# Biosorption Parameter Estimation with Genetic Algorithm

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Parameter Estimation Methods

#### 2.1. Genetic Algorithm Optimization

_{j}is an appropriate weighting factor for observation j, taken to equal unity in this paper, and y

_{pred,j}, y

_{exp,j}are the model-predicted and measured values for observation j, respectively.

#### 2.2. Nonlinear and Linear Regressions

#### 2.3. Goodness-of-Fit Measure

## 3. Results and Discussion

#### 3.1. Equilibrium Isotherms

**Figure 1.**Experimental isotherm for lead biosorption on orange peels; data of Schiewer and Balaria [12].

#### 3.1.1. Langmuir equation

_{e}is the equilibrium sorbed concentration and C

_{e}is the equilibrium solution concentration. The two parameters to be optimized are the saturation capacity q

_{m}and the Langmuir constant b. At sufficiently low sorbed concentrations the Langmuir equation approaches linearity (Henry’s law). At higher loadings the equation tends to a maximum asymptotically. When the product bC

_{e}is large, Equation (3) reduces to the rectangular form typical of highly favorable sorption. The Langmuir equation is derived from a sound theoretical footing and is based on several assumptions [13]. Biosorbents, due to their complex surface structure, rarely satisfy the assumptions made in the Langmuir theory. In this context, the Langmuir equation may be viewed as a convenient tool for reproducing the correct shape of biosorption equilibrium curves rather than a mechanistic model.

_{m}and b can be obtained by least-squares linear regression. Different linearization methods are available, as shown in Table 1. The terminology for the four linearized equations in Table 1 is adopted by extension from analogous linearized versions of the Michaelis-Menten equation used in enzyme kinetics studies. While linearized Michaelis-Menten equations are noted to be only of historical interest, their linearized Langmuir counterparts are still being used in the biosorption field. Plotted in Figure 2 are the linearized data of Figure 1 for the different linearization methods presented in Table 1. The goodness-of-fit indicated by R

^{2}for each plot is also given in the figure. Table 2 summarizes the values of q

_{m}and b obtained from these transformations. COD scores calculated from Equations (2) and (3) with the derived parameters are also shown in Table 2.

Linearization plot | Equation form |
---|---|

Lineweaver-Burk | $\frac{1}{{q}_{e}}=\frac{1}{{q}_{m}b}\frac{1}{{C}_{e}}+\frac{1}{{q}_{m}}\text{}\left(1/{q}_{e}\text{}vs\text{}1/{C}_{e}\right)$ |

Hanes-Woolf | $\frac{{C}_{e}}{{q}_{e}}$ |

Eadie-Hofstee | ${q}_{e}$ |

Scatchard | $\frac{{q}_{e}}{{C}_{e}}$ |

**Figure 2.**Equilibrium data in Figure 1 fitted with the following linearizations:

**(a)**Lineweaver-Burk.

**(b)**Hanes-Woolf.

**(c)**Eadie-Hofstee.

**(d)**Scatchard.

**Table 2.**Parameter estimation in the Langmuir equation by the linear regression, nonlinear regression and GA methods.

Estimation method | q_{m} (mmol/g) | b (L/mmol) | COD |
---|---|---|---|

Linear regression | |||

Lineweaver-Burk | 0.99 | 10.31 | 0.414 |

Hanes-Woolf | 2.37 | 2.17 | 0.967 |

Eadie-Hofstee | 1.60 | 5.20 | 0.809 |

Scatchard | 1.97 | 3.46 | 0.923 |

Nonlinear regression | 3.20 | 1.00 | 0.990 |

Genetic algorithm | 3.20 | 1.00 | 0.990 |

^{2}of 0.976 provided the best fit among the four linearizations. However, it also had the lowest COD (0.414), as shown in Table 2. Note that the R

^{2}value was obtained with the transformed data while the COD value was calculated on the untransformed data. It is evident that the fitted q

_{m}and b based on the transformed data of the Lineweaver-Burk linearization perform poorly when they are substituted back into Equation (3). This phenomenon illustrates the limitations associated with the transformation of data required by the Lineweaver-Burk plot. The main problems with the Lineweaver-Burk linearization are that most of the data points clump near the origin and the slope of the linear plot is extremely sensitive to variability at low values of C

_{e}(high values of 1/C

_{e}) [14], as can be seen in Figure 2a. Although the Hanes-Woolf plot yielded the second highest R

^{2}, it provided the best result as indicated by the highest COD score.

_{m}and b to the GA. Still, the Hanes-Woolf-derived q

_{m}and b were, respectively, 26% smaller and 117% bigger than the GA-generated q

_{m}and b. Therefore, the GA improved the parameter accuracy considerably. Figure 3 compares the performance of the four linearizations and GA in visual terms. The Langmuir equation containing the Lineweaver-Burk-derived parameters systematically underestimated the measured values of q

_{e}at high concentrations, suggesting that the derived parameters are not adequate at these concentration levels. All q

_{e}values calculated with the GA-generated parameters fall close to the 1:1 line (solid line in Figure 3), confirming the superiority of the GA over the four linearizations. Although the limitations of linearized Langmuir equations have been noted for some time [14,15,16], out of inertia they are found to persist in biosorption modeling. There is little doubt that the antiquated practice of linearization has no place in today’s research environment.

**Figure 3.**Comparison between the Figure 1 equilibrium data and q

_{e}calculated from the Langmuir equation (Equation (3)) containing parameters estimated from the following linearizations: Lineweaver-Burk (open circles), Hanes-Woolf (triangles), Eadie-Hofstee (diamonds), Scatchard (squares), and parameters estimated from the GA (filled circles).

#### 3.1.2. Freundlich equation

_{e}, C

_{e}are similarly defined in Equation (3) and K

_{F}, n

_{F}constitute the two unknown parameters. If the sorption is favorable, then n

_{F}< 1. Unlike the Langmuir equation, Equation (4) has neither a proper Henry law behavior at low sorbed concentration nor a finite saturation limit when sorbed concentration is sufficiently high. As a result, it is not applicable over a large range of equilibrium data. The Freundlich equation may be linearized as follows:

**Table 3.**Parameter estimation in the Freundlich equation by the linear regression, nonlinear regression and GA methods.

Estimation method | ${K}_{F}\left(\left({\text{mmol}}^{\left(1-{n}_{F}\right)}\xb7{\text{L}}^{{n}_{F}}\right)/\text{g}\right)$ | n_{F} | COD |
---|---|---|---|

Linear regression (Equation (5)) | 1.554 | 0.609 | >0.999 |

Nonlinear regression | 1.550 | 0.607 | >0.999 |

Genetic algorithm | 1.550 | 0.607 | >0.999 |

#### 3.2. Batch Kinetic Models

**Figure 4.**Experimental uptake curve for lead biosorption on orange peels; data of Schiewer and Balaria [12].

#### 3.2.1. Lagergren equation

_{t}is the sorbed concentration at any time t, q

_{e}is the equilibrium sorbed concentration, and k

_{1}is the first order Lagergren rate constant. The analytical solution of Equation (6) for the initial condition of q

_{t}= 0 at t = 0 can be written as:

_{e}and k

_{1}are fitting parameters. Note that this linear regression procedure requires a value of q

_{e}in order to calculate the left side of Equation (8). The logarithmic term ln(q

_{e}− q

_{t}) dictates that q

_{e}be assigned the maximum measured value. For the Figure 4 data, the maximum value is given by the second last data point measured at 120 min. The q

_{e}term on the left side of Equation (8) was thus assigned the value of this data point (q

_{e}= 0.79 mmol/g). Figure 5 shows the data in Figure 4 plotted according to Equation (8). It can be seen that the linear fit is satisfactory as indicated by the high value of R

^{2}. The two parameters q

_{e}and k

_{1}on the right side of Equation (8) were determined, respectively, from the y-intercept and slope of the linear plot. Listed in Table 4 are the derived parameters.

_{e}was much lower than the assigned value of 0.79 mmol/g for q

_{e}which was used to generate the linear plot in Figure 5. Furthermore, the COD value was rather low (Table 4), indicating significant differences between the calculated and measured q

_{t}. A comparison between the Figure 4 data and q

_{t}calculated from the Lagergren equation (Equation (7)) with the derived parameters is shown in Figure 6 (open circles). The figure includes results calculated from a general nth order rate equation and these will be discussed in the next section. Figure 6 shows that all calculated values of q

_{t}were much smaller than the measured values of q

_{t}. The poor representation of the untransformed data indicates that the linearized Lagergren equation is inadequate for parameter estimation.

**Figure 5.**Kinetic data in Figure 4 plotted according to the linearized Lagergren equation (Equation (8)).

**Table 4.**Parameter estimation in the Lagergren equation by the linear regression, nonlinear regression and GA methods.

Estimation method | q_{e} (mmol/g) | k_{1} (min^{−1}) | COD |
---|---|---|---|

Linear regression (Equation (8)) | 0.32 | 0.028 | 0.512 |

Nonlinear regression | 0.71 | 0.268 | 0.819 |

Genetic algorithm | 0.71 | 0.268 | 0.819 |

_{e}and k

_{1}estimates were, respectively, 122% and 857% larger than the linear regression-generated q

_{e}and k

_{1}. The higher COD and the proximity of calculated q

_{t}to the 1:1 line, as indicated by the filled circles in Figure 6, suggest that the GA was capable of finding realistic parameters that fit the measured data quite well. For this data set, the performance of the linear regression approach was obviously inferior to that of the GA. Although the linearized Lagergren equation ((Equation (8)) is clearly unsatisfactory, it remains the equation of choice for parameter estimation in many biosorption studies that employ the Lagergren equation. The nonlinear Lagergren equation (Equation (7)), by contrast, is often ignored because it is necessary to go beyond linear regression in order to estimate its parameters from measured data.

**Figure 6.**Comparison between the Figure 4 kinetic data and q

_{t}calculated from the Lagergren equation (Equation (7)) with the linear regression-derived parameters (open circles) and GA-generated parameters (filled circles) listed in Table 4. Also shown are q

_{t}calculated from the nth order equation (Equation (10)) with the GA-derived parameters (triangles) given in Table 5.

#### 3.2.2. nth Order Rate Equation

_{t}, q

_{e}, t are similarly defined in Equation (6) and k

_{n}, n indicate the nth order rate constant and reaction order, respectively. When n = 1 we recover the first order Lagergren equation. Note that n as defined in Equation (9) may be a noninteger. The integrated form of Equation (9) for the initial condition of q

_{t}= 0 at t = 0 is given by [19]:

_{n}, n (n ≠ 1) and q

_{e}, are to be determined simultaneously, which of course cannot be estimated using linear regression. Note that Equation (10) may be recovered from a more general solution of the nth order rate law incorporating the concept of fractal kinetics [23].

_{t}calculated from Equation (10) with the GA-derived parameters are compared with the Figure 4 data in Figure 6 (triangles). Both the graphical comparison (Figure 6) and the COD statistics (Table 4 and Table 5) indicate that the three-parameter nth order rate equation provided a better description of the kinetic data compared to the two-parameter Lagergren equation. This is not surprising because models with more adjustable parameters will almost always result in a better fit. However, the fact that Equation (10) cannot be linearized is likely to limit its application in the correlation of biosorption kinetic data. Note that the nonlinear regression method was sensitive to initial parameter guesses. For instance, no convergence difficulties were observed when the starting point of the parameter vector (q

_{e}, k

_{n}, n) was chosen to be (10, 10, 10). However, false convergence was encountered with the starting point (100, 100, 100). As noted above, rather than operating on a single set of parameters, the GA makes use of a population of parameter sets (individuals). For the nth order rate equation, the GA was able to obtain the optimal parameter set within a search range having upper parameter limits as high as (10,000, 10,000, 10,000).

**Table 5.**Parameter estimation in the nth order rate equation by the nonlinear regression and GA methods.

Estimation method | q_{e} (mmol/g) | k_{n} ((mmol/g)^{1−n}/min) | n | COD |
---|---|---|---|---|

Nonlinear regression | 0.900.90 | 0.84 | 3.89 | 0.971 |

Genetic algorithm | 0.84 | 3.89 | 0.971 |

#### 3.3. Fixed Bed Models

**Figure 7.**Experimental breakthrough curve for nickel biosorption on seaweed biomass; data of Borba et al. [25].

#### 3.3.1. Bohart-Adams equation

_{t}is the solution concentration at the fixed bed outlet at time t, C

_{i}is the feed concentration, k

_{BA}is the Bohart-Adams rate constant, N is the sorption capacity of the sorbent per unit volume of the bed, Z is the total bed depth, and u is the superficial velocity. In Equation (11) k

_{BA}and N are fitting parameters. Equation (11) may be rearranged in the following manner to allow parameter estimation by linear regression:

_{i}, Z and u, the two parameters N and k

_{BA}are given by the y-intercept and slope of the plot, respectively.

^{2}of 0.931. Listed in Table 6 are the values of N and k

_{BA}obtained from the linear plot. The values of other variables, used in the calculation, are as follows: Z = 30.5 cm, C

_{i}= 2.12 meq/L and u = 0.097 cm/min. Figure 9 compares the Figure 7 data with C

_{t}/C

_{i}calculated from the Bohart-Adams equation (Equation (11)) with the derived parameters (open circles). It can be seen that the Bohart-Adams equation containing the linear regression-generated parameters underestimated breakthrough concentrations in the low C

_{t}/C

_{i}region. This discrepancy is undesirable because the initial portion of a breakthrough curve determines the breakthrough time for a specified breakthrough concentration.

**Figure 8.**Breakthrough data in Figure 7 plotted according to the linearized Bohart-Adams equation (Equation (12)). Data points denoted by filled circles are excluded from the linear fit.

**Table 6.**Parameter estimation in the Bohart-Adams equation by the linear regression, nonlinear regression and GA methods.

Estimation method | N (meq/L) | k_{BA} (L/meq min) | COD |
---|---|---|---|

Linear regression (Equation (12)) | 5.29 | 0.0035 | 0.963 |

Nonlinear regression | 4.54 | 0.0029 | 0.998 |

Genetic algorithm | 4.54 | 0.0029 | 0.998 |

**Figure 9.**Comparison between the Figure 7 breakthrough data and C

_{t}/C

_{i}calculated from the Bohart-Adams equation (Equation (11)) with the linear regression-derived parameters (open circles) and GA-generated parameters (filled circles) tabulated in Table 6. Also shown are C

_{t}/C

_{i}calculated from the Belter-Cussler-Hu equation (Equation (13)) with the GA-derived parameters (triangles) given in Table 7.

_{t}/C

_{i}calculated from Equation (11) with the GA-derived parameters (filled circles) lie much closer to the 1:1 line for the entire data range. Even with a judicial selection of data points to aid the parameter estimation, the linear regression approach performed worse than the GA. This does not imply that the functional form of the Bohart-Adams equation is inadequate, merely that the suboptimal linear regression-derived parameters impair its correlative capability. An agreement between the Bohart-Adams equation and the breakthrough data can be reached as long as optimal parameter estimates are used in the equation.

_{t}/C

_{i}) of 0.1, the Bohart-Adams equation containing the linear regression-generated parameters predicts a breakthrough time of 488 min. From Figure 7 it is seen that the corresponding experimental breakthrough time is approximately 357 min. The predicted breakthrough time is thus 37% bigger than the observed breakthrough time. In contrast, a much better agreement can be obtained with the GA-derived parameters. In this case, the predicted breakthrough time for C

_{t}/C

_{i}= 0.1 is 313 min, which is only 13% smaller than the observed breakthrough time. From the foregoing discussion, it is clear that the linear regression approach yielded suboptimal parameters which can overestimate the breakthrough time substantially. Additionally, the linear regression approach relied on the use of a subset of the data points to achieve a good fit. The GA and nonlinear regression methods, by contrast, are free from these deficiencies. Despite its shortcomings, the linearized Bohart-Adams equation is a very popular modeling tool. Also shown in Figure 9 are results calculated from the Belter-Cussler-Hu equation (triangles), and these are discussed in the next section.

#### 3.3.2. Belter-Cussler-Hu Equation

_{t}, C

_{i}, t are similarly defined in Equation (11) and t

_{c}(characteristic time), σt

_{c}(standard deviation) are parameters. The quantity erf(x) is the error function of x. Because the Belter-Cussler-Hu model is nonlinear in the parameters, t

_{c}and σ can be found only by search. Equation (13) was fit to the Figure 7 data by using the nonlinear regression and GA methods. Both methods were equally successful in estimating the two parameters from the breakthrough data, as shown in Table 7. Comparing the COD statistics in Table 6 and Table 7 indicates that the Belter-Cussler-Hu equation was marginally better than the Bohart-Adams equation in correlating the breakthrough data. The same conclusion may be seen in Figure 9, which shows computed results of the Belter-Cussler-Hu equation (triangles) and those of the Bohart-Adams equation (filled circles).

**Table 7.**Parameter estimation in the Belter-Cussler-Hu equation by the nonlinear regression and GA methods.

Estimation method | t_{c} (min) | σ | COD |
---|---|---|---|

Nonlinear regression | 670.3 | 0.41 | 0.999 |

Genetic algorithm | 670.3 | 0.41 | 0.999 |

## 4. Conclusions

## List of Symbols, Acronyms and Abbreviations

b | Langmuir constant |

BDST | Bed-depth-service-time |

C_{e} | Equilibrium solution concentration |

C_{i} | Feed solution concentration |

C_{t} | Solution concentration at fixed bed outlet at time t |

COD | Coefficient of determination |

erf(x) | Error function of x |

GA | Genetic algorithm |

k_{1} | Lagergren rate constant |

k_{BA} | Bohart-Adams rate constant |

k_{n} | nth order rate constant |

K_{F} | Freundlich parameter |

n | Reaction order |

n_{F} | Freundlich exponent |

N | Sorption capacity of sorbent per unit volume of fixed bed |

p | Number of observations |

q_{e} | Sorbed concentration at C _{e} |

q_{m} | Langmuir saturation capacity |

q_{t} | Sorbed concentration at time t |

SSE | Sum of squared errors |

t | Time |

t_{c} | Characteristic time |

u | Superficial velocity |

w_{j} | Weighting factor for observation j |

y_{exp,j} | Measured value for observation j |

y_{pred,j} | Model-predicted value for observation j |

${\overline{y}}_{exp}$ | Mean of measured values |

Z | Total bed depth |

σ t_{c} | Standard deviation |

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**MDPI and ACS Style**

Chu, K.H.; Feng, X.; Kim, E.Y.; Hung, Y.-T.
Biosorption Parameter Estimation with Genetic Algorithm. *Water* **2011**, *3*, 177-195.
https://doi.org/10.3390/w3010177

**AMA Style**

Chu KH, Feng X, Kim EY, Hung Y-T.
Biosorption Parameter Estimation with Genetic Algorithm. *Water*. 2011; 3(1):177-195.
https://doi.org/10.3390/w3010177

**Chicago/Turabian Style**

Chu, Khim Hoong, Xiao Feng, Eui Yong Kim, and Yung-Tse Hung.
2011. "Biosorption Parameter Estimation with Genetic Algorithm" *Water* 3, no. 1: 177-195.
https://doi.org/10.3390/w3010177