459

Modeling of culture cells for pharmaceutical industry applications

Monica Roman, Dan Selişteanu, Eugen Bobaşu, Dorin Şendrescu Department of Automation, Electronics and Mechatronics

University of Craiova, Romania {monica, dansel, ebobasu, dorins}@automation.ucv.ro

Abstract—The paper presents the results of mathematical modeling using bond graph methodology applied to cells growing process in pharmaceutical industry. Largely used at industrial scale, the mammalian cell culture represents an important sector with intensive dynamic that requires continuous optimization due to evolution and transformations of medical applications. The research is focused on monoclonal antibodies (mAbs), proteins that are usually produced from mammalian cell cultures and used in biochemistry, biology and medicine. A particular model of mammalian cell culture, generic for this class of bioprocesses, was used. Usually the operation of these processes is mostly based on empirical knowledge and repeated experimental procedures, continuous adjusted using the operators experience. This paper proposes a comprehensible nonlinear dynamic modeling as a tool for this type of bioprocesses characterization.

Keywords- modeling; simulation; pharmaceutical industry

I. INTRODUCTION Monoclonal antibodies (MAbs) are proteins produced from

mammalian cell cultures through biochemical processes based on experimental research and previous practice with limited computation prediction of the results [1], [2]. Commonly used as production platforms for mAbs, the mammalian cell cultures represents a priority for the biopharmaceutical industry concerning the development of accurate models for growing processes to improve the production processes with respect to accuracy, efficiency and lower costs [1], [2], [3]. The MAbs cover a wide range of medical applications from diagnosis to oncology therapies. Different from other processes which are usually easier to model, the bioprocesses are characterized by higher complexity and nonlinearity. To optimize these processes for better quality products and higher efficiency, accurate dynamical models are required to develop model-based applications. Consequently these characteristics generate difficulties in modeling process.

In this paper we will use a practical alternative to the dynamic classical modeling (based on mass balance) namely the bond graph method, which was introduced by Paynter in 1961. The model of this kind of bioprocesses can be obtained by using the mass balance of the components and obeying some modeling rules; the resulting dynamical state-space model resides either using the classical modeling methods, either the bond graph methodology [4], [5], [6], [7], [8]. The classical modeling method for bioprocesses can be found in

several papers and industrial applications [9], [10]. However, due to the diversity of the bioprocesses, there are some problems concerning the development of a unified modeling approach. Also, for control purposes it is necessary to obtain reduced-order models, with some structural properties (linear – nonlinear decoupling, stable – unstable subsystems, the decoupling of uncertain kinetics, etc.).

Taking into account these problems, the bond graph method is considered to be a viable alternative to the classical modeling [11], [12]. The bond graph technique is used in this work for the modeling of a specific mammalian cell culture. More precisely, a process of an Immunoglobulin G - secreting murine hybridoma cultured in a growth medium supplemented with proline, L-asparagine, and L-aspartic acid is taken into consideration (Gao et al. [13]).

II. BOND GRAPH MODELING PROCEDURE FOR BIOPROCESSES

In biotechnology, pseudo bond graph models are accomplished starting with processes reactions schemes and using both base elements of “true” bond graph methodology and pseudo bonds with effort and flow variables as concentrations and mass flows [11], [12]. For the modeling of a batch process, the pseudo bond graph procedure starts from the reaction scheme, tacking into account the transfers inside the batch process. This kind of process is characterized by the absence of input or output flows; therefore the dynamics of the volume V inside the reactor is absent ( 0/ =dtdV ). The steps of the model building procedure are:

Construction of the bond graph model: 1. Introduction of 0-junctions, corresponding to the mass

balance of the involved components; 2. Introduction of capacitive elements C: for the modeling

of mass accumulations of the reaction components; these are attached by using bonds to the 0-junctions;

3. Introduction of transformer elements TF: for the modeling of stoichiometric coefficients;

4. Configurations of the modulated R or R elements MRi (Ri) for the reaction kinetics modeling;

5. Causality assignment.

Achievement of the mathematical model: Starting from the bond graph model obtained in the

previous stage, by using the characteristic equations of the bond graph elements and junctions, and by introducing the

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constructive and process properties of the system, the associated dynamical model of the bond graph model is obtained, as follows:

1. The constitutive relations for the 0-junctions are written: the efforts at all ports are equal, and the algebraic sum of the flows is zero: neee === ...21 ; 0...21 =±±±± nfff

2. The constitutive relations of C elements (that describe the components accumulation) are obtained as follows:

∫ τ== t

t iii dfC

qC

e0

11 , where C is a parameter of the C

element, ni ,1= , and the f variables are explicitly written by using the equations describing the others bond graph elements;

3. The relations that characterize the TF elements are written, and only those that imply f variables are taken into consideration.

4. The relations that characterize the elements used for the modeling of kinetic rates are obtained.

5. Taking into account the significance of bond graph elements, by deriving the equations of C elements and by using some substitutions, the dynamical model is obtained.

III. MODELING OF MAB PRODUCTION BIOPROCESS

In this section, the pseudo bond graph technique is used to In this section, the pseudo bond graph technique is used to model the mAb production bioprocess inside a batch reactor. The reaction scheme of this complex consists of nine macro reactions with 11 extracellular compounds [3], [13]:

LAC2GLC → (1) LAC2ALA2GLU2GLC +→+ (2) LAC2ASP2GLU2GLC +→+ (3) PROGLU → (4) NH3ASPASN +→ (5) GLUASNASPGLN +→+ (6)

BM0.081PRO0.0016GLUASP0201.0

0.006ASN0.0133ALAGLN0577.00.0508GLC→+++

++++ (7)

MAbPRO0148.0GLU0107.0

ASP082.0ASN072.0ALA011.00.0104GLN→++

++++ (8)

3NHGLUGLN +→ (9)

where GLC is the glucose, GLN - glutamine, GLU - glutamate, ASN - asparagine, ASP - aspartate, LAC - lactate, ALA - alanine, PRO - proline, MAb - monoclonal antibody, BM - Biomass, NH3 – Ammonia. For simplicity, the following notations will be used: GLC = S1, GLN = S2, GLU = S3, ASN = S4, ASP = S5, LAC = P1, ALA = P2, PRO = P3, MAb = P4, BM = P5, NH3 = P6.

From the reaction scheme given in equations (1) – (9) and taking into account the mass transfer through the batch reactor, the pseudo bond graph model of the bioprocess is achieved by using the modeling procedure I. The obtained bond graph model is depicted in Fig. 1, in the 20sim environment (registered trademark of Controllab Products B.V. Enschede, Netherlands). The directions of half arrows correspond to the run of the reaction.

In bond graph terms, the mass balances of the components involved in the bioreactor are represented by eleven 0-junctions: 01,2,7,17,40 (mass balance for 1S ), 034,35,42,55,70 (mass balance for 2S ), 09,10,19,24,39,45,58,72 (mass balance for 3S ), 028,29,38,47,60 (mass balance for 4S ), 022,23,31,37,49,62 (mass balance for 5S ), 05,6,13,79 (mass balance for 1P ), 015,16,51,64 (mass balance for 2P ), 026,27,53,66 (mass balance for 3P ), 068,69 (mass balance for 4P ), 055,56 (mass balance for 5P ), 032,33,73 (mass balance for

6P ). Two additional 0-junctions were introduced for the modeling of mass balances of X and Y: 074,75 (mass balance for X ), 076,77 (mass balance for dX ). Thus, the accumulations of

the involved components 65432154321 ,,,,,,,,,, PPPPPPSSSSS ar

e represented by bonds 1, 34, 9, 28, 23, 6, 16, 27, 69, 56, and 33, and they are modeled using capacitive elements C. The constitutive equations of C-elements are:

1S : ( )dtffffC

qC

et∫ −−−−== 401772

11

11

11 (10)

2S : ( )dtffffC

qC

et∫ −−−−== 70554235

3434

3434

11 (11)

3S : ( )dtfffffffC

qC

et∫ +−−+−−−== 72584539241910

99

99

11 (12)

4S : ( )dtffffC

qC

et∫ −−+−== 60473829

2828

2828

11 (13)

5S : ( )dtfffffC

qC

et∫ −−−+== 6249373122

2323

2323

11 (14)

1P : ( )dtfffC

qC

et∫ ++== 79135

66

66

11 (15)

2P : ( )dtfffC

qC

et∫ −−== 645115

1616

1616

11 (16)

3P : ( )dtfffC

qC

et∫ −−== 665326

2727

2727

11 (17)

4P : ( )dtfC

qC

et∫== 68

6969

6969

11 (18)

5P : ( )dtfC

qC

et∫== 55

5656

5656

11 (19)

6P : ( )dtffC

qC

et∫ +== 7332

3333

3333

11 (20)

Also, the accumulations of X and dX are represented by

bonds 74 and 77 and they are modeled using capacitive elements C. The constitutive equations of C-elements are:

( )dtfC

qC

et∫ −== 75

7474

7474

11 (21)

( )dtfC

qC

et∫== 76

7777

7777

11 (22)

461

0

C 1

0

C

0

C

0

C

0

C

0

C

0

C

0

C

0

C

0

C

0

C

TFR1

R2 1

TF

TF

TF

R3

TF

1TF

TF

R4 1

R5 1

R6

1

1

TF

TF R7

TF

TF

TF

TF

TFR8

1

TF

TF

TF

TF

TF

TF

R9

1

0 0R

C C

1 2

3 4 5

6

7 8

10

11

12

13

14

15

16

17

18

19

20 23

9

21 22

24 25 26

27

28 29

30 32

31 33

34

35

36

37

38

39

40

41

42 43 44

45

46

47

48

49

50

51 52

53

54 55

56

57

58

59

60

61

62

63

64

65

66 67

68

55

56

69

70

71

72

73

74

75 76

77

78

79

Figure 1. The pseudo bond graph model of mAb production bioprocess

where 74e , 77e are the concentrations of X and dX , and the parameters corresponding to capacitive elements are equal to the volume of bioreactor: VCC == 7774 .

The signification of the effort variables is as follows: 1e is concentration of the substrate 1S (glucose), 34e - concentration of 2S (glutamine), 9e - concentration of 3S (glutamate), 28e - concentration of 4S (aspargine), 23e - concentration of 5S (aspartate), 6e - concentration of the product 1P (lactate), 16e - concentration of 2P (alanine), 27e - concentration of 3P (proline), 69e - concentration of 4P (monoclonal antibody),

56e - concentration of 5P (biomass), 33e - concentration of 6P (nh3), 74e - viable cell concentration X , 77e - dead cell concentration dX .

The transformer elements TF4,5, TF10,11, TF12,13, TF14,15, TF19,20, TF21,22, TF24,25, TF40,41, TF42,43, TF45,46, TF47,48, TF49,50,

TF51,52, TF53,54, TF55,56, TF58,59, TF60,61, TF62,63, TF64,65, TF66,67, TF78,79 were introduced to model the stoichiometric coefficients, and their parameters 8,1,,, =jik ji are in fact the stoichiometric coefficients of the bioprocess.

For the modeling of the reaction kinetics nine modulated two-port elements were used: R1, R2, R3, R4, R5, R6, R7, R8, R9 due to the dependence of the reaction rates on the concentrations of the components involved in the reactions. The constitutive relations of R1, R2, R3, R4, R5, R6, R7, R8, R9 elements imply that Vf 12 ϕ= , Vf 27 ϕ= , Vf 317 ϕ= ,

Vf 424 ϕ= , Vf 529 ϕ= , Vf 635 ϕ= , Vf 744 ϕ= , Vf 857 ϕ= , Vf 971 ϕ= . Taking into consideration the constitutive relations

of the nine 1-junction elements 13,4, 18,11,12,14, 118,20,21,24, 125,26,

130,31,32, 136,37,38,39, 141,44,46,48,50,52,54,55, 157,59,61,63,65,67,68, 171,72,73 and the signification of bond graph elements from Fig. 1, by deriving the equations (10)-(20) the following dynamical model is obtained:

462

VkVVVdt

dSV 77,13211 ϕ−ϕ−ϕ−ϕ−=

VVkVkVdt

dSV 988,277,262 ϕ−ϕ−ϕ−ϕ−=

VVkVkVVVkVkdt

dSV 988,377,36433,322,3

3 ϕ+ϕ−ϕ−ϕ+ϕ−ϕ−ϕ−=

VkVkVVdt

dSV 88,477,465

4 ϕ−ϕ−ϕ+ϕ−=

VkVkVVVkdt

dSV 88,577,56533,5

5 ϕ−ϕ−ϕ−ϕ+ϕ=

VkVkVkdt

dPV 33,622,611,6

1 ϕ+ϕ+ϕ=

VkVkVkdt

dPV 88,777,722,52 ϕ−ϕ−ϕ=

VkVkVdt

dPV 88,877,84

3 ϕ−ϕ−ϕ=

Vdt

dPV 8

4 ϕ= , Vdt

dPV 7

5 ϕ= , VVdt

dPV 95

6 ϕ+ϕ=

However, these equations can be easily written as follows:

77,13211 ϕ−ϕ−ϕ−ϕ−= k

dtdS

(23)

988,277,262 ϕ−ϕ−ϕ−ϕ−= kk

dtdS

(24)

988,377,36433,322,33 ϕ+ϕ−ϕ−ϕ+ϕ−ϕ−ϕ−= kkkk

dtdS

(25)

88,477,4654 ϕ−ϕ−ϕ+ϕ−= kk

dtdS

(26)

88,577,56533,55 ϕ−ϕ−ϕ−ϕ+ϕ= kkk

dtdS

(27)

33,622,611,61 ϕ+ϕ+ϕ= kkk

dtdP

(28)

88,777,722,72 ϕ−ϕ−ϕ= kkk

dtdP

(29)

88,877,843 ϕ−ϕ−ϕ= kk

dtdP

(30)

84 ϕ=

dtdP

, 75 ϕ=

dtdP

, 956 ϕ+ϕ=

dtdP

(31)

The model (23)-(31) can be written in the compact form:

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ϕϕϕϕϕϕϕϕϕ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−−−−−−−−−−−

−−−−−−−−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

9

8

7

6

5

4

3

2

1

8,87,8

8,77,72,7

3,62,61,6

8,57,53,5

8,47,4

8,37,33,32,3

8,27,26,2

7,1

6

5

4

3

2

1

5

4

3

2

1

1000100000010000000100000000001000000000000000011000011000011010100000

00000111

kkkkk

kkkkkkkkkkkkkkk

k

PPPPPPSSSSS

dtd

(32)

In order to complete the model of mAb production process, it is necessary to add the submodels corresponding to the dynamics of viable cell concentration and dead cell concentration, which are given in the bond graph formalism in the relations (21)-(22). From the experimental observations [13], the model describing viable cell growth changes at t = 54 h to reflect the transition from exponential growth to the decline phase. With this remark, the dynamical model of the viable and cell concentrations evolutions is as follows [13]:

ddd

dd

dd

XXkdt

dX

ttXXkdtdX

ttXXkXdtdX

=

⎪⎪⎩

⎪⎪⎨

⎧

≥−=

<−μ=

exp

exp

for,

for,

(33)

where μ is the specific growth rate of the viable cells, dk is a kinetic (decay) parameter, and expt is the time period of the exponential growth phase.

The reaction kinetics are described by the following expressions [13]

1

111

11SK

SX

,S

*

+ϕ=ϕ

3

3

1

122

2321SK

SSK

SX

,, SS

*

++ϕ=ϕ

3

3

1

133

3331SK

SSK

SX

,, SS

*

++ϕ=ϕ ,

3

344

43SK

SX

,S

*

+ϕ=ϕ

4

455

54SK

SX

,S

*

+ϕ=ϕ ,

5

5

2

266

6,562SK

SSK

SX

SS

*

,++

ϕ=ϕ

2

277

72SK

SX

,S

*

+ϕ=ϕ ,

2

288

82SK

SX

,S

*

+ϕ=ϕ

29

299 SK

SX

S

*

+ϕ=ϕ

In the kinetic rates expressions, iϕ is the reaction rate for

reaction i , *iϕ is the maximum reaction rate for reaction i ,

jS is the concentration of substrate j within the set iS of substrates for reaction i , and

ijSK,

is a kinetic half-saturation constant for substrate j in reaction i .

The dynamical model of mAb production process is given by the equations (32) together with the dynamical models (33) of viable and dead cell evolution in the batch reactor.

IV. SIMULATION RESULTS AND DISCUSSIONS

In order to test the behavior of the obtained bioprocess model, several numerical simulations were performed by using the 20sim environment. The mAb production bioprocess has

463

been simulated for process parameters presented in Table 1 [3], [13]:

TABLE I. TABLE 1. PARAMETERS FOR MAB PRODUCTION BIOPROCESS

Symbol Value *1ϕ [pmol/(cell h)] 8.85×10-4 *2ϕ [pmol/(cell h)] 1.12×106 *3ϕ [pmol/(cell h)] 1.1×105 *4ϕ [pmol/(cell h)] 1.97×10-2 *5ϕ [pmol/(cell h)] 4.95 *6ϕ [pmol/(cell h)] 1.34×105 *7ϕ [pmol/(cell h)] 1.36×103 *8ϕ [pmol/(cell h)] 1×10-5 *9ϕ [pmol/(cell h)] 1.83×104

1,1SK [mM] 1.63×105

2,1SK [mM] 1.08×104

3,1SK [mM] 1.44×104

2,3SK [mM] 9.64×102

3,3SK [mM] 1.04×103

4,3SK [mM] 5.42×10-2

6,2SK [mM] 3.03×103

7,2SK [mM] 6.39×103

8,2SK [mM] 3.73×10-1

9,2SK [mM] 3.23×105

5,4SK [mM] 7.42×103

6,5SK [mM] 4.45×103 μ 3.22×10-2

dk 4.99×10-2

In Figs. 2-10 the time profiles of some of the state variables of this complex model are depicted.

0 20 40 60 80 100 120 1400

0.2

0.4

0.6

0.8

1

Time (h)

106 (mM)

Figure 2. Time evolution of viable cells.

0 20 40 60 80 100 120 1400

0.2

0.4

0.6

0.8

1

Time (h)

106 (mM)

Figure 3. Time evolution of dead cells.

0 20 40 60 80 100 120 1401.5

2

2.5

3

3.5

4

Time (h)

(mM)

Figure 4. Time profile of glucose concentration.

0 20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

Time (h)

(mM)

Figure 5. Time profile of glutamine concentration.

0 20 40 60 80 100 120 1400

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (h)

(mM)

Figure 6. Time profile of glutamate concentration.

464

0 20 40 60 80 100 120 1400.36

0.38

0.4

0.42

0.44

0.46

0.48

Time (h)

(mM)

Figure 7. Time evolution of aspargine concentration.

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

Time (h)

(mM)

Figure 8. Time evolution of aspartate concentration.

0 20 40 60 80 100 120 1400.0339

0.03395

0.034

0.03405

0.0341

0.03415

0.0342

0.03425

0.0343

Time (h)

(mM)

Figure 9. Time evolution of monoclonal antibodies concentration.

0 20 40 60 80 100 120 1400

5

10

15

20

Time (h)

(mM)

Figure 10. Time evolution of biomass concentration.

The simulations were performed for the following values of stoichiometric coefficients:

,0508.07,1 =k ,16,2 =k ,0577.07,2 =k ,0104.08,2 =k ,22,3 =k ,23,3 =k ,0016.07,3 =k

,0107.08,3 =k ,006.07,4 =k ,072.08,4 =k ,23,5 =k ,0201.07,5 =k ,082.08,5 =k ,21,6 =k ,22,6 =k

,23,6 =k ,22,7 =k ,0133.07,7 =k ,011.08,7 =k ,081.07,8 =k .0148.08,8 =k

V. CONCLUSION In this paper a complex mAb production process was

modeled using bond graph methodology. From mathematical point of view, the bond graph approach allows the achievement of the model of mAb production process directly in terms of dynamical state space equations; due to causality assignment it gives the possibility of localization of the state variables. The state variables within the bond graph model are associated with components of the macro-reactions from the reaction scheme. The obtained model can be used in estimation procedures, that are usually developed in order to estimate the kinetic parameters, and also in the design of control techniques.

ACKNOWLEDGMENT

This work was supported by UEFISCDI, project number PN II-RU TE 106/2010, Romania.

REFERENCES [1] D. Fernandez Slezak, C. Suarez, G.A. Cecch, G. Marshall, G.

Stolovitzky, “When the optimal is not the best: Parameter estimation in complex biological models,” PLoS ONE, vol. 5, no. 10, 2000.

[2] C. Kontoravdi, S.P. Asprey, E.N. Pistikopoulos, A. Mantalaris, “Development of a dynamic model of monoclonal antibody production and glycosylation for product quality monitoring,” Comput Chem Eng vol. 31, pp. 392–400, 2007.

[3] A.C. Baughman, X. Huang, S.T. Sharfstein, L.L. Martin, “On the dynamic modeling of mammalian cell metabolism and mAb production,” Comput Chem Eng, vol. 34, pp. 210–222, 2010.

[4] D. Karnopp and R. Rosenberg, System Dynamics: A Unified Approach. New York: John Wiley, 1974.

[5] J. Thoma, Introduction to bond graphs and Their Applications. Pergamon Press, Oxford, 1975.

[6] P. Gawthrop, L. Smith, Metamodelling: Bond graphs and dynamic systems. Hemel: Prentice Hall, 1996.

[7] W. Borutzky, Bond graph methodology. Development and analysis of multidisciplinary dynamic system models. London: Springer-Verlag; 2009.

[8] G. Dauphin-Tanguy, Les bond graphs. Paris: Hermes Sci., 2000. [9] Bastin. G., Dochain D., 1990. On-line Estimation and Adaptive Control

of Bioreactors. Elsevier. [10] D. Dochain, Automatic Control of Bioprocesses. ISTE Publ., 2008. [11] M. Roman, D. Selişteanu, “Pseudo bond graph modelling of wastewater

treatment bioprocesses,” Simul.-T Soc Mod Sim, vol.88, pp. 233–251, 2012.

[12] M. Roman, D. Selişteanu, “Enzymatic synthesis of ampicillin: Nonlinear modeling, kinetics estimation and adaptive control”, J Biomed Biotechnol, ID 512691, 2012.

[13] J Gao, V.M. Gorenflo, J.M. Scharer, H.M. Budman, “Dynamic metabolic modeling for a MAb bioprocess,” Biotechnol Progr. vol. 23, pp. 168–181, 2007.