Skip to main content

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Baldi Antognini A, Zagoraiou M. Exact optimal designs for computer experiments via kriging metamodelling. J Stat Plan Infer. 2010;140(9):2607–17.

    Article  Google Scholar 

  2. Banerjee I, Ierapetritou MG. Design optimization under parameter uncertainty for general black-box models. Ind Eng Chem Res. 2002;41(26):6687–97.

    Article  CAS  Google Scholar 

  3. Banerjee I, Ierapetritou MG. Feasibility evaluation of nonconvex systems using shape reconstruction techniques. Ind Eng Chem Res. 2005;44(10):3638–47.

    Article  CAS  Google Scholar 

  4. Banerjee I, Pal S, et al. Computationally efficient black-box modeling for feasibility analysis. Comput Chem Eng. 2010;34(9):1515–21.

    Article  CAS  Google Scholar 

  5. Bertsimas D, Nohadani O, et al. Robust optimization for unconstrained simulation-based problems. Oper Res. 2010;58(1):161–78.

    Article  Google Scholar 

  6. Biegler LT, Grossmann IE, et al. Systematic methods of chemical process design. Upper Saddle River: Prentice Hall; 1997.

    Google Scholar 

  7. Booker AJ, Dennis JE, et al. A rigorous framework for optimization of expensive functions by surrogates. Struct Multidiscip Optim. 1999;17(1):1–13.

    Article  Google Scholar 

  8. Boukouvala F, Dubey A, et al. Computational approaches for studying the granular dynamics of continuous blending processes, 2-population balance and data-based methods. Macromol Mater Eng. 2012;297(1):9–19.

    Article  CAS  Google Scholar 

  9. Boukouvala F, Ierapetritou MG. Feasibility analysis of black-box processes using an adaptive sampling kriging-based method. Comput Chem Eng. 2012;36:358–68.

    Article  CAS  Google Scholar 

  10. Boukouvala F, Niotis V, et al. An integrated approach for dynamic flowsheet modeling and sensitivity analysis of a continuous tablet manufacturing process. Comput Chem Eng. 2012;42:30–47.

    Article  CAS  Google Scholar 

  11. Boukouvala F, et al. Computer aided design and analysis of continuous pharmaceutical manufacturing processes, In: Computer aided chemical engineering, Elsevier. 2011;29:216–220.

  12. Box GEP, Wilson KB. On the experimental attainment of optimum conditions. J R Stat Soc B (Methodological). 1951;13(1):1–45.

    Google Scholar 

  13. Caballero JA, Grossmann IE. An algorithm for the use of surrogate models in modular flowsheet optimization. AICHE J. 2008;54(10):2633–50.

    Google Scholar 

  14. Crary SB. Design of computer experiments for metamodel generation. Analog Integr Circ Sig Process. 2002;32(1):7–16.

    Article  Google Scholar 

  15. Cressie, N. (1993). Statistics for Spatial Data (Wiley Series in Probability and Statistics), Wiley-Interscience

  16. Davis E, Ierapetritou M. A kriging method for the solution of nonlinear programs with black-box functions. AICHE J. 2007;53(8):2001–12.

    Article  CAS  Google Scholar 

  17. Davis E, Ierapetritou M. A kriging-based approach to MINLP containing black-box models and noise. Ind Eng Chem Res. 2008;47(16):6101–25.

    Article  CAS  Google Scholar 

  18. Davis E, Ierapetritou M. A kriging based method for the solution of mixed-integer nonlinear programs containing black-box functions. J Glob Optim. 2009;43(2–3):191–205.

    Article  Google Scholar 

  19. Davis E, Ierapetritou M. A centroid-based sampling strategy for kriging global modeling and optimization. AICHE J. 2010;56(1):220–40.

    CAS  Google Scholar 

  20. Dec RT, Zavaliangos A, et al. Comparison of various modeling methods for analysis of powder compaction in roller press. Powder Technol. 2003;130(1–3):265–71.

    Article  CAS  Google Scholar 

  21. Engisch, W. and F. J. Muzzio (2010). Hopper refill of loss-in-weight feeding equipment. AIChE Annual Conference, Salt Lake City.

  22. Forrester AIJ, Sóbester A. et al. Engineering design via surrogate modeling—a practical guide. New York: Wiley; 2008.

  23. Fowler K, Jenkins E, et al. Understanding the effects of polymer extrusion filter layering configurations using simulation-based optimization. Optim Eng. 2010;11(2):339–54.

    Article  Google Scholar 

  24. Fu MC. Feature article: optimization for simulation: theory vs. practice. INFORMS J Comput. 2002;14(3):192–215.

    Article  Google Scholar 

  25. Gernaey KV, Cervera-Padrell AE, et al. A perspective on PSE in pharmaceutical process development and innovation. Comput Chem Eng. 2012;42:15–29.

    Article  CAS  Google Scholar 

  26. Gernaey KV, Gani R. A model-based systems approach to pharmaceutical product-process design and analysis. Chem Eng Sci. 2010;65(21):5757–69.

    Article  CAS  Google Scholar 

  27. Gruhn G, Werther J, et al. Flowsheeting of solids processes for energy saving and pollution reduction. J Clean Prod. 2004;12(2):147–51.

    Article  Google Scholar 

  28. Halemane KP, Grossmann IE. Optimal process design under uncertainty. AICHE J. 1983;29(3):425–33.

    Article  CAS  Google Scholar 

  29. Heckel RW. Density–pressure relationships in powder compaction. Trans Metall Soc AIME. 1961;221(4):671–5.

    CAS  Google Scholar 

  30. Horowitz B, Guimarães LJDN, et al. A concurrent efficient global optimization algorithm applied to polymer injection strategies. J Pet Sci Eng. 2010;71(3–4):195–204.

    Article  CAS  Google Scholar 

  31. Huang D. Experimental planning and sequential kriging optimization using variable fidelity data, in Industrial and Systems Engineering. 2005, Ohio State University: Ohio.

  32. Huang D, Allen T, et al. Global optimization of stochastic black-box systems via sequential kriging meta-models. J Glob Optim. 2006;34(3):441–66.

    Article  Google Scholar 

  33. Husain A, Kim K-Y. Enhanced multi-objective optimization of a microchannel heat sink through evolutionary algorithm coupled with multiple surrogate models. Appl Therm Eng. 2010;30(13):1683–91.

    Article  Google Scholar 

  34. Jakobsson S, Patriksson M, et al. A method for simulation based optimization using radial basis functions. Optim Eng. 2010;11(4):501–32.

    Article  Google Scholar 

  35. Jones DR. A taxonomy of global optimization methods based on response surfaces. J Glob Optim. 2001;21(4):345–83.

    Article  Google Scholar 

  36. Jones DR, Schonlau M, et al. Efficient global optimization of expensive black-box functions. J Glob Optim. 1998;13(4):455–92.

    Article  Google Scholar 

  37. Kleijnen JPC. Kriging metamodeling in simulation: a review. Eur J Oper Res. 2009;192(3):707–16.

    Article  Google Scholar 

  38. Leuenberger H. New trends in the production of pharmaceutical granules: batch versus continuous processing. Eur J Pharm Biopharm. 2001;52(3):289–96.

    PubMed  Article  CAS  Google Scholar 

  39. McKenzie P, Kiang S, et al. Can pharmaceutical process development become high tech? AICHE J. 2006;52(12):3990–4.

    Article  CAS  Google Scholar 

  40. Ng KM. Design and development of solids processes—a process systems engineering perspective. Powder Technol. 2002;126(3):205–10.

    Article  CAS  Google Scholar 

  41. Nunnally BK, McConnell JS. Six sigma in the pharmaceutical industry: understanding, reducing, and controlling variation in pharmaceuticals and biologics. Boca Raton: CRC; 2007.

    Google Scholar 

  42. Pedone P, Romano D, Vicario G. New Sampling Procedures in coordinate metrology based on kriging-based adaptive designs. Statistics for Innovation. 2009;103–21.

  43. Pistone G, Vicario G. Design for computer experiments: comparing and generating designs in kriging models. Statistics for Innovation. 2009;91–102.

  44. Plumb K. Continuous processing in the pharmaceutical industry: changing the mind set. Chem Eng Res Des. 2005;83(6):730–8.

    Article  CAS  Google Scholar 

  45. Queipo NV, Haftka RT, et al. Surrogate-based analysis and optimization. Prog Aerosp Sci. 2005;41(1):1–28.

    Article  Google Scholar 

  46. Regis RG. Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions. Comput Oper Res. 2011;38(5):837–53.

    Article  Google Scholar 

  47. Sacks J, Welch WJ, et al. Design and analysis of computer experiments. Stat Sci. 1989;4(4):409–23.

    Article  Google Scholar 

  48. Sankaran S, Audet C, et al. A method for stochastic constrained optimization using derivative-free surrogate pattern search and collocation. J Comput Phys. 2010;229(12):4664–82.

    Article  CAS  Google Scholar 

  49. Schaber SD, Gerogiorgis DI, et al. Economic analysis of integrated continuous and batch pharmaceutical manufacturing: a case study. Ind Eng Chem Res. 2011;50(17):10083–92.

    Article  CAS  Google Scholar 

  50. Schonlau M, Welch W. Screening the input variables to a computer model via analysis of variance and visualization. Screening. 2006;308–27.

  51. Seider WD. Product and process design principles: synthesis, analysis, and evaluation. Hoboken: Wiley; 2009.

    Google Scholar 

  52. Sen M, Ramachandran R. A multi-dimensional population balance model approach to continuous powder mixing processes. Adv Powder Technol. 2013;24(1):51–9.

    Article  Google Scholar 

  53. Sen M, Singh R, et al. Multi-dimensional population balance modeling and experimental validation of continuous powder mixing processes. Chem Eng Sci. 2012;80:349–60.

    Article  CAS  Google Scholar 

  54. Soh JLP, Wang F, et al. Utility of multivariate analysis in modeling the effects of raw material properties and operating parameters on granule and ribbon properties prepared in roller compaction. Drug Dev Ind Pharm. 2008;34(10):1022–35.

    PubMed  Article  CAS  Google Scholar 

  55. Stephanopoulos G, Reklaitis GV. Process systems engineering: from Solvay to modern bio- and nanotechnology. A history of development, successes and prospects for the future. Chem Eng Sci. 2011;66(19):4272–306.

    CAS  Google Scholar 

  56. Vanarase AU, Alcala M, et al. Real-time monitoring of drug concentration in a continuous powder mixing process using NIR spectroscopy. Chem Eng Sci. 2010;65(21):5728–33.

    Article  CAS  Google Scholar 

  57. Villemonteix J, Vazquez E, et al. Global optimization of expensive-to-evaluate functions: an empirical comparison of two sampling criteria. J Glob Optim. 2009;43(2):373–89.

    Article  Google Scholar 

  58. Villemonteix J, Vazquez E, et al. An informational approach to the global optimization of expensive-to-evaluate functions. J Glob Optim. 2009;44(4):509–34.

    Article  Google Scholar 

  59. Wan X, Pekny JF, et al. Simulation-based optimization with surrogate models—application to supply chain management. Comput Chem Eng. 2005;29(6):1317–28.

    Article  CAS  Google Scholar 

  60. Werther J, Reimers C, et al. Flowsheet simulation of solids processes—data reconciliation and adjustment of model parameters. Chem Eng Process. 2008;47(1):138–58.

    Article  Google Scholar 

  61. Werther J, Reimers C, et al. Design specifications in the flowsheet simulation of complex solids processes. Powder Technol. 2009;191(3):260–71.

    Article  Google Scholar 

  62. Yin J, Ng SH, et al. Kriging metamodel with modified nugget-effect: the heteroscedastic variance case. Comput Ind Eng. 2011;61(3):760–77.

    Article  Google Scholar 

  63. Yu L. Pharmaceutical quality by design: product and process development, understanding, and control. Pharm Res. 2008;25(4):781–91.

    PubMed  Article  CAS  Google Scholar 

  64. Yuan J, Wang K, et al. Reliable multi-objective optimization of high-speed WEDM process based on Gaussian process regression. Int J Mach Tools Manuf. 2008;48(1):47–60.

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to acknowledge the funding provided by the ERC (NSF-0504497, NSF-ECC 0540855).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marianthi G. Ierapetritou.

Appendix

Appendix

  1. 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)
  2. 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)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Boukouvala, F., Ierapetritou, M.G. Surrogate-Based Optimization of Expensive Flowsheet Modeling for Continuous Pharmaceutical Manufacturing. J Pharm Innov 8, 131–145 (2013). https://doi.org/10.1007/s12247-013-9154-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12247-013-9154-1

Keywords

  • Surrogate-based optimization
  • Simulation-based optimization
  • Kriging
  • Pharmaceutical manufacturing
  • Flowsheet simulation