Skip to main content

Advertisement

SpringerLink
Log in

A Computationally Efficient Surrogate-Based Reduction of a Multiscale Comill Process Model

  • Original Article
  • Published:
Journal of Pharmaceutical Innovation Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Purpose

Particle breakage in milling operations is often modeled using population balance models (PBMs). A discrete element method (DEM) model can be coupled with a PBM in order to explicitly identify the effect of material properties on breakage rate. However, the DEM-PBM framework is computationally expensive to evaluate due to high-fidelity DEM simulations. This limits its application in continuous process modeling for dynamic simulation, optimization, or control purposes.

Methods

The current work proposes the use of surrogate modeling (SM) techniques to map mechanistic data obtained from DEM simulations as a function of processing conditions. To demonstrate the benefit of the SM-PBM approach in developing integrated process models for continuous pharmaceutical manufacturing, a comill-tablet press model integration utilizing the proposed framework is presented.

Results

The SM-PBM approach is in excellent agreement with the DEM-PBM approach to predict particle size distributions (PSDs) and dynamic holdup, with a maximum sum of square errors of 0.0012 for PSD in volume fraction and 0.93 for holdup in grams. In addition, the time taken to run a DEM simulation is in the order of days whereas the proposed hybrid model takes few seconds. The SM-PBM approach also enables comill-tablet press model integration to predict tablet properties such as weight and hardness.

Conclusions

The proposed hybrid framework compares well with a DEM-PBM framework and addresses limitations on computational expense, thus enabling its use in continuous process modeling.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Abbreviations

E :

specific energy, J/kg

F :

force, N

v :

velocity, m/s

M(x, t):

particle mass of size x and at time t, kg

ℝ form :

rate of formation of particles, kg/s

ℝ dep :

rate of depletion of particles, kg/s

\( {\dot{M}}_{in} \) :

rate of particles entering the mill, kg/s

\( {\dot{M}}_{out} \) :

rate of particles exiting the mill, kg/s

b :

breakage distribution function, dimensionless

u :

daughter particle volume, mm3

w :

parent particle volume, mm3

n :

breakage distribution function parameter, dimensionless

σ :

log normal distribution standard deviation, mm3

C :

breakage distribution function parameter for mass conservation, mm6

K :

breakage kernel, 1/s

f mat :

material strength parameter, kg/J m

E const :

size-independent threshold energy, J m/kg

f coll :

collision frequency per particle per second, 1/s

E min :

threshold energy, J/kg

f d :

size-dependent parameter to define screen model, dimensionless

Δ :

critical screen size, μm

δ :

critical screen size ratio

d screen :

screen size, μm

v imp :

impeller speed, rpm

v imp, min :

minimum impeller speed, rpm

ε :

critical size ratio coefficient, dimensionless

α :

critical size ratio exponent, dimensionless

Δt :

time interval, s

\( \widehat{f}\left({x}^i\right) \) :

kriging model predictor for a d dimensional input xi

\( \widehat{\mu}\left({x}^i\right) \) :

regression term in the kriging model predictor

\( \widehat{\varepsilon}\left({x}^i\right) \) :

error term in the kriging model predictor

R :

correlation model in kriging, size mxm

θ :

vector of kriging correlation model parameters [θ1, θ2, θ3, θ4]

β :

kriging regression model parameter, constant

γ :

kriging model correlation factors, size mx1

σ krig :

kriging model variance parameter

Sx :

kriging model scaling factors for design sites, size 2 × 3

Sy :

kriging model scaling factors for responses, size 2 × 1

x new :

vector representing untried point, size 3 × 1

x scale,new :

scaled vector representing untried point, size 3 × 1

b I → H :

ANN model biases, input to hidden layer, size 10 × 1

w I → H :

ANN model weights, input to hidden layer, size 10 × 3

b H → O :

ANN model bias, hidden to output layer, size 1 × 1

w H → O :

ANN model weight, hidden to output layer, size 10 × 1

L :

maximum likelihood function

SSE PSD :

sum of square errors of steady state particle size distribution defined by ns bins

SSE holdup :

sum of square errors of dynamic holdup computed for pre-defined time intervals

ρ bulk :

granule bulk density, kg/m3

a i :

bulk density regression model parameters, i = 0,1,2,3,4

ρ true :

granule true density, kg/m3

ρ r :

relative density of material in the die, kg/m3

ρ r,cr :

critical relative density of material in the die, dimensionless

D tablet :

diameter of the tablet, m

L tablet :

length of the tablet, m

FD :

fill depth in the die, m

H :

tablet hardness, N

H max :

maximum tablet hardness, N

References

  1. Administration, F.a.D. Guidance for Industry, Q8 (R2) Pharmaceutical Development. MD, USA: Silver Spring; 2009.

    Google Scholar 

  2. Liu XY, et al. DEM study on the surface mixing and whole mixing of granular materials in rotary drums. Powder Technol. 2017;315:438–44.

    CAS  Google Scholar 

  3. Barrasso D, Ramachandran R. Qualitative assessment of a multi-scale, compartmental PBM-DEM model of a continuous twin-screw wet granulation process. J Pharm Innov. 2016;11(3):231–49.

    Google Scholar 

  4. Loreti S, Wu CY, Reynolds G, Mirtič A, Seville J. DEM-PBM modeling of impact dominated ribbon milling. AICHE J. 2017;63(9):3692–705.

    CAS  Google Scholar 

  5. Mateo-Ortiz D, Mendez R. Microdynamic analysis of particle flow in a confined space using DEM: the feed frame case. Adv Powder Technol. 2016;27(4):1597–606.

    Google Scholar 

  6. Ketterhagen WR, Curtis JS, Wassgren CR, Hancock BC. Predicting the flow mode from hoppers using the discrete element method. Powder Technol. 2009;195(1):1–10.

    CAS  Google Scholar 

  7. Kretz D, Callau-Monje S, Hitschler M, Hien A, Raedle M, Hesser J. Discrete element method (DEM) simulation and validation of a screw feeder system. Powder Technol. 2016;287:131–8.

    CAS  Google Scholar 

  8. Coetzee C. Particle upscaling: calibration and validation of the discrete element method. Powder Technol. 2019;344:487–503.

    CAS  Google Scholar 

  9. Barrasso D, Eppinger T, Pereira FE, Aglave R, Debus K, Bermingham SK, et al. A multi-scale, mechanistic model of a wet granulation process using a novel bi-directional PBM–DEM coupling algorithm. Chem Eng Sci. 2015;123:500–13.

    CAS  Google Scholar 

  10. Barrasso D, Ramachandran R. Multi-scale modeling of granulation processes: bi-directional coupling of PBM with DEM via collision frequencies. Chem Eng Res Des. 2015;93:304–17.

    CAS  Google Scholar 

  11. Dosta M, Antonyuk S, Heinrich S. Multiscale simulation of the fluidized bed granulation process. Chem Eng Technol. 2012;35(8):1373–80.

    CAS  Google Scholar 

  12. Ketterhagen WR. Simulation of powder flow in a lab-scale tablet press feed frame: effects of design and operating parameters on measures of tablet quality. Powder Technol. 2015;275:361–74.

    CAS  Google Scholar 

  13. Barrasso D, el Hagrasy A, Litster JD, Ramachandran R. Multi-dimensional population balance model development and validation for a twin screw granulation process. Powder Technol. 2015;270:612–21.

    CAS  Google Scholar 

  14. Sen M, Barrasso D, Singh R, Ramachandran R. A multi-scale hybrid CFD-DEM-PBM description of a fluid-bed granulation process. Processes. 2014;2(1):89–111.

    CAS  Google Scholar 

  15. Barrasso D, Tamrakar A, Ramachandran R. Model order reduction of a multi-scale PBM-DEM description of a wet granulation process via ANN. Procedia Engineering. 2015;102:1295–304.

    Google Scholar 

  16. Chaudhury A, Wu H, Khan M, Ramachandran R. A mechanistic population balance model for granulation processes: effect of process and formulation parameters. Chem Eng Sci. 2014;107:76–92.

    CAS  Google Scholar 

  17. Metta N, Ierapetritou M, Ramachandran R. A multiscale DEM-PBM approach for a continuous comilling process using a mechanistically developed breakage kernel. Chem Eng Sci. 2018;178:211–21.

    CAS  Google Scholar 

  18. Capece M, Bilgili E, Dave RN. Formulation of a physically motivated specific breakage rate parameter for ball milling via the discrete element method. AICHE J. 2014;60(7):2404–15.

    CAS  Google Scholar 

  19. Frouzakis CE, Kevrekidis YG, Lee J, Boulouchos K, Alonso AA. Proper orthogonal decomposition of direct numerical simulation data: data reduction and observer construction. Proc Combust Inst. 2000;28(1):75–81.

    CAS  Google Scholar 

  20. Xiao M, Breitkopf P, Filomeno Coelho R, Knopf-Lenoir C, Sidorkiewicz M, Villon P. Model reduction by CPOD and kriging. Struct Multidiscip Optim. 2010;41(4):555–74.

    Google Scholar 

  21. Akkisetty PK, et al. Population balance model-based hybrid neural network for a pharmaceutical milling process. J Pharm Innov. 2010;5(4):161–8.

    Google Scholar 

  22. Rogers A, Ierapetritou MG. Discrete element reduced-order modeling of dynamic particulate systems. AICHE J. 2014;60(9):3184–94.

    CAS  Google Scholar 

  23. Boukouvala F, Niotis V, Ramachandran R, Muzzio FJ, Ierapetritou MG. An integrated approach for dynamic flowsheet modeling and sensitivity analysis of a continuous tablet manufacturing process. Comput Chem Eng. 2012;42:30–47.

    CAS  Google Scholar 

  24. Wang Z, et al., Surrogate-based optimization for pharmaceutical manufacturing processes, in Computer Aided Chemical Engineering, A. Espuña, M. Graells, and L. Puigjaner, Editors. 2017, Elsevier. p. 2797–2802.

  25. Reynolds GK. Modelling of pharmaceutical granule size reduction in a conical screen mill. Chem Eng J. 2010;164(2–3):383–92.

    CAS  Google Scholar 

  26. Metta N, Verstraeten M, Ghijs M, Kumar A, Schafer E, Singh R, et al. Model development and prediction of particle size distribution, density and friability of a comilling operation in a continuous pharmaceutical manufacturing process. Int J Pharm. 2018;549(1):271–82.

    CAS  PubMed  Google Scholar 

  27. Deng XL, et al. Discrete element method simulation of a conical screen mill: a continuous dry coating device. Chem Eng Sci. 2015;125:58–74.

    CAS  Google Scholar 

  28. Hertz HJ reine angew. Math. 92, 156. Reprinted. English, in Hertz’s Miscellanenous paper, 1881.

  29. Mindlin RD. Compliance of elastic bodies in contact. ASME Journal of Applied Mechanics. 1949;16:259–68.

    Google Scholar 

  30. Delaney GW, et al. Predicting breakage and the evolution of rock size and shape distributions in ag and SAG mills using DEM. Miner Eng. 2013;50–51:132–9.

    Google Scholar 

  31. Datta A, Rajamani RK. A direct approach of modeling batch grinding in ball mills using population balance principles and impact energy distribution. Int J Miner Process. 2002;64(4):181–200.

    CAS  Google Scholar 

  32. Mishra BK, Rajamani RK. Simulation of charge motion in ball mills. 1. experimental verifications. Int J Miner Process. 1994;40(3–4):171–86.

    CAS  Google Scholar 

  33. Weerasekara NS, Powell MS, Cleary PW, Tavares LM, Evertsson M, Morrison RD, et al. The contribution of DEM to the science of comminution. Powder Technol. 2013;248:3–24.

    CAS  Google Scholar 

  34. Cleary PW. Recent advances in dem modelling of tumbling mills. Miner Eng. 2001;14(10):1295–319.

    CAS  Google Scholar 

  35. O’Sullivan C, Cui L. Micromechanics of granular material response during load reversals: combined DEM and experimental study. Powder Technol. 2009;193(3):289–302.

    Google Scholar 

  36. Meng W, Kotamarthy L, Panikar S, Sen M, Pradhan S, Marc M, et al. Statistical analysis and comparison of a continuous high shear granulator with a twin screw granulator: effect of process parameters on critical granule attributes and granulation mechanisms. Int J Pharm. 2016;513(1):357–75.

    CAS  PubMed  Google Scholar 

  37. Marigo M, et al. A numerical comparison of mixing efficiencies of solids in a cylindrical vessel subject to a range of motions. Powder Technol. 2012;217(Supplement C):540–7.

    CAS  Google Scholar 

  38. Flores-Johnson EA, Wang S, Maggi F, el Zein A, Gan Y, Nguyen GD, et al. Discrete element simulation of dynamic behaviour of partially saturated sand. Int J Mech Mater Des. 2016;12(4):495–507.

    CAS  Google Scholar 

  39. Hovad E, Spangenberg J, Larsen P, Walther JH, Thorborg J, Hattel JH. Simulating the DISAMATIC process using the discrete element method—a dynamical study of granular flow. Powder Technol. 2016;303:228–40.

    CAS  Google Scholar 

  40. Maione R, Kiesgen de Richter S, Mauviel G, Wild G. Axial segregation of a binary mixture in a rotating tumbler with non-spherical particles: experiments and DEM model validation. Powder Technol. 2017;306:120–9.

    CAS  Google Scholar 

  41. Vogel L, Peukert W. Breakage behaviour of different materials—construction of a mastercurve for the breakage probability. Powder Technol. 2003;129(1–3):101–10.

    CAS  Google Scholar 

  42. Barrasso D, Oka S, Muliadi A, Litster JD, Wassgren C, Ramachandran R. Population balance model validation and prediction of CQAs for continuous milling processes: toward QbDin pharmaceutical drug product manufacturing. J Pharm Innov. 2013;8(3):147–62.

    Google Scholar 

  43. Capece M, Bilgili E, Dave R. Identification of the breakage rate and distribution parameters in a non-linear population balance model for batch milling. Powder Technol. 2011;208(1):195–204.

    CAS  Google Scholar 

  44. Klimpel RR, Austin LG. The back-calculation of specific rates of breakage from continuous mill data. Powder Technol. 1984;38(1):77–91.

    CAS  Google Scholar 

  45. Meier M, John E, Wieckhusen D, Wirth W, Peukert W. Generally applicable breakage functions derived from single particle comminution data. Powder Technol. 2009;194(1):33–41.

    CAS  Google Scholar 

  46. Epstein B. Logarithmico-normal distribution in breakage of solids. Ind Eng Chem. 1948;40(12):2289–91.

    CAS  Google Scholar 

  47. Capece M, Bilgili E, Dave R. Insight into first-order breakage kinetics using a particle-scale breakage rate constant. Chem Eng Sci. 2014;117:318–30.

    CAS  Google Scholar 

  48. Bouhlel MA, Bartoli N, Otsmane A, Morlier J. Improving kriging surrogates of high-dimensional design models by partial least squares dimension reduction. Struct Multidiscip Optim. 2016;53(5):935–52.

    Google Scholar 

  49. Forrester AIJ, Keane AJ. Recent advances in surrogate-based optimization. Prog Aerosp Sci. 2009;45(1):50–79.

    Google Scholar 

  50. Bhosekar A, Ierapetritou M. Advances in surrogate based modeling, feasibility analysis, and optimization: a review. Comput Chem Eng. 2018;108:250–67.

    CAS  Google Scholar 

  51. Shvartsman SY, et al., Order reduction of nonlinear dynamic models for distributed reacting systems. Dynamics & Control of Process Systems 1998, Volumes 1 and 2, ed. C Georgakis 1999. 637–644.

  52. Wang Z, Escotet-Espinoza MS, Ierapetritou M. Process analysis and optimization of continuous pharmaceutical manufacturing using flowsheet models. Comput Chem Eng. 2017;107:77–91.

    CAS  Google Scholar 

  53. Wang Z, Ierapetritou M. A novel feasibility analysis method for black-box processes using a radial basis function adaptive sampling approach. AICHE J. 2017;63(2):532–50.

    CAS  Google Scholar 

  54. Rogers A, Ierapetritou M. Feasibility and flexibility analysis of black-box processes part 1: surrogate-based feasibility analysis. Chem Eng Sci. 2015;137:986–1004.

    CAS  Google Scholar 

  55. Wang Z, Ierapetritou M. Surrogate-based feasibility analysis for black-box stochastic simulations with heteroscedastic noise. J Glob Optim. 2018:1–29.

  56. Jia Z, Davis E, Muzzio FJ, Ierapetritou MG. Predictive modeling for pharmaceutical processes using kriging and response surface. J Pharm Innov. 2009;4(4):174–86.

    Google Scholar 

  57. Lang YD, Malacina A, Biegler LT, Munteanu S, Madsen JI, Zitney SE. Reduced order model based on principal component analysis for process simulation and optimization. Energy Fuel. 2009;23(3–4):1695–706.

    CAS  Google Scholar 

  58. Fei Y, et al. Evaluation of the potential of retrofitting a coal power plant to oxy-firing using CFD and process co-simulation. Fuel Process Technol. 2015;131(Supplement C):45–58.

    CAS  Google Scholar 

  59. Smith JD, Neto AA, Cremaschi S, Crunkleton DW. CFD-based optimization of a flooded bed algae bioreactor. Ind Eng Chem Res. 2013;52(22):7181–8.

    CAS  Google Scholar 

  60. Boukouvala F, Gao Y, Muzzio F, Ierapetritou MG. Reduced-order discrete element method modeling. Chem Eng Sci. 2013;95:12–26.

    CAS  Google Scholar 

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

    Google Scholar 

  62. Basheer IA, Hajmeer M. Artificial neural networks: fundamentals, computing, design, and application. J Microbiol Methods. 2000;43(1):3–31.

    CAS  PubMed  Google Scholar 

  63. Sacks J, Welch WJ, Mitchell TJ, Wynn HP. Design and analysis of computer experiments. Stat Sci. 1989;4(4):409–23.

    Google Scholar 

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

    Google Scholar 

  65. Wesolowski M, Suchacz B. Artificial neural networks: theoretical background and pharmaceutical applications: a review. J AOAC Int. 2012;95(3):652–68.

    CAS  PubMed  Google Scholar 

  66. Levenberg K. A method for the solution of certain non-linear problems in least squares. Q Appl Math. 1944;2(2):164–8.

    Google Scholar 

  67. Marquardt DW. An algorithm for least-squares estimation of nonlinear parameters. J Soc Ind Appl Math. 1963;11(2):431–41.

    Google Scholar 

  68. Looney CG. Advances in feedforward neural networks: demystifying knowledge acquiring black boxes. IEEE Trans Knowl Data Eng. 1996;8(2):211–26.

    Google Scholar 

  69. MATLAB, Neural Network Toolbox User’s Guide, version 10.0 (R2017a). 2017: The Mathworks Inc.

  70. Yan Z, Wilkinson SK, Stitt EH, Marigo M. Discrete element modelling (DEM) input parameters: understanding their impact on model predictions using statistical analysis. Comp Particle Mech. 2015;2(3):283–99.

    Google Scholar 

  71. Rogers AJ, Inamdar C, Ierapetritou MG. An integrated approach to simulation of pharmaceutical processes for solid drug manufacture. Ind Eng Chem Res. 2014;53(13):5128–47.

    CAS  Google Scholar 

  72. Galbraith, S.C., et al. Flowsheet modeling of a continuous direct compression tableting process at production scale. 2016.

    Google Scholar 

  73. Seo-Young Park SCG, Liu H, Lee HW, Cha B, Huang Z, O’Connor T, et al. Prediction of critical quality attributes and optimization of continuous dry granulation process via flowsheet modeling and experimental validation. Powder Technol. 2018;330:461–70.

    Google Scholar 

  74. Boukouvala F, Chaudhury A, Sen M, Zhou R, Mioduszewski L, Ierapetritou MG, et al. Computer-aided flowsheet simulation of a pharmaceutical tablet manufacturing process incorporating wet granulation. J Pharm Innov. 2013;8(1):11–27.

    Google Scholar 

  75. Kuentz M, Leuenberger H. A new model for the hardness of a compacted particle system, applied to tablets of pharmaceutical polymers. Powder Technol. 2000;111(1–2):145–53.

    CAS  Google Scholar 

  76. Gabbott IP, Al Husban F, Reynolds GK. The combined effect of wet granulation process parameters and dried granule moisture content on tablet quality attributes. Eur J Pharm Biopharm. 2016;106:70–8.

    CAS  PubMed  Google Scholar 

  77. Miyamoto Y, et al. Optimization of the granulation process for designing tablets. Chem Pharm Bull. 1998;46(9):1432–7.

    CAS  PubMed  Google Scholar 

  78. Sandler N, Wilson D. Prediction of granule packing and flow behavior based on particle size and shape analysis. J Pharm Sci. 2010;99(2):958–68.

    CAS  PubMed  Google Scholar 

  79. Yajima T, et al. Optimization of size distribution of granules for tablet compression. Chem Pharm Bull. 1996;44(5):1056–60.

    CAS  Google Scholar 

  80. Lophaven, S.N., H.B. Nielsen, and J. Søndergaard, DACE: A MATLAB Kriging Toolbox, Version 2.0. 2002.

  81. Beale, M.H., M.T. Hagan, and H.B. Demuth, Neural Netwrok Toolbox User’s Guide. 2017.

    Google Scholar 

  82. Pantelides CC, Barton PI. Equation-oriented dynamic simulation current status and future perspectives. Comput Chem Eng. 1993;17:S263–85.

    CAS  Google Scholar 

  83. Jin Y, Li J, du W, Qian F. Adaptive sampling for surrogate modelling with artificial neural network and its application in an industrial cracking furnace. Can J Chem Eng. 2016;94(2):262–72.

    CAS  Google Scholar 

  84. Kohavi, R., A study of cross-validation and bootstrap for accuracy estimation and model selection, in Proceedings of the 14th international joint conference on Artificial intelligence - Volume 2. 1995, Morgan Kaufmann publishers Inc.: Montreal. p. 1137–1143.

  85. Wolpert DH. Stacked generalization. Neural Netw. 1992;5(2):241–59.

    Google Scholar 

  86. Rogers A, Hashemi A, Ierapetritou M. Modeling of particulate processes for the continuous manufacture of solid-based pharmaceutical dosage forms. Processes. 2013;1(2):67–127.

    CAS  Google Scholar 

  87. Sampat C, Bettencourt F, Baranwal Y, Paraskevakos I, Chaturbedi A, Karkala S, et al. A parallel unidirectional coupled DEM-PBM model for the efficient simulation of computationally intensive particulate process systems. Comput Chem Eng. 2018;119:128–42.

    CAS  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Stephen Cole for insights into DEM model setup. The authors also gratefully acknowledge support from Process Systems Enterprise and EDEM solutions for providing academic licenses.

Funding

This work is supported by the US Food and Drug Administration (FDA), through grant 11695471, and a Consortium Agreement between Janssen Pharmaceutica, University of Ghent, and Rutgers University.

Author information

Authors and Affiliations

  1. Department of Chemical and Biochemical Engineering, Rutgers University, 98 Brett Road, Piscataway, NJ, 08854, USA

    Nirupaplava Metta, Rohit Ramachandran & Marianthi Ierapetritou

Authors
  1. Nirupaplava Metta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rohit Ramachandran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marianthi Ierapetritou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marianthi Ierapetritou.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Table 8 Kriging model parameters for case study 2
Full size table
Table 9 ANN model parameters for case study 2
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Metta, N., Ramachandran, R. & Ierapetritou, M. A Computationally Efficient Surrogate-Based Reduction of a Multiscale Comill Process Model. J Pharm Innov 15, 424–444 (2020). https://doi.org/10.1007/s12247-019-09388-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12247-019-09388-2

Keywords

Search

Navigation