Castilho Leda R., Moraes Angela Maria (Ed.) Animal Cell Technology: From Biopharmaceuticals to Gene Therapy

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the model is evaluated and, if necessary, one should try new values for the

parameters to improve the forecasts so that they resemble more and more

the experimental values. This approach is named model ﬁtting and intends

to ﬁnd the optimal parameters that minimize the difference between the

experimental data and that generated by the model. The ﬁnal result of this

ﬁtting is the ﬁnal model parameter estimation, also called parameter

optimization.

The estimation of model parameters can be reduced to the solution of

an optimization problem. This means that once the objective is deﬁned, a

class of mathematical functions that best match the speciﬁed objectives

must be found.

The optimization can be carried out by several methods of linear and

nonlinear regression. The mathematical methods must be chosen with

criteria to ﬁt the calculation of the applied objective functions. The most

widely applied methods of nonlinear regression can be separated into two

categories: methods with or without using partial derivatives of the

objective function to the model parameters. The most widely employed

nonderivative methods are zero order, such as the methods of direct search

and the Simplex (Himmelblau, 1972). The most widely used derivative

methods are ﬁrst order, such as the method of indirect search, Gauss-

Seidel or Newton, gradient method, and the Marquardt method.

The objective functions employed in the parameter optimization can

assume different forms (Equations 84 to 89):

OF ¼

( ~yy

 y

)

(84)

where:

OF ¼ objective function;

¼ experimental value of variable i.

OF ¼

~yy

)



)



(85)

where:

~yy

¼ calculated value of variable i;

)

¼ maximum experimental value of variable i.

OF ¼

~yy

 y

~yy



(86)

OF ¼

~yy

 y

~yy

 y



(87)

212 Animal Cell Technology

OF ¼

r  1

a  1

(88)

where:

r ¼ correlation coefﬁcient of the linear regression (experimental vs.

calculated variables);

a ¼ angular coefﬁcient of the linear regression (experimental vs.

calculated variables).

OF ¼ r  1

þ a  1

(89)

Whatever the form of calculation of the objective function or the

method employed in parameter optimization, the numerical solution of

the set of differential equations can be obtained by applying algorithms of

initial values. Examples of these algorithms are the implicit integration

methods (Euler, Crank-Nicolson, Adams-Moulton), or prediction-correc-

tion methods (Runge-Kutta, Michelsen, Gear) or bounded value problem

algorithms, such as ﬁnite difference methods, Jacobi’s polynomial or

orthogonal collocation methods (Rice and Do, 1995; Constantinides and

Mostouﬁ, 1999).

In the Appendix of this book, there is an example of parameter ﬁtting

proposed for the case of mathematical modeling of mAb production,

adapted from Lee (2003).

8.3.4 Model validation

The next step to model ﬁtting is its validation. Statistical analysis is

employed for the veriﬁcation of the experimental error, determination of

the best model, to identify any systematic deviation between the ﬁtted

model and the experimental data, as well as to verify whether the model is

statistically consistent, that is, whether it shows an error in the real data

consistent with the experimental error.

The most widely employed statistical test for validation of a mathema-

tical model is the F test.

For the application of this test, the experimental error must be known.

One way of determining the experimental error is to carry out a set of

tests, conducted in identical conditions. The relationship between the error

from model ﬁtting and the experimental error estimate is given by Equa-

tion 90:

(90)

where:

¼ model error for variance estimate;

¼ experimental error variance estimate.

The form of calculation of these variances is given by Equations 91 and 92.

Mathematical models for growth and product synthesis in animal cell culture 213

i¼1

j¼1

( ~yy

 y

)

(nv)

 p

(91)

i¼1

j¼1

 y

)

(nv)

 v

(92)

where:

n ¼ number of points for each variable;

v ¼ number of state variables deﬁned by the model;

(nv)

¼ number of ﬁtted points (all tests and variables);

(nv)

¼ number of experimental points (all repeated tests and state

variables);

p ¼ number of parameters from the model;

~yy

¼ variable values calculated by the model;

¼ experimental variable values;

¼ average variable values for repeated tests.

For model validation, F

, 1ors

, s

For more than one model, the relationship of the model to experimental

data can be determined by the 

Bartlett statistical test (Fromment and

Bischoff, 1990).

8.4 Structured and non-segregated models

The structured and non-segregated models, as mentioned before, consider

the cell population as homogeneous, but the intracellular structures are

separated, in a way that the cell is considered complex and treated as a

heterogeneous structure. Thus, it is possible to characterize the intrinsic

cell behavior and to forecast its interaction. As indicated in Chapter 5,

media for animal cells culture are complex, with components that include

sugar, amino acids, salts, and growth factors. Glucose and glutamine added

to media as carbon, nitrogen, and energy sources can affect cellular growth

and byproducts (such as lactate and ammonia) can accumulate in the

culture medium causing inhibition.

The difﬁculty of understanding all these interactive phenomena from

the cell and environment, which are in permanent transition, demands a

detail of the cellular structure – the characteristic of the structured models.

Depending on the quantity of detail and physiological structures consid-

ered, these structured models have varying ability for forecasting and

become highly valuable tools for understanding the cell physiology.

One of the pioneers of structured models in animal cell culture used a

single-cell model (Batt and Kompala, 1989). Based on hybridoma metabo-

lism (Figure 8.6), the model was based on the formulation of four

compartments: amino acids (including the TCA precursors), the nucleo-

tides (including DNA and RNA), the proteins, and lipids. The excreted

byproducts (lactate and ammonia) and the excreted product (mAb) were

also considered. However, although ﬂexible for simulation of different

214 Animal Cell Technology

operating strategies, the model lacked detail concerning inhibition by

secreted byproducts.

Different structured models formulated for hybridoma growth have

been shown (Linardos et al., 1992; Cazzador and Mariani, 1993; Martens

et al., 1995; Sanderson et al., 1999), as well as production of mAbs (Suzuki

and Ollis, 1990).

Besides being more robust than the unstructured models, these struc-

tured models describe the cellular response under a narrow band of

conditions. To forecast the cellular response to operation strategies of

various bioreactors, the model can be extended by an increasing number

of structures, to accumulate metabolic information (Wu et al., 1992) or

associate the metabolic ﬂuxes to the structured model (Follstad et al.,

1999). The methodologies for ﬂux balances, which employ a series of

reactions from glucose metabolism (Figure 8.7), and calculate all the meta-

bolic reactions rates during the stationary state, allow a better under-

standing of the cellular behavior and the prediction of other more efﬁcient

operating conditions.

Evidently, a higher number of structures implies that a higher number

of equations and parameters must be employed to describe the process

kinetics, increasing the difﬁculties in formulation and quantiﬁcation of the

intracellular components. Even so, the capacity for identifying these

intrinsic cellular phenomena and their incorporation into the models

increases the ability for prediction and forecasting. This forms an impor-

tant part of the study and development of the kinetic models.

8.5 Unstructured and segregated models

The representation of a property variation (mass, age, volume, etc.) within

a cellular population is made using population balance models (PBM).

These models allow the calculation of the variation of the rate of a

population characteristic considering the rates of increase and disappear-

ance of the speciﬁc characteristic within the cell population. Thus, they

allow a calculation of one special feature in the population. Mathematical

models that were originally unable to consider the diversity of character-

AMINO ACIDS GLUCOSE

nucleotides

lipids

amino acids

proteins

Cell

antibodies

GLUTAMINE

ammonia

lactate

Figure 8.6

Hybridoma metabolism scheme (based on Batt and Kompala, 1989).

Mathematical models for growth and product synthesis in animal cell culture 215

istics can now take this into account. The inclusion of cellular diversity

makes the model more complex and it is necessary to use non-trivial

mathematics to solve the equations, including the integration of partial

differential equations. Sidoli et al. (2004) have written a review of PBMs

for animal cells, indicating the major segregated models and their charac-

teristics. These models are classiﬁed into single- or multi-variables,

depending on the number of properties considered in the model, as well as

single- or multi-staged models, depending on the number of phases con-

sidered in the cellular life cycle. If the mass is the property variation to be

considered, then the model is called mass structured, and if the cellular

age is the property to be considered then the model is called aged

structured.

As an example, the single-variable population balance model, single-

staged, mass structured is indicated by Equation 93.

@Nm,t

ðÞ

zﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄ{

Accumulation

¼

@ rm,S

ðÞ

Nm,t

ðÞ



zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{

Single cell growth

ˆ m, S

ðÞ

Nm,t

ðÞ

zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{

Celullar division rate

þ 2



ˆ m9,S

ðÞ

pm,m9,S

ðÞ

Nm9,t

ðÞ

zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{

Birth rate

(93)

This equation states that accumulation of cells with a mass m is the

consequence of three rates that determine the dynamics of the mass

variation in the population: the rate of reduction of the number of

elements with mass m due to the mass increase that results from cell

Glucose

G6P

GAP

Lactate

Pyruvate

Biomass G6P

Biomass R5P

Biomass

Gly

Ser

Lys

Leu

AcCoA Biomass

Biomass

Asn

Asp

Biomass

OAA

Fum

TCA

cycle

a-KG

SuCoA

Val Thr Ile

Ala

Glu

Pro

Gln

CO2

NADH

NADH 0.502⫹

FADH2 0.502⫹

BiomassCO2

BiomassNADH

3ATP

2ATP

Figure 8.7

Hybridoma metabolism scheme (based on Follstad et al., 1999).

216 Animal Cell Technology

growth; the rate of reduction of the number of cells due cellular division,

and the rate of generation of new cells within this mass m.

In this equation N(m, t) is the distribution indicating the number of

cells with mass m, a function that depends on the mass of the cell, because

there is a mass distribution in the population, and on time, since there are

factors that continuously modify this distribution over time. Thus, N(m,

t) is a distribution function of a property in the population, in this in case

the cellular mass, over time.

The growth of cells of mass m is indicated by function r(m, S), in

Equation 93. It is a cell growth rate dependent on the cell mass and also on

the limiting substrate concentration. If a cell increases its mass, as a

consequence it leaves the class of the cells of mass m.

The cellular division also causes the reduction of the number of cell with

mass m and this reduction is determined by a division rate, ˆ(m, S) in

Equation 93. The product ˆ.N indicates the division rate of all the cells

with a mass m . Cell division gives rise to two daughter cells from the

mother cells of mass m9. The information on how the parent cell partitions

its mass off is contained in the partition function p(m, m9, S), that

expresses the probability of a parent cell of mass m 9 giving rise to a

daughter cell of mass m. In this way, 2.p.ˆ.N indicates the rate of

appearance of new cells with mass m, originating from parent cells of mass

m9. The parent cells have a continuous distribution of mass that goes from

a minimum mass, m

min

, to an unlimited value (theoretically to an inﬁnite

mass). Consequently, the total rate of cell generation with mass m is

calculated by the integration of cell generation rates for all intervals of

variation of the mass in the population. The division rate for the whole

distribution is the third term of Equation 93.

To solve Equation 93 it is necessary to know the characteristic functions

of the cellular physiology, that is, the growth rate r(m, S), the cellular

division rate ˆ(m, S), and the cell mass partition function p(m, m9, S). Since

these functions are substrate concentration- dependent, the substrate

consumption rate must also be deﬁned. These substrate concentration

variations are calculated using the yield coefﬁcient Y

X=S

, that establishes a

relationship between the growth rates and the substrate consumption rate.

The consumption rate is indicated by Equation 94.

Yx=s

min

rm,S

ðÞ

3 Nm,t

ðÞ

dm (94)

As pointed out before when discussing structured models, physiological

functions like the single cell growth rate r(m, S), cellular division rate ˆ(m,

S), and cell mass partition function p(m, m9, S) are difﬁcult to measure and

this adds to the complexity of using the model. The mathematical

complexity is also a limitation for optimization and control purposes that

demand online measurements and calculations.

The difﬁculties of characterizing the cell physiological properties can

explain the difﬁculties in applying these models and why they have not

been widely used in the solution of biochemical engineering problems. On

the other hand, the advances in monitoring, especially in ﬂow cytometry,

and in computer science development, that increase the capacity for model

Mathematical models for growth and product synthesis in animal cell culture 217

prediction, have provided new stimulating factors that have made these

models an option for some researchers.

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220 Animal Cell Technology

Bior eactors for animal cells

Ernesto Chico Ve

liz, Gryssell Rodrı

guez,

and Alvio Figueredo Cardero

9.1 Introduction

Around the end of the 1970s it became apparent that cells could be

cultivated in vitro on a large scale for the production of a wide range of

human health products. Since then, studies aimed at deﬁning the relevant

environmental conditions needed for efﬁcient cell proliferation in vitro

have been increasingly carried out. This has motivated work on the

development of chemically deﬁned media and on the determination of

optimal physicochemical conditions (pH, temperature, dissolved oxygen,

etc.) for culturing cells for large-scale operations. As a practical conse-

quence of these studies, several types of equipment to control environ-

mental culture conditions have been developed, giving rise to the

bioreactor systems currently used in the biopharmaceutical industry.

To ensure adequate cell proliferation and product biosynthesis, bioreac-

tors have to meet the following requirements:

(i) control the acid-base equilibrium (pH) of the culture medium;

(ii) control temperature;

(iii) provide gas exchange to appropriately supply oxygen to the cells and

remove excess carbon dioxide;

(iv) allow the supply of nutrients through the use of adequate formula-

tions of culture media;

(v) supply enough surface for adhesion of cells, in the case of anchorage-

dependent cells;

(vi) maintain aseptic conditions, avoiding contamination by microorgan-

isms, viruses, or other cells.

The basic challenge regarding the operation of bioreactors is to guaran-

tee a homogeneous physicochemical environment for the whole cell popu-

lation, preventing the occurrence of large gradients and regions with

inadequate conditions, which could cause the undesired death of part of

the cell population. The maintenance of adequate environmental condi-

tions requires the use of sensors and control systems, as discussed in detail

in Chapter 10.

9.2 Inoculum propagation and small-scale culture systems

To ensure consistency and reproducibility, especially in industrial produc-

tion processes, it is important always to use raw materials with identical