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Estimation of optimal feeding strategies for fed-batch bioprocesses

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Bioprocess and Biosystems Engineering Aims and scope Submit manuscript

A Publisher’s Erratum to this article was published on 14 September 2005

Abstract

A generic methodology for feeding strategy optimization is presented. This approach uses a genetic algorithm to search for optimal feeding profiles represented by means of artificial neural networks (ANN). Exemplified on a fed-batch hybridoma cell cultivation, the approach has proven to be able to cope with complex optimization tasks handling intricate constraints and objective functions. Furthermore, the performance of the method is compared with other previously reported standard techniques like: (1) optimal control theory, (2) first order conjugate gradient, (3) dynamical programming, (4) extended evolutionary strategies. The methodology presents no restrictions concerning the number or complexity of the state variables and therefore constitutes a remarkable alternative for process development and optimization.

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Abbreviations

ANN:

Artificial neural network

DP:

Dynamical programming

ES:

Evolutionary strategy

FOCG:

First order conjugate gradient

GA:

Genetic algorithm

IDP+SQP:

Iterative dynamical programming + sequential quadratic programming

NNMO:

Neural network model optimization

OCT:

Optimal control theory

Amm:

Ammonia concentration (mM)

F 1 :

Volumetric feed rate of glucose (dm3 day−1)

F 2 :

Volumetric feed rate of glutamine (dm3 day−1)

Glc:

Glucose concentration (mM)

Glcin :

Glucose concentration in the feed stream (mM)

Gln:

Glutamine concentration (mM)

Glnin :

Glutamine concentration in the feed stream (mM)

J :

Optimization goal (mg Mab)

k d :

First order death rate (day−1)

k μ :

Kinetic constant (day−1)

Lac:

Lactate concentration (mM)

Mab:

Concentration of monoclonal antibodies (mM)

q glc :

Specific use rate of glucose (mM cells−1 day−1)

q gln :

Specific use rate of glutamine (mM cells−1 day−1)

q amm :

Specific production rate of ammonia (mM cells−1 day−1)

q lac :

Specific production rate of lactate (mM cells−1 day−1)

q Mab :

Specific production rate of monoclonal antibodies (mg cells−1 day−1)

t :

Time (day)

t f :

Time at the end of fermentation (day)

V :

Culture reaction volume (dm3)

V max :

Maximal culture reaction volume (dm3)

X v :

Concentration of viable cells (cells cm−3)

Y xv/glc :

Cell yield coefficient for glucose (cells mM−1)

Y xv/gln :

Cell yield coefficient for glutamine (cells mM−1)

Y lac/glc :

Yield coefficient lactate/glucose (mM mM−1)

Y amm/gln :

Yield coefficient ammonia/glutamine (mM mM−1)

α0 :

Maximal specific Mab production rate (mg cells−1 d−1)

β:

Kinetic constant (mg cells−1 d−1)

μ:

First order growth rate (d−1)

μmax :

1.09 day−1

kd max :

0.69 day−1

Y xv/glc :

1.09·10−8 cells mM−1

Y xv/gln :

3.8·10−8 cells mM−1

m glc :

0.17 mM·10−8 cells−1 day−1

km glc :

19.0 mM

k glc :

1.0 mM

k gln :

0.3 mM

α0 :

2.57 mg ·10−8 cells−1 day−1

k μ :

0.02 day−1

β:

0.35 mg ·10−8 cells−1 day−1

kd lac :

0.01 day−1 mM−1

kd amm :

0.06 day−1 mM−1

kd gln :

0.02 mM

Y lac/glc :

1.8 mM mM−1

Y amm/gln :

0.85 mM mM−1

References

  1. Park S, Ramirez WF (1988) Optimal production of secreted protein in fed-batch bioreactors. AIChE Journal 34:1550–1558

    Article  CAS  Google Scholar 

  2. De Tremblay M, Perrier M, Chavarie C, Archambault J (1992) Optimization of fed-batch culture of hybridoma cells using dynamic programming: single and multi feed cases. Bioprocess Eng 7:229–234

    Article  Google Scholar 

  3. Wang FS, Shyu CH (1997) Optimal feed policy for fed-batch fermentation of ethanol production by Zymomous mobilis. Bioprocess Eng 17:63–68

    Article  Google Scholar 

  4. Patkar A, Seo JH, Lim HC (1993) Modelling and optimization of cloned invertase expression in Saccharomyces cerevisiae. Biotech Bioeng 41:1066–1074

    Article  CAS  Google Scholar 

  5. Roubos JA, de Gooijer CD, van Straten G, van Boxtel AJB (1997) Comparison of optimization methods for fed-batch cultures of hybridoma cells. Bioprocess Eng 17:99–102

    Article  CAS  Google Scholar 

  6. Waldraff W, King R, Gilles ED (1997) Optimal feeding strategies by adaptive mesh selection for fed-batch bioprocesses. Bioprocess Eng 17:221–227

    Article  CAS  Google Scholar 

  7. Chaudhuri B, Modak JM (1998) Optimization of fed-batch bioreactor using neural network model. Bioprocess Eng 19:71–79

    Article  CAS  Google Scholar 

  8. Lee A (2001) A nonsingular optimization approach to the feed rate profile optimization of fedbatch cultures. Bioproc Biosyst Eng 24:115–121

    CAS  Google Scholar 

  9. Roubos JA, van Straten G, van Boxtel AJB (1999) An evolutionary strategy for fed-batch bioreactor optimization: concepts and performance. J Biotechnol 67:173–187

    Article  CAS  Google Scholar 

  10. Ronen M, Shabtai Y, Guterman H (2002) Optimization of feeding profile for a fed-batch bioreactor by an evolutionary algorithm. J Biotechnol 97:253–263

    Article  CAS  Google Scholar 

  11. Na JG, Chang YK, Chung BH, Lim HC (2002) Adaptive optimization of fed-batch culture of yeast by using genetic algorithms. Bioproc Biosyst Eng 24:299–308

    CAS  Google Scholar 

  12. Sarkar D, Modak JM (2004) Genetic algorithms with filters for optimal control problems in fed-batch bioreactors. Bioproc Biosyst Eng 26:295–306

    CAS  Google Scholar 

  13. Pontryagin LS, Boltyanskii YG, Gramkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New York

    Google Scholar 

  14. Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading

    Google Scholar 

  15. Michalewicz Z (1996) Genetic algorithms + data structures = evolution programs, 3rd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  16. Weuster-Botz D (2000) Experimental design for fermentation media development: statistical design or global random search? J Biosci Bioeng 5:473–483

    Google Scholar 

  17. Karim MN, Rivera SL (1992) Artificial neural networks in bioprocess state estimation. Adv Biochem Eng 46:1–33

    CAS  Google Scholar 

  18. Lübbert A, Simutis R (1994) Adequate use of measuring data in bioprocess modelling and control. Trend Biotechnol 12:304–311

    Google Scholar 

  19. Montague G, Morris J (1994) Neural-network contributions in biotechnology. Trend Biotechnol 12:312–324

    Article  CAS  Google Scholar 

  20. Simutis R, Havlik I, Schneider F, Dors M, Lübbert A (1995) Artificial neural networks of improved reliability for industrial process supervision. IFAC Computer Applications in Biotechnology :59–65

    Google Scholar 

  21. Dors M, Simutis R, Lübbert A (1995) Advanced supervision of mammalian cell cultures using hybrid process models. Artificial neural networks of improved reliability for industrial process supervision. IFAC Computer Applications in Biotechnology :72–77

    Google Scholar 

  22. Hilera JR, Martinez VJ (1995) Redes neuronales artificiales. Fundamentos, modelos y aplicaciones. Addison-Wesley, Iberoamericana

    Google Scholar 

  23. Chipperfield AJ, Fleming PJ, Fonseca CM (1994) Genetic algorithm tools for control systems engineering. Presented at adaptive computing in engineering design and control. Plymouth, UK

    Google Scholar 

  24. Chipperfield AJ, Fleming PJ (1995) The MATLAB genetic algorithm toolbox. Presented at IEE colloqium on applied control techniques using MATLAB, Digest No. 1995/014

  25. The MathWorks (2003) Getting started with MATLAB. Online version. The MathWorks, Inc.

  26. The MathWorks (2003) Using simulink. Online version. The MathWorks, Inc.

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Authors and Affiliations

  1. Biochemical Engineering, Munich University of Technology, Boltzmannstr. 15, 85748, Garching, Germany

    Ezequiel Franco-Lara & Dirk Weuster-Botz

Authors
  1. Ezequiel Franco-Lara
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  2. Dirk Weuster-Botz
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Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00449-005-0017-0.

Appendix

Appendix

Hybridoma cells model developed by [2] and examined in [5, 9]

$$\frac{{{\text{d}}X_{\rm v}}}{{{\text{d}}t}} = (\mu - k_{\rm d})X_{\rm v} - \frac{{(F_1 + F_2 )}}{V}X_{\rm v},$$
$$\frac{{{\text{dGlc}}}}{{{\text{d}}t}} = \frac{{F_1 }}{V}{\text{Glc}}_{\rm in} - \frac{{(F_1 + F_2 )}}{V}{\text{Glc}} - q_{\rm glc} X_{\rm v},$$
$$ \frac{{{\text{d}} {\text{Gln}}}} {{{\text{d }}t}}{\text{ }} = \frac{{F2}} {V}{\text{ Gln}}_{{\text{in}}} {\text{ }} - {\text{ }}\frac{{{\text{( }}F1{\text{ }} + F{\text{2 )}}}} {V}{\text{ Gln}} - {\text{ }}q_{{\text{gln}}} Xv, $$
$$ \frac{{{\text{d}} {\text{Lac}}}} {{{\text{d }}t}}{\text{ }} = {\text{ }}q_{{\text{lac}}} X_V{\text{ - }}\frac{{{\text{( }}F1{\text{ }} + F{\text{2 )}}}} {V}{\text{ Lac}}, $$
$$\frac{{{\text{dAmm}}}}{{{\text{d}}t}} = q_{{\text{amm}}} X_{\rm v} - \frac{{(F_1 + F_2 )}}{V}{\text{Amm}}, $$
$$\frac{{{\text{dMab}}}}{{{\text{d}}t}} = q_{{\text{Mab}}} X_{\rm v} - \frac{{(F_1 + F_2 )}}{V}{\text{Mab}}, $$
$$\frac{{{\text{d}}V}}{{{\text{d}}t}} = (F_1 + F_2 ),$$

where,

$$ {\text{ }}\mu {\text{ }} = {\text{ }}\mu _{{\text{max}}} {\text{ }}\frac{{{\text{Glc}}}} {{k_{{\text{glc}}} + {\text{Glc}}}}{\text{ }}\frac{{{\text{Gln}}}} {{k_{{\text{gln}}} + {\text{Gln}}}}, $$
$$ k_d = \frac{{kd_{\text{max}} }} {{\left( {\mu _{{\text{max}}} - kd_{{\text{max}}} {\text{Lac}}} \right)}} \frac{{\text{1}}} {{\left( {\mu _{{\text{max}}} - kd_{\text{max}} {\text{Amm}}} \right)}} \frac{{kd_{\text{Gln}} }} {{\left( {kd_{\text{gln}} + {\text{Gln}}} \right)}}, $$
$$ q_{{\text{glc}}} = \frac{\mu } {{Y_{{\text{xv}}/{\text{glc}}} }} + m_{{\text{glc}}} \left( {\frac{{{\text{Glc}}}} {{km_{{\text{glc}}} + {\text{Glc}}}}} \right) $$
$$ q_{{\text{gln}}} = \frac{\mu } {{Y_{{\text{xv/gln}}} }} ,\quad q_{{\text{lac}}} = Y_{{\text{lac/glc}}} q_{{\text{glc}}} ,\quad q_{{\text{amm}}} = Y_{{\text{amm/gln}}} q_{{\text{gln}}} , $$
$$ q_{{\text{Mab}}} = \frac{{\alpha _0 }} {{k_\mu + \mu }} \cdot \mu + \beta, $$

The profit function to be minimized:

$$ J\left( {t_{\rm f}} \right) = \int_0^{t_{\rm f}} {L\;{\rm d}t \to \min } $$

subject to the constrains,

$$ L = - q_{{\text{Mab}}} X_{\rm v}\left( t \right)V\left( t \right)\quad {\text{if}}\;V(t) \leq V_{{\text{max}}} $$
$$ L = 0\quad {\text{if}}\;V(t) > V_{{\text{max}}} . $$
$$ V_{{\text{max}}} = 2.0\,L $$
$$ 0 \leqslant F_1 {\text{ or }}F_2 \leqslant {\text{ 0}}{\text{.5}} {\text{ (dm}}^{\text{3}} {\text{/day)}}. $$

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Franco-Lara, E., Weuster-Botz, D. Estimation of optimal feeding strategies for fed-batch bioprocesses. Bioprocess Biosyst Eng 27, 255–262 (2005). https://doi.org/10.1007/s00449-005-0415-3

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