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Solubility prediction of anthracene in binary and ternary solvents by artificial neural networks (ANNs)
Fluid Phase Equilibria 225 (2004) 133–139
Solubility prediction of anthracene in binary and ternary solvents by artificial neural networks (ANNs) Abolghasem Jouybana,∗ , Mir-Reza Majidib , Farnaz Jabbaribarc , Karim Asadpour-Zeynalib a
School of Pharmacy and Drug Applied Research Centre, Tabriz University of Medical Sciences, Tabriz 51664, Iran b Department of Analytical Chemistry, Faculty of Chemistry, University of Tabriz, Tabriz 51664, Iran c Kimia Research Institute, P.O. Box 51665-171, Tabriz, Iran Received 9 August 2003; received in revised form 29 March 2004; accepted 20 August 2004
Abstract Solubility of anthracene in binary and ternary solvent systems was modeled by artificial neural networks (ANNs) technique. The results obtained using the ANN method indicated that the solubility of anthracene in mixed solvents could be calculated using the mole fraction solubilities in pure solvents, mole/volume fraction of solvents and solvent’s solubility parameters. The topology of neural network was optimized empirically and optimum topology was a 6-6-1 architecture for binary and 9-6-1 for ternary mixtures. The solubility of anthracene in mixed solvents was estimated by means of ANN and the predicted solubility was compared with experimental solubility data. The overall absolute percentage mean deviation (OPMD) for trained ANNs using all data points in 25 binary and 30 ternary solvent systems were 0.16 and 0.20%, respectively. A minimum number of data points from binary and ternary solvents have been employed to train the ANN and solubility at other solvent compositions has been predicted. The OPMD obtained for solubility in binary and ternary solvents were 0.67 and 0.27%, respectively. The trained network with 25 binary data sets was applied to predict the solubility in 16 other binary solvent systems and the OPMD obtained is 15.32%. The results of ANNs were also compared with similar numerical analyses carried out using multiple linear regression models and found that the ANN method is generally promising more accurate calculations. © 2004 Elsevier B.V. All rights reserved. Keywords: Artificial neural networks; Anthracene; Solubility; Prediction
1. Introduction The use of artificial neural networks (ANNs) in chemical and pharmaceutical areas has been increased recently [1]. The wide applicability of ANNs stems from their flexibility and ability to model linear and non-linear systems without prior knowledge of an empirical model [2]. This gives ANNs an advantage over traditional fitting methods for some chemical applications. The solubility is one of most important physicochemical properties of chemical/pharmaceutical compounds. ∗
Corresponding author. Tel.: +98 411 3392584; fax: +98 411 3344798. E-mail address: [email protected] (A. Jouyban).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.08.031
Solvent mixing (cosolvency) is a common method in chemical and pharmaceutical industries to enhance the solubility of a poorly soluble compound. Determination of solubility in all possible solvent compositions in mixed solvents by experiments is hardly possible and also time consuming, since there is an infinite number of solvent compositions. However, a number of mathematical methods have been used to predict the solubility in mixed solvents using a minimum number of experimental data points [3–5]. To the best of our knowledge, ANNs have not been employed for modeling solute solubility data in mixed solvents. In this work, a simple feed-forward artificial neural network is presented to model the solubility of anthracene in binary and ternary solvent mixtures. The ob-
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tained results from ANNs were compared with other methods based on least square analysis.
The ANNs are able to acquire information and provide models even when the information and data are complex, noise contaminated, non-linear or incomplete. The goal of ANN is to map a set of input patterns onto a corresponding set of output patterns. The network accomplishes this mapping by learning from a series of past examples and defining the input and output sets for a given system. The network then applies what it has been learned to a new input pattern to predict the appropriate output [6]. A network comprises an input layer, a number of hidden layers and an output layer (as schematically illustrated in Fig. 1). Many different types of ANNs have been developed [6,7]. The multilayer feed-forward network with backpropagation (BP) learning method is the most popular network in chemical applications [8,9] and is used in this work. The BP networks include one input layer, at least one hidden layer and one output layer. The number of neurons in the input and output layers are defined by system’s properties. The number of neurons in the hidden layer could be considered as an adjustable parameter, which should be optimized. The input layer receives the experimental information, experimental parameters, topological descriptors, etc. The output layer produces the calculated values of dependent variable. The mathematical relationship between inputs and outputs can be expressed as Eq. (1) for a multilayer network: 1 1 + e−α Netj
(2)
i=1
2. Computational methods
Outj =
the transfer function and Netj is given by Eq. (2): n Netj = Wij Outi + θj
(1)
where Outj is the transfer function as output from the jth neuron connected to n neuron in previous layer, α which is usually equal to one denotes the slope of the raising part of
Fig. 1. Schematic architecture of the ANN used for prediction of anthracene solubility in binary solvents, Xm is the solubility of anthracene in binary solvents X1 and X2 are the mole fraction solubilities in pure solvents 1 and 2, f1 and f2 denote the mole/volume fraction of solvents 1 and 2, δ1 and δ2 are solubility parameters of solvents 1 and 2, respectively.
where Outi is the output from ith neuron in previous layer, Wij represents the weights applied to the connection from ith to jth neuron and θ j is a bias term. In this work the numerical value of α is considered equal to 1. BP networks operate in the supervised learning mode. In the training step, known data is given to the network and using the BP algorithm, the network iteratively adjusts connection weights (Wij ) and biases (θ j ). Training of an ANN is usually done by starting with random connection weights. The computed output (Opm ) is compared to target value (Tpm ) and an error term (Tpm − Opm )2 is determined. The mean square error (MSE) is used as a criterion for finalizing the learning process and computed using Eq. (3): P M 1 (Tpm − Opm )2 MSE = P ×M
(3)
p=1 m=1
where M is the number of neurons in output layer and P denotes the number of patterns. The weights are adjusted in the learning stage for all the inter-connections and could be used to predict of output values of each data point in the prediction and test sets. All the ANNs calculations were carried out using Matlab mathematical software with artificial neural network toolbox for windows running on a personal computer (Pentium III 640 MHz). The networks were generated by using the mole fraction solubility of anthracene in pure solvents (X1 , X2 , X3 ), mole/volume fraction of solvents 1–3 (f1 , f2 , f3 ) in the mixture and solubility parameters of the solvents (δ1 , δ2 , δ3 ) as inputs. Solubility of anthracene in mixed solvents (Xm ) was the output. A three-layer network with sigmoidal transfer function in hidden and output layers was designed by using BP learning algorithm. The transfer function possesses minimum and maximum values of 0 and 1, respectively. The inputs and targets were normalized between 0.1 and 0.9, which allows the network to slightly exceed the minimum and maximum values that were given in the original data file [10]. The number of neurons in hidden layer was optimized by plotting the MSE versus the number of neurons in the hidden layer (as shown in Fig. 2) and six neurons were found as the best. Therefore, the optimum topologies of the networks were 6-6-1 and 9-6-1 for solubility in binary and ternary solvent mixtures, respectively. Number of training epoch was optimized and 2000 and 2500 epoch was selected for binary and ternary systems, respectively. The training process was stopped manually when the MSE of the test set remained constant after successive iteration. Since there are several local minima where the model could arrive, the algorithm ran from different starting values for initial weights to find the optimum, but nearly the same results were obtained. The training function used here was traingdm.
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where Mi is the model constant estimated by regressing ln Xm − f1 ln X1 −f2 ln X2 against f1 f2 (f1 − f2 )i terms using a no (zero) intercept least square analysis [12]. This equation requires a minimum number of data points (at least five points) to be able to predict the solubility at all solvent compositions [3]. To further reduce the number of data points, a relationship between the model constants, i.e. Mi , and solvents’ solubility parameters has been proposed as: Mi = K1i (δ1 − δs ) + K2i (δ2 − δs ) + K3i (δ1 − δs )2 Fig. 2. Variation of MSE vs. number of neurons in hidden layer for solubility in binary and ternary solvents.
Other training parameters used in this work were default values. The accuracy of the ANN method is compared with those of previously reported multiple linear regression models which is briefly reviewed here. The combined nearly ideal binary solvent/Redlich–Kister (CNIBS/R–K) equation was presented [11] to calculate solute solubility in binary solvents as: 2 ln Xm = f1 ln X1 + f2 ln X2 + f1 f2 Mi (f1 − f2 )i i=0
+ K4i (δ2 − δs )2
(5)
in which δs is the solute’s solubility parameter and K1i − K4i are the model constants. Using this relationship, the solubility in pure solvents 1 and 2 are the unknown values for predicting the solubility at all solvent compositions [3]. The CNIBS/R–K model was extended to calculate solubility in ternary solvents as: ln Xm = f1 ln X1 + f2 ln X2 + f3 ln X3 + f1 f2
n
Bi (f1 − f2 )i + f1 f3
i=0
+ f2 f3
(4)
n i=0
n i=0
Bi (f1 − f3 )i
Bi (f2 − f3 )i
(6)
Table 1 Details of solubility data, the absolute percentage mean deviation (APMD) and overall APMD (OPMD) for solubility of anthracene in binary solvents No.
Solvent 1
Solvent 2
δ1 a
δ2 a
APMDb
APMDc
APMDd
Ne
Ref.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Heptane Heptane Heptane Heptane Heptane Heptane 1-Propanol 2-Propanol 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane
2-Butoxyethanol 1-Propanol 2-Propanol 2-Butoxyethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 2-Butoxyethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 2-Butoxyethanol Cyclohexane 1-Propanol 2-Propanol 2-Butoxyethanol 2-Butoxyethanol 1-Butanol 2-Butanol 2-Butoxyethanol Cyclohexane 1-Propanol 2-Propanol
11.29 11.29 11.29 11.13 11.13 11.13 8.20 8.20 8.20 8.20 8.20 7.50 7.50 7.50 7.50 7.50 7.50 11.99 11.50 6.86 6.86 6.86 6.86 6.86 6.86 OPMD
9.88 11.99 11.50 9.88 11.99 11.50 11.29 11.13 9.88 11.99 11.50 11.29 11.13 9.88 8.20 11.99 11.50 9.88 9.88 11.29 11.13 9.88 8.20 11.99 11.50 0.16
0.08 0.24 0.20 0.27 0.19 0.14 0.13 0.10 0.09 0.07 0.11 0.09 0.21 0.14 0.05 0.38f 0.36 0.24 0.10 0.23 0.28 0.17 0.09 0.06 0.09 0.67
0.36 0.52 0.66 0.90 0.36 0.26 0.98 0.71 0.61 0.78 0.80 0.13 0.57 0.41 0.84 0.64 0.68 0.91 0.57 0.63 0.81 1.53f 0.59 0.58 0.88 1.83
1.76 3.05 3.08 3.43f 1.06 3.31 2.14 0.26 0.54 1.03 0.54 1.02 1.58 1.02 2.85 0.91 0.92 2.94 2.74 1.27 1.82 1.54 3.18 0.80 2.92
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 9 9 9 9 9 9 7 9 9
[13] [14] [14] [13] [14] [14] [15] [16] [17] [15] [18] [15] [16] [17] [19] [15] [18] [13] [13] [15] [16] [17] [19] [15] [18]
a b c d e f
δ1 and δ2 are the solubility parameters of solvents 1 and 2. APMD is calculated using back-calculated solubilities by trained models with all data points in each set. APMD is calculated using predicted solubilities by trained models with five data points in each set. APMD is calculated using predicted solubilities by trained model with 125 data points from all 25 sets. N is the number of reported experimental solubilities in each set. Maximum APMD values.
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where f3 stands for the mole/volume fraction of the solvent 3 in the mixed solvent, Bi , Bi and Bi are the sub-binary interaction terms [4]. In a recent work, the ternary interaction terms (Ti ) have been included in Eq. (6) as shown below [3] ln Xm = f1 ln X1 + f2 ln X2 + f3 ln X3 + f1 f2
n
Bi (f1 − f2 )i + f1 f3
i=0
+ f2 f3
n i=0
+ f1 f2 f3
n i=0
Bi (f1 − f3 )i
Bi (f2 − f3 )i
n
Ti (f1 − f2 − f3 )i
(7)
i=0
The goal of this work is to predict anthracene solubility in mixed solvents using ANNs. To provide an accuracy criterion comparable with experimentally obtained relative standard deviation, the absolute percentage mean deviation (APMD) has been calculated using Eq. (8): 100 (Xm )cal. − (Xm )obs. APMD = (8) N (Xm )obs. Overall APMD (OPMD) was used to compare the models and defined by Eq. (9): APMD OPMD = (9) number of data sets 3. Results and discussion All data points of anthracene solubilities in each binary solvent were employed to train the ANN models and the backcalculated solubilities were used to compute APMD. Details of solubility data and APMD values were shown in Table 1. Anthracene solubility in heptane + cyclohexane with 0.05 and heptane + 1-propanol with 0.38% produced the lowest and highest APMDs whereas OPMD for 25 binary sets studied is 0.16% (see column 6 in Table 1). The similar OPMD calculated in this work using Eq. (4), which is presented in a previous paper [4], is 0.24%. These two OPMDs are compared using paired t-test and the results (P < 0.03) showed that the ANN method is able to provide better correlation for solute solubility in binary solvents. Each binary solvent system includes nine data points (except in two cases) and divided to three subsets: training, test and validation. Overfitting is avoided by using two sets of samples; thus, weights were calculated from training set while the solubility of test set was being simultaneously predicted. In binary solvent systems the five solubility data points were selected with nearly constant volume/mole fraction intervals for training of the network and other data points (from test and validation sets) were predicted using trained network. Solubility in heptane + 1-butanol and 2,2,4-trimethylpentane + 2-butoxyethanol produced the lowest and highest APMD
Fig. 3. Plot of calculated vs. experimental solubility for binary mixed solvents for (a) training set and (b) prediction set. The training set includes a minimum number of three solubility data points in binary solvents and X1 –X2 values from each set and the solubility at other solvent compositions in each set were used as prediction set.
values respectively with 0.13 and 1.53%. The OPMD for ANN method from this work (see column 7 in Table 1) and the CNIBS/R–K model from a previous work [3] were 0.67 and 0.40%, respectively, and the difference was statistically significant (paired t-test, P < 0.0005). Fig. 3 shows the calculated solubilities by ANN method against experimental values and also coefficients of determination, slopes and intercepts for the linear relationships observed for training and prediction sets. All data points (N = 223) from 25 binary sets shown in Table 1, were used to train an ANN model. Then the trained network was employed to predict the solubility of anthracene in 16 solubility data sets listed in Table 2. Anthracene solubility in 1-butanol + dibutyl ether with 4.56 and 1-pentanol + 2-butoxyethanol with 28.35% showed the lowest and highest APMDs, respectively. The produced APMDs were slightly high and could not be considered within experimental uncertainty. The OPMD for 16 data sets was 15.32%. A similar numerical analysis has been reported for CNIBS/R–K equation with binary interaction terms calculated using Eq. (5) and the OPMD was 12.87%. There is no significant difference between two OPMDs, i.e. 15.32 and 12.87% (paired
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Table 2 Details of solubility data, the absolute percentage mean deviation (APMD) and overall APMD (OPMD) for solubility of anthracene in binary solvents No.
Solvent 1
Solvent 2
δ1 a
δ2 a
APMDb
Ref.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol 1-Octanol 1-Octanol 1-Octanol 1-Pentanol 1-Propanol 1-Propanol 1-Propanol 2-Propanol 2-Propanol 2-Propanol
Dibutyl ether 1,4-Dioxan 2-Propanol Dibutyl ether 1,4-Dioxan 2-Propanol 2-Butoxyethanol Dibutyl ether 1,4-Dioxan 2-Butoxyethanol 1,4-Dioxan 2-Propanol Dibutyl ether Dibutyl ether 1,4-Dioxan 2-Propanol
11.29 11.29 11.29 11.13 11.13 11.13 10.23 10.23 10.23 10.59 11.99 11.99 11.99 11.50 11.50 11.50
7.79 10.01 10.59 7.79 10.01 10.59 9.88 7.79 10.01 9.88 10.01 10.59 7.79 7.79 10.01 10.59 OPMD
4.56 13.28 22.31 7.22 6.52 20.73 25.76 19.12 24.86 28.35c 8.30 14.42 13.72 10.25 8.52 17.27 15.32
[20] [21] [21] [20] [21] [22] [13] [20] [21] [13] [21] [21] [20] [20] [21] [22]
a b c
δ1 and δ2 are the solubility parameters of solvents 1 and 2. APMD is calculated using the trained network with solubility data listed in Table 1. Maximum APMD value.
t-test, P > 0.93). Different solubility parameters for 16 data sets in comparison with 25 training data sets could be considered as a possible reason for such high APMD values for both numerical methods. It should be noted that the APMD obtained for this analysis is still less than other predictive methods reported in the literature. A minimum number of five data points including X1 and X2 from 25 binary sets (N = 125) were used to train an ANN model and solubility at other solvent compositions were predicted. The OPMD obtained for ANN model (see column 8 in Table 1) was 1.83% whereas the corresponding value was 6.83% for CNIBS/R–K model (when the model constants calculated using Eq. (5)). The OPMD difference was statistically significant (paired t-test, P < 0.0005). All data points from each solubility data in ternary solvents listed in Table 3 were used to train the network and the backcalculated solubilities were employed to calculate APMDs (see column 8 in Table 3). The APMDs varied between 0.08 (set number 9) and 0.31% (set number 11) and the OPMD was 0.20%. A similar OPMD for Eq. (7) was 0.38% [3] and the difference between these OPMDs was statistically significant (paired t-test, P < 0.0005). A minimum number of 11 data points from each ternary set and also X1 , X2 and X3 values were used to train the networks. Then the other eight points in each ternary solvent system were predicted using trained networks and the OPMD obtained was 0.27%. There was no significant difference between OPMD obtained from ANN model trained with all data points (0.20%) and by using a minimum number of data points (paired t-test, P > 0.07). This means that it is possible to use a smaller number of training points to train the ANN model and predict the solubility at other solvent compositions in ternary solvent systems with error less than experimental uncertainty. The calculated solubilities by ANN using a minimum number of data points in each ternary
Fig. 4. Plot of calculated vs. experimental solubility for ternary mixed solvents for (a) training set and (b) prediction set. The training set includes a minimum number of 11 solubility data points in ternary solvents and X1 –X3 values from each set and the solubility at other solvent compositions in each set were used as prediction set.
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Table 3 Details of solubility data, the absolute percentage mean deviation (APMD) and overall APMD (OPMD) for solubility of anthracene in ternary solvents Solvent 1
Solvent 2
Solvent 3
δ1 a
δ2 a
δ3 a
APMDb
APMDc
APMDd
Ref.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol
Heptane 2,2,4-Trimethylpentane Heptane 2,2,4-Trimethylpentane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Heptane Heptane 1-Propanol 1-Propanol 2-Propanol 2-Propanol 1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol Heptane 2,2,4-Trimethylpentane 1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol Heptane 2,2,4-Trimethylpentane
Cyclohexane Cyclohexane Cyclohexane Cyclohexane 1-Propanol 2-Propanol Heptane 2,2,4-Trimethylpentane 1-Propanol 2-Propanol 1-Butanol 2-Butanol 1-Butanol 2-Butanol Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Cyclohexane Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Cyclohexane
11.29 11.29 11.13 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 11.99 11.99 11.99 11.99 11.99 11.99 11.99 11.99 11.50 11.50 11.50 11.50 11.50 11.50 11.50 11.50 OPMD
7.50 6.86 7.50 6.86 8.20 8.20 8.20 8.20 7.50 7.50 11.99 11.99 11.50 11.50 11.29 11.29 11.29 11.13 11.13 11.13 7.50 6.86 11.29 11.29 11.29 11.13 11.13 11.13 7.50 6.86 0.20
8.20 8.20 8.20 8.20 11.99 11.50 7.50 6.86 11.99 11.50 11.29 11.13 11.29 11.13 8.20 7.50 6.86 8.20 7.50 6.86 8.20 8.20 7.30 7.50 6.86 7.30 7.50 6.86 7.30 7.30 0.27
0.09 0.20 0.25 0.20 0.24 0.11 0.27 0.17 0.08 0.18 0.31e 0.16 0.17 0.22 0.26 0.23 0.15 0.25 0.30 0.26 0.19 0.25 0.11 0.21 0.19 0.21 0.27 0.14 0.23 0.26 1.90
0.24 0.16 0.23 0.17 0.30 0.26 0.38 0.68 0.56 0.31 0.26 0.25 0.15 0.60 0.20 0.11 0.10 0.21 0.14 0.17 0.17 0.21 0.19 0.78e 0.12 0.16 0.17 0.10 0.34 0.40
1.48 2.04 1.27 1.15 1.63 1.15 1.92 1.24 1.87 3.27 1.48 2.68 3.37e 3.10 1.50 2.30 0.98 1.27 2.48 2.64 1.57 1.09 1.46 2.25 2.49 1.68 2.02 2.71 1.38 1.47
[23] [24] [23] [24] [25] [25] [26] [26] [25] [25] [27] [27] [27] [27] [28] [29] [30] [28] [29] [30] [23] [24] [28] [29] [30] [28] [29] [30] [23] [24]
a b c d e
δ1 –δ3 are the solubility parameters of solvents 1–3. APMD is calculated using back-calculated solubilities by trained models with all data points in each set. APMD is calculated using predicted solubilities by trained models with a minimum number of data points in each set. APMD is calculated using predicted solubilities by trained model with a minimum number of data points from 30 data set. Maximum APMD values.
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set against experimental values reported in the literature for training and prediction sets are shown in Fig. 4. As indicated by high values of coefficient of determinations, the ANN is able to provide accurate calculations. A minimum number of 11 data points from all 30 ternary data sets was employed to train a network (total training points is 330) and the trained network was used to calculate the solubilities at other ternary solvent compositions (total number of predicted points is 240). The obtained results were shown in column ten of Table 3. The OPMD is 1.90%, which was a comparable accuracy with experimental RSD value. In conclusion, the ANN method is generally provided more accurate calculations for the solubility of anthracene in non-aqueous binary and ternary solvents in comparison with the multiple linear regression models and could be used to model such data in industry. List of symbols ANN artificial neural network APMD absolute percentage mean deviation BP back-propagation CNIBS/R–K combined nearly ideal binary solvent/Redlich–Kister f1 , f2 , f3 mole/volume fraction of solvents 1, 2, 3 in the solvent mixture K1i , K2i , K3i , K4i model constants of Eq. (5) M number of neurons in output layer Mi , Bi , Bi , Bi , Bi , Ti model constants of CNIBS/R–K model MSE mean square error N number of data points in each set OPMD overall absolute percentage mean deviation Opm computed output P number of patterns RSD relative standard deviation Tpm target value Wij weight terms in Eq. (2) Xm mole fraction solubility of anthracene in mixed solvent X1 , X2 , X3 mole fraction solubility of anthracene in pure solvents 1, 2, 3 Greek letters α slope of the transfer function δs solubility parameter of anthracene δ1 , δ2 solubility parameter of solvents 1 and 2 bias term θi
Acknowledgments Financial supports of Drug Applied Research Centre, Tabriz University of Medical Sciences and Re-
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search Affair of Tabriz University are gratefully acknowledged.
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Solubility prediction of anthracene in binary and ternary solvents by artificial neural networks (ANNs) Abolghasem Jouybana,∗ , Mir-Reza Majidib , Farnaz Jabbaribarc , Karim Asadpour-Zeynalib a
School of Pharmacy and Drug Applied Research Centre, Tabriz University of Medical Sciences, Tabriz 51664, Iran b Department of Analytical Chemistry, Faculty of Chemistry, University of Tabriz, Tabriz 51664, Iran c Kimia Research Institute, P.O. Box 51665-171, Tabriz, Iran Received 9 August 2003; received in revised form 29 March 2004; accepted 20 August 2004
Abstract Solubility of anthracene in binary and ternary solvent systems was modeled by artificial neural networks (ANNs) technique. The results obtained using the ANN method indicated that the solubility of anthracene in mixed solvents could be calculated using the mole fraction solubilities in pure solvents, mole/volume fraction of solvents and solvent’s solubility parameters. The topology of neural network was optimized empirically and optimum topology was a 6-6-1 architecture for binary and 9-6-1 for ternary mixtures. The solubility of anthracene in mixed solvents was estimated by means of ANN and the predicted solubility was compared with experimental solubility data. The overall absolute percentage mean deviation (OPMD) for trained ANNs using all data points in 25 binary and 30 ternary solvent systems were 0.16 and 0.20%, respectively. A minimum number of data points from binary and ternary solvents have been employed to train the ANN and solubility at other solvent compositions has been predicted. The OPMD obtained for solubility in binary and ternary solvents were 0.67 and 0.27%, respectively. The trained network with 25 binary data sets was applied to predict the solubility in 16 other binary solvent systems and the OPMD obtained is 15.32%. The results of ANNs were also compared with similar numerical analyses carried out using multiple linear regression models and found that the ANN method is generally promising more accurate calculations. © 2004 Elsevier B.V. All rights reserved. Keywords: Artificial neural networks; Anthracene; Solubility; Prediction
1. Introduction The use of artificial neural networks (ANNs) in chemical and pharmaceutical areas has been increased recently [1]. The wide applicability of ANNs stems from their flexibility and ability to model linear and non-linear systems without prior knowledge of an empirical model [2]. This gives ANNs an advantage over traditional fitting methods for some chemical applications. The solubility is one of most important physicochemical properties of chemical/pharmaceutical compounds. ∗
Corresponding author. Tel.: +98 411 3392584; fax: +98 411 3344798. E-mail address: [email protected] (A. Jouyban).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.08.031
Solvent mixing (cosolvency) is a common method in chemical and pharmaceutical industries to enhance the solubility of a poorly soluble compound. Determination of solubility in all possible solvent compositions in mixed solvents by experiments is hardly possible and also time consuming, since there is an infinite number of solvent compositions. However, a number of mathematical methods have been used to predict the solubility in mixed solvents using a minimum number of experimental data points [3–5]. To the best of our knowledge, ANNs have not been employed for modeling solute solubility data in mixed solvents. In this work, a simple feed-forward artificial neural network is presented to model the solubility of anthracene in binary and ternary solvent mixtures. The ob-
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tained results from ANNs were compared with other methods based on least square analysis.
The ANNs are able to acquire information and provide models even when the information and data are complex, noise contaminated, non-linear or incomplete. The goal of ANN is to map a set of input patterns onto a corresponding set of output patterns. The network accomplishes this mapping by learning from a series of past examples and defining the input and output sets for a given system. The network then applies what it has been learned to a new input pattern to predict the appropriate output [6]. A network comprises an input layer, a number of hidden layers and an output layer (as schematically illustrated in Fig. 1). Many different types of ANNs have been developed [6,7]. The multilayer feed-forward network with backpropagation (BP) learning method is the most popular network in chemical applications [8,9] and is used in this work. The BP networks include one input layer, at least one hidden layer and one output layer. The number of neurons in the input and output layers are defined by system’s properties. The number of neurons in the hidden layer could be considered as an adjustable parameter, which should be optimized. The input layer receives the experimental information, experimental parameters, topological descriptors, etc. The output layer produces the calculated values of dependent variable. The mathematical relationship between inputs and outputs can be expressed as Eq. (1) for a multilayer network: 1 1 + e−α Netj
(2)
i=1
2. Computational methods
Outj =
the transfer function and Netj is given by Eq. (2): n Netj = Wij Outi + θj
(1)
where Outj is the transfer function as output from the jth neuron connected to n neuron in previous layer, α which is usually equal to one denotes the slope of the raising part of
Fig. 1. Schematic architecture of the ANN used for prediction of anthracene solubility in binary solvents, Xm is the solubility of anthracene in binary solvents X1 and X2 are the mole fraction solubilities in pure solvents 1 and 2, f1 and f2 denote the mole/volume fraction of solvents 1 and 2, δ1 and δ2 are solubility parameters of solvents 1 and 2, respectively.
where Outi is the output from ith neuron in previous layer, Wij represents the weights applied to the connection from ith to jth neuron and θ j is a bias term. In this work the numerical value of α is considered equal to 1. BP networks operate in the supervised learning mode. In the training step, known data is given to the network and using the BP algorithm, the network iteratively adjusts connection weights (Wij ) and biases (θ j ). Training of an ANN is usually done by starting with random connection weights. The computed output (Opm ) is compared to target value (Tpm ) and an error term (Tpm − Opm )2 is determined. The mean square error (MSE) is used as a criterion for finalizing the learning process and computed using Eq. (3): P M 1 (Tpm − Opm )2 MSE = P ×M
(3)
p=1 m=1
where M is the number of neurons in output layer and P denotes the number of patterns. The weights are adjusted in the learning stage for all the inter-connections and could be used to predict of output values of each data point in the prediction and test sets. All the ANNs calculations were carried out using Matlab mathematical software with artificial neural network toolbox for windows running on a personal computer (Pentium III 640 MHz). The networks were generated by using the mole fraction solubility of anthracene in pure solvents (X1 , X2 , X3 ), mole/volume fraction of solvents 1–3 (f1 , f2 , f3 ) in the mixture and solubility parameters of the solvents (δ1 , δ2 , δ3 ) as inputs. Solubility of anthracene in mixed solvents (Xm ) was the output. A three-layer network with sigmoidal transfer function in hidden and output layers was designed by using BP learning algorithm. The transfer function possesses minimum and maximum values of 0 and 1, respectively. The inputs and targets were normalized between 0.1 and 0.9, which allows the network to slightly exceed the minimum and maximum values that were given in the original data file [10]. The number of neurons in hidden layer was optimized by plotting the MSE versus the number of neurons in the hidden layer (as shown in Fig. 2) and six neurons were found as the best. Therefore, the optimum topologies of the networks were 6-6-1 and 9-6-1 for solubility in binary and ternary solvent mixtures, respectively. Number of training epoch was optimized and 2000 and 2500 epoch was selected for binary and ternary systems, respectively. The training process was stopped manually when the MSE of the test set remained constant after successive iteration. Since there are several local minima where the model could arrive, the algorithm ran from different starting values for initial weights to find the optimum, but nearly the same results were obtained. The training function used here was traingdm.
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135
where Mi is the model constant estimated by regressing ln Xm − f1 ln X1 −f2 ln X2 against f1 f2 (f1 − f2 )i terms using a no (zero) intercept least square analysis [12]. This equation requires a minimum number of data points (at least five points) to be able to predict the solubility at all solvent compositions [3]. To further reduce the number of data points, a relationship between the model constants, i.e. Mi , and solvents’ solubility parameters has been proposed as: Mi = K1i (δ1 − δs ) + K2i (δ2 − δs ) + K3i (δ1 − δs )2 Fig. 2. Variation of MSE vs. number of neurons in hidden layer for solubility in binary and ternary solvents.
Other training parameters used in this work were default values. The accuracy of the ANN method is compared with those of previously reported multiple linear regression models which is briefly reviewed here. The combined nearly ideal binary solvent/Redlich–Kister (CNIBS/R–K) equation was presented [11] to calculate solute solubility in binary solvents as: 2 ln Xm = f1 ln X1 + f2 ln X2 + f1 f2 Mi (f1 − f2 )i i=0
+ K4i (δ2 − δs )2
(5)
in which δs is the solute’s solubility parameter and K1i − K4i are the model constants. Using this relationship, the solubility in pure solvents 1 and 2 are the unknown values for predicting the solubility at all solvent compositions [3]. The CNIBS/R–K model was extended to calculate solubility in ternary solvents as: ln Xm = f1 ln X1 + f2 ln X2 + f3 ln X3 + f1 f2
n
Bi (f1 − f2 )i + f1 f3
i=0
+ f2 f3
(4)
n i=0
n i=0
Bi (f1 − f3 )i
Bi (f2 − f3 )i
(6)
Table 1 Details of solubility data, the absolute percentage mean deviation (APMD) and overall APMD (OPMD) for solubility of anthracene in binary solvents No.
Solvent 1
Solvent 2
δ1 a
δ2 a
APMDb
APMDc
APMDd
Ne
Ref.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Heptane Heptane Heptane Heptane Heptane Heptane 1-Propanol 2-Propanol 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane 2,2,4-Trimethylpentane
2-Butoxyethanol 1-Propanol 2-Propanol 2-Butoxyethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 2-Butoxyethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 2-Butoxyethanol Cyclohexane 1-Propanol 2-Propanol 2-Butoxyethanol 2-Butoxyethanol 1-Butanol 2-Butanol 2-Butoxyethanol Cyclohexane 1-Propanol 2-Propanol
11.29 11.29 11.29 11.13 11.13 11.13 8.20 8.20 8.20 8.20 8.20 7.50 7.50 7.50 7.50 7.50 7.50 11.99 11.50 6.86 6.86 6.86 6.86 6.86 6.86 OPMD
9.88 11.99 11.50 9.88 11.99 11.50 11.29 11.13 9.88 11.99 11.50 11.29 11.13 9.88 8.20 11.99 11.50 9.88 9.88 11.29 11.13 9.88 8.20 11.99 11.50 0.16
0.08 0.24 0.20 0.27 0.19 0.14 0.13 0.10 0.09 0.07 0.11 0.09 0.21 0.14 0.05 0.38f 0.36 0.24 0.10 0.23 0.28 0.17 0.09 0.06 0.09 0.67
0.36 0.52 0.66 0.90 0.36 0.26 0.98 0.71 0.61 0.78 0.80 0.13 0.57 0.41 0.84 0.64 0.68 0.91 0.57 0.63 0.81 1.53f 0.59 0.58 0.88 1.83
1.76 3.05 3.08 3.43f 1.06 3.31 2.14 0.26 0.54 1.03 0.54 1.02 1.58 1.02 2.85 0.91 0.92 2.94 2.74 1.27 1.82 1.54 3.18 0.80 2.92
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 9 9 9 9 9 9 7 9 9
[13] [14] [14] [13] [14] [14] [15] [16] [17] [15] [18] [15] [16] [17] [19] [15] [18] [13] [13] [15] [16] [17] [19] [15] [18]
a b c d e f
δ1 and δ2 are the solubility parameters of solvents 1 and 2. APMD is calculated using back-calculated solubilities by trained models with all data points in each set. APMD is calculated using predicted solubilities by trained models with five data points in each set. APMD is calculated using predicted solubilities by trained model with 125 data points from all 25 sets. N is the number of reported experimental solubilities in each set. Maximum APMD values.
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where f3 stands for the mole/volume fraction of the solvent 3 in the mixed solvent, Bi , Bi and Bi are the sub-binary interaction terms [4]. In a recent work, the ternary interaction terms (Ti ) have been included in Eq. (6) as shown below [3] ln Xm = f1 ln X1 + f2 ln X2 + f3 ln X3 + f1 f2
n
Bi (f1 − f2 )i + f1 f3
i=0
+ f2 f3
n i=0
+ f1 f2 f3
n i=0
Bi (f1 − f3 )i
Bi (f2 − f3 )i
n
Ti (f1 − f2 − f3 )i
(7)
i=0
The goal of this work is to predict anthracene solubility in mixed solvents using ANNs. To provide an accuracy criterion comparable with experimentally obtained relative standard deviation, the absolute percentage mean deviation (APMD) has been calculated using Eq. (8): 100 (Xm )cal. − (Xm )obs. APMD = (8) N (Xm )obs. Overall APMD (OPMD) was used to compare the models and defined by Eq. (9): APMD OPMD = (9) number of data sets 3. Results and discussion All data points of anthracene solubilities in each binary solvent were employed to train the ANN models and the backcalculated solubilities were used to compute APMD. Details of solubility data and APMD values were shown in Table 1. Anthracene solubility in heptane + cyclohexane with 0.05 and heptane + 1-propanol with 0.38% produced the lowest and highest APMDs whereas OPMD for 25 binary sets studied is 0.16% (see column 6 in Table 1). The similar OPMD calculated in this work using Eq. (4), which is presented in a previous paper [4], is 0.24%. These two OPMDs are compared using paired t-test and the results (P < 0.03) showed that the ANN method is able to provide better correlation for solute solubility in binary solvents. Each binary solvent system includes nine data points (except in two cases) and divided to three subsets: training, test and validation. Overfitting is avoided by using two sets of samples; thus, weights were calculated from training set while the solubility of test set was being simultaneously predicted. In binary solvent systems the five solubility data points were selected with nearly constant volume/mole fraction intervals for training of the network and other data points (from test and validation sets) were predicted using trained network. Solubility in heptane + 1-butanol and 2,2,4-trimethylpentane + 2-butoxyethanol produced the lowest and highest APMD
Fig. 3. Plot of calculated vs. experimental solubility for binary mixed solvents for (a) training set and (b) prediction set. The training set includes a minimum number of three solubility data points in binary solvents and X1 –X2 values from each set and the solubility at other solvent compositions in each set were used as prediction set.
values respectively with 0.13 and 1.53%. The OPMD for ANN method from this work (see column 7 in Table 1) and the CNIBS/R–K model from a previous work [3] were 0.67 and 0.40%, respectively, and the difference was statistically significant (paired t-test, P < 0.0005). Fig. 3 shows the calculated solubilities by ANN method against experimental values and also coefficients of determination, slopes and intercepts for the linear relationships observed for training and prediction sets. All data points (N = 223) from 25 binary sets shown in Table 1, were used to train an ANN model. Then the trained network was employed to predict the solubility of anthracene in 16 solubility data sets listed in Table 2. Anthracene solubility in 1-butanol + dibutyl ether with 4.56 and 1-pentanol + 2-butoxyethanol with 28.35% showed the lowest and highest APMDs, respectively. The produced APMDs were slightly high and could not be considered within experimental uncertainty. The OPMD for 16 data sets was 15.32%. A similar numerical analysis has been reported for CNIBS/R–K equation with binary interaction terms calculated using Eq. (5) and the OPMD was 12.87%. There is no significant difference between two OPMDs, i.e. 15.32 and 12.87% (paired
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Table 2 Details of solubility data, the absolute percentage mean deviation (APMD) and overall APMD (OPMD) for solubility of anthracene in binary solvents No.
Solvent 1
Solvent 2
δ1 a
δ2 a
APMDb
Ref.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol 1-Octanol 1-Octanol 1-Octanol 1-Pentanol 1-Propanol 1-Propanol 1-Propanol 2-Propanol 2-Propanol 2-Propanol
Dibutyl ether 1,4-Dioxan 2-Propanol Dibutyl ether 1,4-Dioxan 2-Propanol 2-Butoxyethanol Dibutyl ether 1,4-Dioxan 2-Butoxyethanol 1,4-Dioxan 2-Propanol Dibutyl ether Dibutyl ether 1,4-Dioxan 2-Propanol
11.29 11.29 11.29 11.13 11.13 11.13 10.23 10.23 10.23 10.59 11.99 11.99 11.99 11.50 11.50 11.50
7.79 10.01 10.59 7.79 10.01 10.59 9.88 7.79 10.01 9.88 10.01 10.59 7.79 7.79 10.01 10.59 OPMD
4.56 13.28 22.31 7.22 6.52 20.73 25.76 19.12 24.86 28.35c 8.30 14.42 13.72 10.25 8.52 17.27 15.32
[20] [21] [21] [20] [21] [22] [13] [20] [21] [13] [21] [21] [20] [20] [21] [22]
a b c
δ1 and δ2 are the solubility parameters of solvents 1 and 2. APMD is calculated using the trained network with solubility data listed in Table 1. Maximum APMD value.
t-test, P > 0.93). Different solubility parameters for 16 data sets in comparison with 25 training data sets could be considered as a possible reason for such high APMD values for both numerical methods. It should be noted that the APMD obtained for this analysis is still less than other predictive methods reported in the literature. A minimum number of five data points including X1 and X2 from 25 binary sets (N = 125) were used to train an ANN model and solubility at other solvent compositions were predicted. The OPMD obtained for ANN model (see column 8 in Table 1) was 1.83% whereas the corresponding value was 6.83% for CNIBS/R–K model (when the model constants calculated using Eq. (5)). The OPMD difference was statistically significant (paired t-test, P < 0.0005). All data points from each solubility data in ternary solvents listed in Table 3 were used to train the network and the backcalculated solubilities were employed to calculate APMDs (see column 8 in Table 3). The APMDs varied between 0.08 (set number 9) and 0.31% (set number 11) and the OPMD was 0.20%. A similar OPMD for Eq. (7) was 0.38% [3] and the difference between these OPMDs was statistically significant (paired t-test, P < 0.0005). A minimum number of 11 data points from each ternary set and also X1 , X2 and X3 values were used to train the networks. Then the other eight points in each ternary solvent system were predicted using trained networks and the OPMD obtained was 0.27%. There was no significant difference between OPMD obtained from ANN model trained with all data points (0.20%) and by using a minimum number of data points (paired t-test, P > 0.07). This means that it is possible to use a smaller number of training points to train the ANN model and predict the solubility at other solvent compositions in ternary solvent systems with error less than experimental uncertainty. The calculated solubilities by ANN using a minimum number of data points in each ternary
Fig. 4. Plot of calculated vs. experimental solubility for ternary mixed solvents for (a) training set and (b) prediction set. The training set includes a minimum number of 11 solubility data points in ternary solvents and X1 –X3 values from each set and the solubility at other solvent compositions in each set were used as prediction set.
138
Table 3 Details of solubility data, the absolute percentage mean deviation (APMD) and overall APMD (OPMD) for solubility of anthracene in ternary solvents Solvent 1
Solvent 2
Solvent 3
δ1 a
δ2 a
δ3 a
APMDb
APMDc
APMDd
Ref.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 2-Butoxyethanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 1-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol 2-Propanol
Heptane 2,2,4-Trimethylpentane Heptane 2,2,4-Trimethylpentane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Heptane Heptane 1-Propanol 1-Propanol 2-Propanol 2-Propanol 1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol Heptane 2,2,4-Trimethylpentane 1-Butanol 1-Butanol 1-Butanol 2-Butanol 2-Butanol 2-Butanol Heptane 2,2,4-Trimethylpentane
Cyclohexane Cyclohexane Cyclohexane Cyclohexane 1-Propanol 2-Propanol Heptane 2,2,4-Trimethylpentane 1-Propanol 2-Propanol 1-Butanol 2-Butanol 1-Butanol 2-Butanol Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Cyclohexane Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Heptane 2,2,4-Trimethylpentane Cyclohexane Cyclohexane
11.29 11.29 11.13 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 9.88 11.99 11.99 11.99 11.99 11.99 11.99 11.99 11.99 11.50 11.50 11.50 11.50 11.50 11.50 11.50 11.50 OPMD
7.50 6.86 7.50 6.86 8.20 8.20 8.20 8.20 7.50 7.50 11.99 11.99 11.50 11.50 11.29 11.29 11.29 11.13 11.13 11.13 7.50 6.86 11.29 11.29 11.29 11.13 11.13 11.13 7.50 6.86 0.20
8.20 8.20 8.20 8.20 11.99 11.50 7.50 6.86 11.99 11.50 11.29 11.13 11.29 11.13 8.20 7.50 6.86 8.20 7.50 6.86 8.20 8.20 7.30 7.50 6.86 7.30 7.50 6.86 7.30 7.30 0.27
0.09 0.20 0.25 0.20 0.24 0.11 0.27 0.17 0.08 0.18 0.31e 0.16 0.17 0.22 0.26 0.23 0.15 0.25 0.30 0.26 0.19 0.25 0.11 0.21 0.19 0.21 0.27 0.14 0.23 0.26 1.90
0.24 0.16 0.23 0.17 0.30 0.26 0.38 0.68 0.56 0.31 0.26 0.25 0.15 0.60 0.20 0.11 0.10 0.21 0.14 0.17 0.17 0.21 0.19 0.78e 0.12 0.16 0.17 0.10 0.34 0.40
1.48 2.04 1.27 1.15 1.63 1.15 1.92 1.24 1.87 3.27 1.48 2.68 3.37e 3.10 1.50 2.30 0.98 1.27 2.48 2.64 1.57 1.09 1.46 2.25 2.49 1.68 2.02 2.71 1.38 1.47
[23] [24] [23] [24] [25] [25] [26] [26] [25] [25] [27] [27] [27] [27] [28] [29] [30] [28] [29] [30] [23] [24] [28] [29] [30] [28] [29] [30] [23] [24]
a b c d e
δ1 –δ3 are the solubility parameters of solvents 1–3. APMD is calculated using back-calculated solubilities by trained models with all data points in each set. APMD is calculated using predicted solubilities by trained models with a minimum number of data points in each set. APMD is calculated using predicted solubilities by trained model with a minimum number of data points from 30 data set. Maximum APMD values.
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set against experimental values reported in the literature for training and prediction sets are shown in Fig. 4. As indicated by high values of coefficient of determinations, the ANN is able to provide accurate calculations. A minimum number of 11 data points from all 30 ternary data sets was employed to train a network (total training points is 330) and the trained network was used to calculate the solubilities at other ternary solvent compositions (total number of predicted points is 240). The obtained results were shown in column ten of Table 3. The OPMD is 1.90%, which was a comparable accuracy with experimental RSD value. In conclusion, the ANN method is generally provided more accurate calculations for the solubility of anthracene in non-aqueous binary and ternary solvents in comparison with the multiple linear regression models and could be used to model such data in industry. List of symbols ANN artificial neural network APMD absolute percentage mean deviation BP back-propagation CNIBS/R–K combined nearly ideal binary solvent/Redlich–Kister f1 , f2 , f3 mole/volume fraction of solvents 1, 2, 3 in the solvent mixture K1i , K2i , K3i , K4i model constants of Eq. (5) M number of neurons in output layer Mi , Bi , Bi , Bi , Bi , Ti model constants of CNIBS/R–K model MSE mean square error N number of data points in each set OPMD overall absolute percentage mean deviation Opm computed output P number of patterns RSD relative standard deviation Tpm target value Wij weight terms in Eq. (2) Xm mole fraction solubility of anthracene in mixed solvent X1 , X2 , X3 mole fraction solubility of anthracene in pure solvents 1, 2, 3 Greek letters α slope of the transfer function δs solubility parameter of anthracene δ1 , δ2 solubility parameter of solvents 1 and 2 bias term θi
Acknowledgments Financial supports of Drug Applied Research Centre, Tabriz University of Medical Sciences and Re-
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search Affair of Tabriz University are gratefully acknowledged.
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