Surrogate-Based Optimization of Expensive Flowsheet Modeling for Continuous Pharmaceutical Manufacturing

Abstract

Simulation-based optimization is a research area that is currently attracting a lot of attention in many industrial applications, where expensive simulators are used to approximate, design, and optimize real systems. Pharmaceuticals are typical examples of high-cost products which involve expensive processes and raw materials while at the same time must satisfy strict quality regulatory specifications, leading to the formulation of challenging and expensive optimization problems. The main purpose of this work was to develop an efficient strategy for simulation-based design and optimization using surrogates for a pharmaceutical tablet manufacturing process. The proposed approach features surrogate-based optimization using kriging response surface modeling combined with black-box feasibility analysis in order to solve constrained and noisy optimization problems in less computational time. The proposed methodology is used to optimize a direct compaction tablet manufacturing process, where the objective is the minimization of the variability of the final product properties while the constraints ensure that process operation and product quality are within the predefined ranges set by the Food and Drug Administration.

Appendix

• A. Deterministic Kriging Modeling

  For a set of m observations for an n-dimensional space, the collected samples are X = {x ⁽¹⁾,…,x ^(m)} ^T and the response measured at these points is Y = {Y(x ⁽¹⁾),…,Y(x ^(m))} ^T.

  Consider the correlation matrix between all observed data points (m).

  $$ \mathbf{R}=\left[ {\begin{array}{*{20}c} {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(1) }},{{\mathbf{x}}^{(1) }}} \right)} \right]} & \ldots & {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(1) }},{{\mathbf{x}}^{(m) }}} \right)} \right]} \\ \vdots & \ddots & \vdots \\ {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(m) }},{{\mathbf{x}}^{(1) }}} \right)} \right]} & \ldots & {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(m) }},{{\mathbf{x}}^{(m) }}} \right)} \right]} \\ \end{array}} \right]=\left[ {\begin{array}{*{20}c} 1 & \ldots & {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(1) }},{{\mathbf{x}}^{(m) }}} \right)} \right]} \\ \vdots & \ddots & \vdots \\ {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(m) }},{{\mathbf{x}}^{(1) }}} \right)} \right]} & \ldots & 1 \\ \end{array}} \right] $$

  By definition, the covariance matrix will be equal to

  $$ \operatorname{Cov}\left( {\mathbf{Y,Y}} \right)={\sigma^2}\mathbf{R} $$

  (11)

  where σ ² is the standard deviation of the data. Any two points are correlated based on the chosen basis function (i.e., Eq. 12)

  $$ \operatorname{Cor}\left[ {\mathbf{Y} \left( {{{\mathbf{x}}^{(i) }}} \right),\mathbf{Y} \left( {{{\mathbf{x}}^{(j) }}} \right)} \right]=\exp \left( {-\sum\limits_{k=1}^n {{\theta_k}} {{{\left| {x_k^{(i) }-x_k^{(j) }} \right|}}^{{{p_k}}}}} \right) $$

  (12)

  Based on a set of observed data, in order to build a kriging model, it is required to minimize the error between the observed response Y and the predicted kriging response. This can be expressed as maximizing the likelihood of Y, which is given in Eq. 13.

  $$ L=\frac{1}{{{{{\left( {2\pi {\sigma^2}} \right)}}^{{{m \left/ {2} \right.}}}}\sqrt{{\left| \mathbf{R} \right|}}}}\exp \left[ {-\frac{{{{{\left( {\mathbf{y}-\mathbf{1}\mu } \right)}}^T}{{\mathbf{R}}^{-1 }}\left( {\mathbf{y}-\mathbf{1}\mu } \right)}}{{2{\sigma^2}}}} \right] $$

  (13)

  which gives the solution of the maximum-likelihood estimators of the mean ($\widehat{\mu}$) and variance (${{\widehat{\sigma}}^2}$) of the observed data for the optimum parameters θ and p.

  $$ \begin{array}{*{20}c} {\hat{\mu}=\frac{{{{\mathbf{1}}^T}{{\mathbf{R}}^{-1 }}\mathbf{y}}}{{{{\mathbf{1}}^T}{{\mathbf{R}}^{-1 }}\mathbf{1}}}} \hfill \\ {{{\hat{\sigma}}^2}=\frac{{{{{\left( {\mathbf{y}-\mathbf{1}\mu } \right)}}^T}{{\mathbf{R}}^{-1 }}\left( {\mathbf{y}-\mathbf{1}\mu } \right)}}{m}} \hfill \\ \end{array} $$

  (14)

  For any new point, the objective is to maximize the likelihood of the sampled data and the prediction, given the parameters obtained by the model construction step. For this purpose, the correlation matrix is augmented by the correlation between the sampled points and the new point (r), which has unknown Y.

  $$ {{\mathbf{R}}^{{\left( {\operatorname{aug}} \right)}}}=\left[ {\begin{array}{*{20}c} \mathbf{R} \hfill & \mathbf{r} \hfill \\ {{{\mathbf{r}}^T}} \hfill & 1 \hfill \\ \end{array}} \right] $$

  (15)

  where

  $$ \mathbf{r}=\left[ {\begin{array}{*{20}c} {\operatorname{cor}\left( {Y\left( {{{\mathbf{x}}^{(1) }}} \right),Y\left( {{{\mathbf{x}}^{{\left( {\operatorname{new}} \right)}}}} \right)} \right)} \\ \vdots \\ {\operatorname{cor}\left( {Y\left( {{{\mathbf{x}}^{(m) }}} \right),Y\left( {{{\mathbf{x}}^{{\left( {\operatorname{new}} \right)}}}} \right)} \right)} \\ \end{array}} \right] $$

  Maximizing the likelihood of the augmented data leads to the solution of Eq. 1.

  $$ \widehat{y}\left( {{{\mathbf{x}}^{{\left( {\operatorname{new}} \right)}}}} \right)=\widehat{\mu}+{{\mathbf{r}}^T}{{\mathbf{R}}^{-1 }}\left( {\mathbf{y}-1\widehat{\mu}} \right) $$

  (16)

• B. Modified Kriging for Noisy Data

  In the case where it is not desired to purely interpolate the experimental data, a nugget effect parameter (w) is added to the diagonal of R, such that in the case where the distance between two points approaches zero, the correlation is no longer equal to 1 (Eq. 17).

  $$ \left| {{{\mathbf{x}}^{(i) }}-{{\mathbf{x}}^{{\left( {\operatorname{new}} \right)}}}} \right|\to 0\Rightarrow \operatorname{cor}\left( {{{\mathbf{x}}^{(i) }},{{\mathbf{x}}^{{\left( {\operatorname{new}} \right)}}}} \right)\to 1+w $$

  (17)
  $$ {{\mathbf{R}}^{{(\bmod )}}}=\mathbf{R}+w\mathbf{I}=\left[ {\begin{array}{*{20}c} {1+w} & \ldots & {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(1) }},{{\mathbf{x}}^{(m) }}} \right)} \right]} \\ \vdots & \ddots & \vdots \\ {\operatorname{cor}\left[ {Y\left( {{{\mathbf{x}}^{(m) }},{{\mathbf{x}}^{(1) }}} \right)} \right]} & \ldots & {1+w} \\ \end{array}} \right] $$

  (18)

  Employing the same exact procedure to obtain the kriging prediction leads to the y ^(mod).

  $$ \begin{array}{*{20}c} {{{\widehat{y}}^{{\left( {\bmod } \right)}}}={{\widehat{\mu}}^{{\left( {\bmod } \right)}}}+{{\mathbf{r}}^T}\left( {\mathbf{R}+w\mathbf{I}} \right)\left( {\mathbf{y}-1{{\widehat{\mu}}^{{\left( {\bmod } \right)}}}} \right)} \hfill \\ {\operatorname{where}} \hfill \\ {{{\widehat{\mu}}^{{\left( {\bmod } \right)}}}=\frac{{{1^T}{{{\left( {\mathbf{R}+w\mathbf{I}} \right)}}^{-1 }}\mathbf{y}}}{{{1^T}{{{\left( {\mathbf{R}+w\mathbf{I}} \right)}}^{-1 }}1}}} \hfill \\ \end{array} $$

  (19)