Application of artificial neural network for prediction of particle size in pharmaceutical cocrystallization using mechanochemical synthesis

Abstract

Milling by mechanical means is vital unit operation in pharmaceutical processing which can be used for controlling particle size reduction. This approach can be also used for mechanochemical synthesis of pharmaceutical cocrystals. However, controlling the particle size of cocrystal during the milling process is challenging due to complexity of the process and different mechanisms involved. In this study, artificial neural network (ANN) approach was performed to predict the size of particles during mechanochemical synthesis of pharmaceutical cocrystals in ball milling operation. Theophylline was used as active pharmaceutical ingredient (API) and 4-aminobenzoic acid as co-former in the experiments. Different types of excipients including hydroxypropylmethylcellulose (HPMC), lactose, microcrystalline cellulose (MCC) polyvinylpyrrolidone (PVP) and polyethylene glycol (PEG) were used to see the effect of excipients on the cocrystals particle size. ANN was developed considering excipient type, excipient percentage, jar size, milling time as inputs, while median particle size (d50) was considered as the response, and representative of particle size. Two hidden layers were considered in developing ANN, and a combination of linear and nonlinear functions was used as transfer function. ANN was trained and validated with measured data, and R² greater than 0.99 was obtained for the training and validation, with RMSE values of 1.06 and 3.89 for training and validation, respectively. The results were used to provide a design space for understanding the cocrystal particle size variation during ball milling process. It was indicated that the largest particles were formed using PEG as excipient, and increase in particle size over the milling time. The latter was attributed to the electrostatic attraction forces between the particles, thus aggregation and agglomeration of particles as the milling time exceeds a threshold. Furthermore, larger particles were obtained with the larger jar size. It turned out that faster cocrystallization rate occurred after only 5 min of milling in 25 mL jar compared to 25 min in 10 mL milling jar. The developed methodology has been shown to be robust and can be used as predictive tool for designing pharmaceutical cocrystallization using ball milling, and controlling the particle size during size reduction process.

Appendix

1.1 The fitting results of ANN using Tansig transfer function

The training, testing and validation results for the Tansig transfer function are shown in Fig. 16.

Fig. 16

Training, testing and validation regression for the best ANN run using Hyperbolic tangent sigmoid transfer function (Tansig) as transfer function

Also, MSE (mean squared error) versus Epoch is plotted and represented in Fig. 17 for Tansig transfer function. The comparisons between the predicted and measured data in terms of d50 are indicated in Fig. 18 for the case of Tansig transfer function.

Fig. 17

The best performance of learning process using hyperbolic tangent sigmoid (Tansig) as transfer function

Fig. 18

Comparison of predicted d50 as output of ANN with initial data (pharmaceutical co-crystallization) using hyperbolic tangent sigmoid (Tansig) as transfer function

1.2 The fitting results of ANN using Logsig transfer function

The training, testing and validation results for the Logsig transfer function are shown in Fig. 19. Also, MSE versus Epoch is plotted and represented in Fig. 20 for Logsig transfer function. The comparisons between the predicted and measured data in terms of d50 are indicated in Fig. 21 for the case of Logsig transfer function.

Fig. 19

Training, testing and validation regression for the best ANN run using log-sigmoid transfer function (Logsig)

Fig. 20

The best performance of learning process using log-sigmoid transfer function (Logsig)

Fig. 21

Comparisons of predicted d50 as output of ANN with initial data (pharmaceutical co-crystallization) using log-sigmoid transfer function (Logsig)

1.3 The fitting results of ANN using Softmax transfer function

The training, testing and validation results for the Softmax transfer function are shown in Fig. 22. Also, MSE versus Epoch is plotted and represented in Fig. 23 for Softmax transfer function. The comparisons between the predicted and measured data in terms of d50 are indicated in Fig. 24 for the case of Softmax transfer function.

Fig. 22

Training, testing and validation regression for the best ANN run using Soft max transfer function (Softmax)

Fig. 23

The best performance of learning process using Soft max transfer function (Softmax)

Fig. 24

Comparisons of predicted d50 as output of ANN with initial data (pharmaceutical co-crystallization) using Soft max transfer function (Softmax)

The summary of ANN models is listed in Table 3.

Table 3 Information of ANN considering equal in evaluation of changes in transfer function

1.4 Simulation using SPOCU transfer function

Finally, SPOCU (scaled polynomial constant unit activation function) which was developed by Kiselak et al. [37] was used in the ANN simulations. It was indicated to perform satisfactorily on a variety of problems because of genuine normalization of the output of layers. The SPOCU activation function is written as [37]:

$$s\left( x \right) = \alpha h\left( {\frac{x}{\gamma } + \beta } \right) - \alpha h\left( \beta \right)$$

(2)

where $\beta \in \left( {0,1} \right), \alpha , \gamma > 0$ and:

$$h\left( x \right) = \left\{ {\begin{array}{*{20}l} {r\left( c \right),} \hfill & {\quad x \ge c,} \hfill \\ {r\left( x \right),} \hfill & {\quad x \in \left[ {0,c} \right),} \hfill \\ {0,} \hfill & {\quad x < 0,} \hfill \\ \end{array} } \right.$$

(3)

with $r\left( x \right) = x^{3} \left( {x^{5} - 2x^{4} + 2} \right)$ and $1 \le c < \infty$.

A code was written in MATLAB for implementing SPOCU transfer function which can be found in the Supplementary file. The results of ANN simulation for the test data are shown in Fig. 25. Also, the MSE results versus iterations are provided in Fig. 26. The SPOCU function was implemented on a network with the properties as reported in Table 4.

Table 4 Parameters of the neural network used for applying SPOCU activation function

Fig. 25

Comparisons of test data with measured data using SPOCU activation function

Fig. 26

MSE calculated using SPOCU activation function

A comparison between various activation functions in terms of R² for the testing/validation dataset is listed in Table 5.

Table 5 Comparisons between different activation functions