18 20.81 15.85 27.91 0.68 5.24 17.32 7.22 21.15 0.00 14.93 58.04 0.00 2.10 7.18 20.81 0.00 7.17 20.78 0.00 7.18 20.81 0.00

3.81 1.21 −92.89 −125.46 −0.51 3.56 6.11 −7.53 −41.34 −0.04 −102.97 −454.53 −0.03 −20.56 −13.53 −58.39 −0.04 −89.4 × 103 −259 × 103 −600.26 −14.21 −60.97 −0.04

10.55 40.42 124.58 181.29 1.87 6.91 28.53 21.97 83.64 0.04 132.84 570.62 0.03 24.76 27.89 100.01 0.04 89.5 × 103 259 × 103 600.26 28.57 102.58 0.04

1.66 9.65 53.00 74.75 0.58 0.83 5.51 7.25 30.72 0.02 57.88 251.64 0.01 11.13 10.18 38.93 0.02 44.0 × 103 127 × 103 295.04 10.51 40.19 0.02

model

a

Units: μ′max = mg·L−1·h−1; KS = mg·L−1; KI = L·mg−1; K′S = (mg·L−1)n; K′I = (L·mg−1)n.

Table 3. Parameter Estimation Results of Toluene Biodegradationa Fobj

parameter

θ

θmin

θmax

standard deviation

(1) Monod

60.15

(2) Moser

59.28

(3) Tessier

61.68

(4) Andrews

60.06

(5) Wu

60.21

(6) modified Monod

60.06

(7) Webb

60.14

(8) Eduards

60.04

μmax ′ KS μmax ′ K′S n μ′max KS μ′max KS KI μ′max KS K′I n μ′max KS KI μ′max KS KI μ′max KS KI

5.57 8.39 6.81 7.38 0.76 4.58 9.11 6.05 9.75 0.00 7.14 11.13 0.20 0.09 6.16 9.90 0.00 5.38 8.15 0.01 6.34 10.38 0.00

4.45 3.12 −0.79 2.22 −0.10 3.97 5.33 0.59 −6.08 −0.02 −3130 −4920 −504.48 −145.44 −1.21 −9.28 −0.02 −4.70 −2.21 −0.34 −0.82 −8.59 −0.02

6.70 13.66 14.40 12.54 1.62 5.18 12.89 11.51 25.57 0.02 3140 4940 504.87 145.62 13.52 29.08 0.03 15.45 18.52 0.35 13.49 29.35 0.02

0.55 2.57 3.70 2.51 0.42 0.29 1.85 2.66 7.71 0.01 1530 2400 245.52 70.80 3.59 9.35 0.01 4.91 5.05 0.17 3.49 9.24 0.01

model

a

Units: μ′max = mg·L−1·h−1; KS = mg·L−1; KI = L·mg−1; KS′ = (mg·L−1)n; KI′ = (L·mg−1)n.

despite the initial part of each of the experiments, that is, at the beginning of the reaction. The difference between the experimental and model values at the beginning of the reaction time can be explained by two different reasons. As discussed in the Methodology section, initial concentration values are not precisely known, and these values were estimated in the parameter estimation procedure. Besides,

microorganisms may have a short period for adaptation, in which the degradation kinetics is not completely stabilized. Despite this, the Monod model adjustment to experimental data can be considered adequate to represent BTX biodegradation along the reaction time. Parameter estimation of biodegradation kinetics for each of the BTX compounds was performed considering models presented 19420



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Table 4. Parameter Estimation Results of Xylene Biodegradationa Fobj

parameter

θ

θmin

θmax

standard deviation

(1) Monod

198.56

(2) Moser

198.48

(3) Tessier

198.81

(4) Andrews

198.51

(5) Wu

198.56

(6) modified Monod

198.51

(7) Webb

198.54

(8) Eduards

198.50

μmax ′ KS μ′max K′S n μ′max KS μ′max KS KI μ′max KS K′I n μ′max KS KI μ′max KS KI μ′max KS KI

4.49 15.81 4.87 15.31 0.93 3.31 13.43 4.90 18.00 0.00 9.66 34.86 0.96 0.05 5.11 18.76 0.00 4.01 14.49 0.01 5.13 18.87 0.00

2.71 2.06 −2.81 0.84 −0.26 2.41 5.24 −4.42 −32.24 −0.05 −64.6 × 103 −233 × 103 −13.0 × 103 −271.79 −10.55 −55.55 −0.05 −3.96 −3.13 −0.27 −8.90 −48.96 −0.04

6.27 29.55 12.55 29.79 2.11 4.20 21.62 14.22 68.23 0.05 64.6 × 103 233 × 103 13.0 × 103 271.88 20.77 93.06 0.06 11.98 32.12 0.30 19.16 86.70 0.05

0.88 6.79 3.79 7.14 0.59 0.44 4.04 4.59 24.77 0.02 31.8 × 103 114 × 103 6.40 × 103 133.90 7.72 36.64 0.03 3.93 8.69 0.14 6.92 33.45 0.02

model

a

Units: μ′max = mg·L−1·h−1; KS = mg·L−1; KI = L·mg−1; K′S = (mg·L−1)n; K′I = (L·mg−1)n.

Figure 2. Elliptical and likelihood confidence regions of the Monod model for benzene (a), toluene (b), and xylene (c) and of Tessier model for benzene (d), toluene (e), and xylene (f).

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Figure 3. Likelihood confidence regions of the Moser model for benzene biodegradation.

substrate inhibition is present in concentrations up to 60 mg·L−1. It can be concluded that the microorganism variety and the meticulous adaptation and immobilization procedures used in this work lead to an improved pool of microorganisms able to consume BTX compounds, even at concentrations that were considered inhibitory by other works. To provide a better evaluation of the precision of parameter estimates, confidence regions were built for some models. Figure 2 presents the confidence region of the Monod and Tessier models. In this figure, the elliptical confidence region, computed according eq 5, and likelihood confidence region, computed according eq 6, can be compared. Elliptic confidence regions obtained for the Monod model for benzene and xylene substrates indicates that parameter KS is not a significant parameter, since the zero value is included in the confidence region. However, when likelihood confidence regions are considered, the zero value is not included in the confidence region of any case presented in Figure 3. It also must be observed that, although both confidence regions present a prolongation in the same direction, the likelihood confidence regions are not symmetric with respect to the optimum estimated parameter values, indicating that the parameter deviations are higher for higher parameter values. For example, parameter μmax′ of Monod model for benzene substrate has its optimum value equal to 7.18 mg· L−1·h−1 and according to the likelihood confidence region, its lower bound is around 6 mg·L−1·h−1 and its upper bound is slightly higher that 16 mg·L−1·h−1. It can be clearly observed that positive deviations are much larger than negative deviations. It must be kept in mind that the likelihood confidence regions provide a much better representation of the estimated parameter uncertainties than the elliptical approach.25,26 Since the elliptical region forces the confidence region to be symmetric, elliptical regions are shifted toward the lower values of μmax′ and KS. It is important to emphasize that, as observed for the two cases presented in Figure 2 (cases “a” and “c”), the elliptical confidence regions include the zero value. On the other side, the likelihood confidence regions do not include the zero value, since the real confidence regions are shifted in the positive region direction. If only the elliptical confidence region was considered, one would

in Table 1 and the results are shown in Tables 2, 3, and 4 for each one of substrates. It can be observed that the objective functions of all models for each substrate are practically equal, indicating that all models lead to an equivalent quality of model adjustment to the BTX biodegradation experimental data. Also, an increase of the number of model parameters, as in Model 2 and Model 5, did not lead to a significant decrease of the objective function and, therefore, did not result in a better adjustment to experimental data. Besides, models that take into account inhibition by the substrate did not provide better representation of experimental values. This result indicates that, at least in the concentration range of each substrate used in this work, the biodegradation process did not present significant inhibition by any substrate used in this work. Tables 2, 3 and 4 also present the limits (minimum and maximum) values (calculated using a t-Student distribution with 95% confidence). Only estimated parameter values of models 1 and 3, that is, Monod and Tessier models, can be considered physically significant, since its confidence intervals do not contain the zero value. All other models present at least two not significant estimated parameter values. This result clearly indicates that two parameter models are sufficiently adequate to represent our BTX degradation experimental data. Also, an increase in the number of model parameters leads to an enlargement of the standard deviation and, consequently, larger confidence intervals of the parameter estimates. Therefore, estimated parameter values become statistically not significant, and the reliability of model predictions are strongly reduced. It can also be observed that estimated values for the inhibition parameter KI are equal (or very close) to zero. These results corroborate with the conclusion that substrate inhibition is not significant in the range of substrate concentrations and with the microorganisms present in the biofilm used in this work. It is important to notice that some literature34,35 works report that inhibition is present in experiments with initial concentrations greater than 40 mg·L−1. These authors concluded that the Andrews model gives better kinetic parameters, because the BTX concentration was close to values considered toxic for cell growth. Despite that, results presented here showed that no 19422



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Figure 4. Likelihood confidence regions of the Moser model for toluene biodegradation.

Figure 5. Likelihood confidence regions of the Moser model for xylene biodegradation.

probably conclude that some of the parameter estimates are not significant. However, observing the likelihood confidence regions, it is clear that parameter estimates are significant and must be considered in BTX biodegradation models. It also can be observed that both confidence regions become more similar as the adjustment of the model to the experimental data becomes better. The major similarity among the elliptical and likelihood confidence regions is obtained with the data of toluene degradation, as showed in Figure 2 panels b and e. It also can be observed in Table 3 that the fits for toluene degradation data present lower objective function values when compared with the fits of the other substrate degradation data presented in Tables 2 and 4. That is, good model fit leads to good elliptical approximation of the confidence regions. The biochemical kinetic models that have three or four parameters presented confidence regions that are very different

from the elliptical ones. To illustrate this behavior, the likelihood confidence regions of the Moser model were constructed for benzene, toluene, and xylene degradation, as presented in Figures 3, 4, and 5. These confidence regions are nonconvex and are unbounded, since in some directions the confidence region grows continuously in the direction of the higher values of the parameters. Analysis of Figures 3, 4, and 5 clearly demonstrates that confidence of the parameters must not be evaluated through the elliptical approach. It is important to observe that the zero value of any parameter is not enclosed in the confidence regions, despite the fact that the confidence regions are very large, indicating parameter estimates with high uncertainty. These large confidence regions are obtained owing to inclusion of an additional parameter that does not lead to a significant improvement of model adjustment 19423



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biodegradation process did not present inhibition by any substrate, which can be attributed to the microorganisms and the meticulous adaptation procedure used in this work. Although some models present a slightly better model performance, such as the Moser model that presents a lower value of the objective function, the inclusion of an additional parameter to be estimated led to large confidence regions of the parameter estimate, deteriorating the quality of parameter estimates and producing nonconvex and unbounded confidence regions. Besides, evaluation of the confidence regions must be performed through the likelihood method, since this method leads to confidence region shapes very close to the actual ones. The results presented here showed that the confidence regions can be nonelliptical and unsymmetrical with respect to the optimum estimated parameter values, even when simple two parameter models are considered. With an increase in the model complexity and the number of model parameters, the confidence regions tend to become very large and with strange shapes. Therefore, it is clearly shown in this work that the evaluation of biochemical models for the kinetics of BTX biodegradation must be performed through rigorous statistical procedures in order to provide the selection of an adequate kinetic model with good estimated parameter values.

to experimental data, causing an increase in parameter estimate uncertainties. In Figures 3−5 it also can be observed that the confidence regions for parameter estimates of toluene biodegradation (Figure 4) are smaller than the ones obtained for benzene (Figure 3) and xylene (Figure 5) biodegradation, a similar behavior observed in the two parameter models. In Figures 3−5 confidence regions present an interesting behavior. If the confidence region for parameters KS and n are considered, when values of parameter KS are high, there are two separated intervals where n parameter values are inside the confidence region. For example, in Figure 5, when the KS value is about 100 (mg·L−1)n, parameter n is approximately in the interval (0.50, 0.85) and (1.70, 3.00). That is, n values from 0.85 to 1.70 are not in the confidence region for this specific value of KS. It can be observe that for the first n interval, that is (0.50, 0.85), μ′max presents values higher than 10 mg·L−1·h−1. By the other side, for the second n interval, that is (1.70, 3.00), μ′max presents values lower than 5 mg·L−1·h−1. This behavior clearly shows the high dependence among estimated parameter values. This result seems strange, probably because we expect very narrow ellipses for confidence regions of highly correlated parameters. However, because of the nonlinear characteristic of the models used in this work, elliptic approximations of the confidence regions are very different from the real ones, as shown in Figures 3−5. It is also important to notice that the general practice that an estimated parameter value is not significant when the zero value is enclosed in its confidence interval is not a proper technique to state the confidence of an estimated parameter. As shown in Figures 3−5, those parameter confidence regions do not include the value zero for any parameter, but even so, these parameters are very uncertain. Besides, the benefit of the use of the likelihood method for the determination of parameter confidence regions is clear, providing a more rigorous and actual statistical evaluation of the parameter estimation results. Considering only the parameter estimates obtained with the Monod model, it can be observed that the apparent maximum specific rate of substrate consumption μ′max presents the higher value for benzene biodegradation, and the lower value is obtained for the xylene biodegradation. Alternately, toluene is the substrate with the lower value for the KS parameter, and benzene presents the higher value. The parameter KS describes the overall affinity of a microorganism for its growth-limiting substrate.27 Besides, according to the Monod model, a high value for KS decreases the rate of substrate consumption. Consequently, even toluene degradation presents an intermediate value for μ′max, the low value for KS leading to the higher rates of degradation, which are in concordance with the decrease in the toluene concentration presented in Figure 1b. Although the comparison among the parameter estimates seems a little confusing, it must also be noted that the high correlation between the parameters μmax ′ and KS complicates the comparison of estimated parameter values and also leads to large confidence intervals, that in fact lead us to conclude that the parameter values cannot be distinguished. Once again, the benefits of a rigorous statistical evaluation of the parameter estimation results seem clear.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 55-55-32208448. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank ANP (National Petroleum Agency) and FINEP (Financial Agency of Studies and Projects) through the Program of Human Resources Formation of ANP for the Petroleum and Natural Gas Sector (PRH09-ANP/MME/MCT) for financial support to this research and for providing scholarships. The authors also thank CAPES and FAPERGS for providing scholarships.

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REFERENCES

(1) Robledo-Ortíz, J. R.; Ramírez-Arreola, D. E.; Pérez-Fonseca, A. A.; Gómez, C.; González-Reynoso, O.; Ramos-Quirarte, J.; GonzálezNúñez, R. Benzene, toluene, and o-xylene degradation by free and immobilized P. putida F1 of postconsumer agave-fiber/polymer foamed composites. Int. Biodeterior. Biodegrad. 2011, 65, 539−546. (2) Otenio, M. H.; Silva, M. T. L.; Marques, M. L. O.; Roseiro, J. C.; Bidoia, E. D. Benzene, toluene and xylene biodegradation by Putida CCMI 852. Braz. J. Microbiol. 2005, 36, 258−261. (3) Trigueros, D. E. G.; Módenes, A. N.; Kroumov, A. D.; EspinozaQuiñ ones, F. R. Modeling of biodegradation process of BTEX compounds: Kinetic parameters estimation by using Particle Swarm Global Optimizer. Process Biochem. 2010, 45, 1355−1361. (4) Mathur, A. K.; Majumder, C. B.; Chatterjee, S. Combined removal of BTEX in air stream by using mixture of sugar cane bagasse, compost and GAC as biofilter media. J. Hazard. Mater. 2007, 148, 64−74. (5) Jo, M. S.; Rene, E. R.; Kim, S. H.; Park, H. S. An analysis of synergistic and antagonistic behavior during BTEX removal in batch system using response surface methodology. J. Hazard. Mater. 2008, 152, 1276−1284. (6) Littlejohns, J. V.; Daugulis, A. J. Kinetics and interactions of BTEX compounds during degradation by a bacterial consortium. Process Biochem. 2008, 43, 1068−1076.

4. CONCLUSIONS The use of statistical procedures for the analysis of a biochemical process is widely proven to be relevant. Among eight considered models, the Monod model could be established as an appropriate choice to represent the kinetics of the BTX biodegradation process, since in the range of concentrations evaluated, the 19424



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(7) Mello, J. M. M.; Brandão, H. L.; Souza, A. A. U.; Silva, A.; Souza, S.M.A.G.U. Biodegradation of BTEX compounds in a biofilm reactor Modeling and simulation. J. Pet. Sci. Eng. 2010, 70, 131−139. (8) Kryst, K.; Karamanev, D. G. Aerobic phenol biodegradation in an inverse fluidized-bed biofilm reactor. Ind. Eng. Chem. Res. 2001, 40, 5436−5439. (9) Xavier, J. B.; Picioreanu, C.; Almeida, J. S.; van Loosdrecht, M. C. M. Monitorizaçaõ e modelaçaõ da estrutura de biofilmes. Bol. Biotecnol. 2003, 76, 2−13. (10) Massalha, N.; Basherr, S.; Sabbah, I. Effect of adsorption and bed size of immobilized biomass on the rate of biodegradation of phenol at high concentration levels. Ind. Eng. Chem. Res. 2007, 46, 6820−6824. (11) Nardi, I. R.; Ribeiro, R.; Zaiat, M.; Foresti, E. Anaerobic packedbed reactor for bioremediation of gasoline-contaminated aquifers. Process Biochem. 2005, 40, 587−592. (12) Souza, A. A. U.; Brandão, H. L.; Zamporlini, I. M.; Soares, H. M.; Souza, S.M.A.G.U. Application of a fluidized bed bioreactor for COD reduction in the textile industry effluents. Resour. Conserv. Recycl. 2008, 52, 511−521. (13) Ahmad, F.; Jameel, A. T.; Kamarudin, M. H.; Mel, M. Study of growth kinetic and modeling of ethanol production by Saccharomyces cerevisae. Afr. J. Biotechnol. 2011, 16, 18842−18846. (14) Beyenal, H.; Chen, S. N.; Lewandowski, Z. The double substrate growth kinetics of Pseudomonas aeruginosa. Enzyme Microb. Technol. 2003, 32, 92−98. (15) Wang, S.; Loh, K. Modeling the role of metabolic intermediates in kinetics of phenol biodegradation. Enzyme Microb. Technol. 1999, 25, 177−184. (16) Lima, U. A.; Aquarone, E.; Borzani, W.; Schmidell, W. Biotecnologia industrial: Processos Fermentativos e Enzimáticos; Edgard Blucher: São Paulo, Brazil, 2001; Vol. 3. (17) Nuhoglu, A.; Yalcin, B. Modelling of phenol removal in a batch reactor. Process Biochem. 2005, 40, 1233−1239. (18) Gokulakrishnan, S.; Gummadi, S. N. Kinetics of cell growth and caffeine utilization by Pseudomonas sp.GSC1182. Process Biochem. 2006, 41, 1417−1421. (19) Schwaab, M.; Pinto, J. C. Análise de Dados Experimentais I: ́ Fundamentos de Estatistica e Estimaçaõ de Parâmetros; E-Papers: Rio de Janeiro, Brasil, 2007. (20) Schwaab, M.; Queipo, C. Q.; Silva, F. M.; Barreto, A. G., Jr.; Nele, M.; Pinto, J. C. A new approach for sequential experimental design for model discrimination. Chem. Eng. Sci. 2006, 61, 5791−5806. (21) Buzzi-Ferraris, G.; Forzatti, P.; Emig, G.; Hofmann, H. Sequential experimental design for model discriminating in the case of multiresponse models. Chem. Eng. Sci. 1984, 39, 81−85. (22) Schwaab, M.; Monteiro, J. L.; Pinto, J. P. Sequential experimental design for model discrimination. Taking into account the posterior covariance matrix of differences between model predictions. Chem. Eng. Sci. 2008, 63, 2408−2419. (23) Draper, N. R.; Smith, H. Applied Regression Analysis; Wiley: New York, USA, 1998. (24) Draper, N. R.; Guttman, I. Confidence intervals versus regions. Statistician 1995, 44, 399−403. (25) Donaldson, J. R.; Schnabel, R. B. Computational experience with confidence-regions and confidence-intervals for nonlinear least-squares. Technometrics 1987, 29, 67−82. (26) Schwaab, M.; Biscaia, E. C., Jr.; Monteiro, J. L.; Pinto, J. P. Nonlinear parameter estimation through particle swarm optimization. Chem. Eng. Sci. 2008, 63, 1542−1552. (27) Snoep, J. L.; Mrwebi, M.; Schuurmans, J. M.; Rohwer, J. M.; Mattos, M. J. T. Control of specific growth rate in Saccharomyces cerevisiae. Microbiology 2009, 155, 1699−1707. (28) Petzold, L. R. DASSL Code (Differential Algebraic System Solver); Computing and Mathematics Research Division, Lawrence Livermore National Laboratory: Livermore, CA, USA, 1989. (29) Prata, D. M.; Schwaab, M.; Lima, E. L.; Pinto, J. P. Nonlinear dynamic data reconciliation and parameter estimation through particle swarm optimization: Application for an industrial polypropylene reactor. Chem. Eng. Sci. 2009, 64, 3953−3967.

(30) Prata, D. M.; Schwaab, M.; Lima, E. L.; Pinto, J. P. Simultaneous robust data reconciliation and gross error detection through particle swarm optimization for an industrial polypropylene reactor. Chem. Eng. Sci. 2010, 65, 4943−4954. (31) Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia, 4, 1942−1948, 1995. (32) Da Ros, S.; Colusso, G.; Weschenfelder, T. A.; Terra, L. M.; Castilhos, F.; Corazza, M. L.; Schwaab, M. A comparison among stochastic optimization algorithms for parameter estimation of biochemical kinetic models. Appl. Soft Comput. 2013, 13, 2205−2214. (33) Noronha, F. B.; Pinto, J. C.; Monteiro, J. L.; Lobão, M. W.; Santos, T. J. ESTIMA; Technical Report PEQ/COPPE/UFRJ, Rio de Janeiro, RJ, Brazil, 1993. (34) Bielefeldt, A. R.; Stensel, H. D. Modeling competitive inhibition effects during biodegradation of BTEX mixtures. Water Res. 1999, 33, 707−714. (35) Shim, H.; Hwang, B.; Lee, S.-S.; Kong, S.-H. Kinetics of BTEX biodegradation by a coculture of Pseudomonas putida and Pseudomonas fluorescens under hypoxic conditions. Biodegradation 2005, 16, 319−327.

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Statistical Evaluation of Biochemical Kinetic Models for BTX

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