Probing Control for PWL Approximation of Nonlinear Cellular Growth

Eduardo Mojica, Alain Gauthier and Naly Rakoto-Ravalontsalama

Abstract— In this paper we develop and improve a piecewise-linear approximations of nonlinear cellular growth using or-thonormal canonical piecewise linear functions [5], and it istested by a probing control strategy [13] for the feed rate.The work is with the mammalian cells BHK (Baby HamsterKidney) in bioreactor in batch, fed-batch and continuous modeoperation. Preliminary results show that this piecewise linearapproximation is well suited for modeling such nonlineardynamics.

Index Terms— Bioreactor, nonlinear dynamics, piecewise-linear (PWL) models.

I. INTRODUCTION

The mammalian cells of Baby Hamster Kidney (BHK)

cultured on a well-defined medium comprising glucose,

glutamine and other amino acids produce cells mass, mono-

clonal antibodies and waste metabolites such as lactate, ala-

nine and ammonia, these cells are used in the production of

the vaccine against the foot-and-mouth disease. Historically,

the control problem of a bioreactor has been approached

by many authors using different approximations that can be

classified openly under strategies opened loop or closed loop

control. The operation in opened loop involves calculations

off-line of substrate feed strategies, while the control in

closed-loop involves follow-up in line of the reference path

in presence of disturbances [15]. The task of the controller

is to determine, in every instant, the best action of control

(the best feed substrate) being based on the compilation

of information on-line of the sensor. Subsequent works in

[16] have reinforced the work on the control adaptive of

bioreactors exposed initially in [1].

Mammalian cells display multiple steady states with

widely varying concentrations of cell mass, desired products

as well as waste metabolites [9],[2],[12]. This means that for

identical input conditions to a fed-batch reactor, the outlet

conditions change depending on how the culture is made

fed-batch. These multiple states are manifestations of the

complex interaction between cells and their environment.

Cells through different metabolic routes take substrate to

generate new cell mas and energy and in the generation

process excrete different waste products. What make this

process difficult is the additional level of complexity present

in biological systems because of the genetic information

present in living cells. Several nonlinear models have been

develop for mammaliam cells [12]. Here we address the

E. Mojica is with Ecole des Mines de Nantes, France, and Universidadde los Andes, Colombia. E-mail: emojica@emn.fr.

A. Gauthier is with Universidad de los Andes, Colombia. E-mail: agau-thie@uniandes.edu.co

N. Rakoto-Ravalontsalama is with Ecole des Mines de Nantes, France.Email: rakoto@emn.fr

peculiar features of mammalian cell growth in batch, fed-

batch and continuous operating conditions and construct a

mathematical model capable to simulate the phenomenon.

The proposed approximation methodology allows us to build

up a hybrid dynamical model is the piecewise-linear (PWL)

approximation technique developed in the past few years

by Julian et al [4],[5] and improvement after in [6] for

bifurcation analysis in academic examples, in [10] some

ideas of [4] are used for control design of a class of

nonlinear system. The model approximation is tested by a

new control methodology, the probing control strategy, which

was developed in [13] for E. coli cultivations.

In this paper, we develop a mathematical model using the

PWL technique mentioned above with Orthonormal High

Level Canonical Piecewise Linear functions [5]. In particular,

the class of all continuous PWL functions defined over a

compact domain in Rm, partitioned with a simplicial bound-

ary configuration is considered. This choice is motivated by

several facts. First of all, this class of functions uniformly

approximate any continuous nonlinear function defined over

a compact domain Rn (see [5]). In addition, the canonical

expression introduced in [5] uses the minimum and exact

number of parameters. As a consequence of this, an efficient

characterization is obtained from the viewpoint of memory

storage and numerical evaluation [11]. On the other hand,

the approximation can be used in real implementations,

the points taken from nonlinear model may be replaced

for points taken from sensors or data directly from the

process, namely, address the problem of finding a piecewise-

linear (PWL) approximation of system where a reasonable

number of measure samples of the vector field is available

(regression set)[14]. The probing control strategy that avoids

acetate accumulation while maintaining a high growth rate

[13] is implemented in simulations for this mathematical

model with interesting results. This control strategy has

been recently investigated but some issues have not been

completely developed, and some open problem persist.

This paper is organized as follows. In section II we present

the process description, a brief description about a nonlinear

model developed previously in [2], we deal with a nonlinear

model of fourth order (4th) to prove the efficiency of the

new PWL approximation. In section III we present basic

definition about PWL models. In section IV we present

the approximation structure proposed for biological models

based on Canonical piecewise-linear (CPWL) models for a

4th order nonlinear model. In section V we give simulations

for different dilution rates, and we verify the phenomenon

of multiple states with the CPWL model through transient

analysis and in section VI some conclusions are drawn.

16th IEEE International Conference on Control ApplicationsPart of IEEE Multi-conference on Systems and ControlSingapore, 1-3 October 2007

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II. PROCESS DESCRIPTION

The mammalian cells of Baby Hamster Kidney are used

in the production of the vaccine against the foot-and-mouth

disease, they are cultivated in bioreactor, which is located

in the biotechnology laboratory LIMOR de Colombia S.A,

in Bogota, Colombia. It is a 2500 liters bioreactor, which

operates in fed-batch mode, have temperature and agitation

speed controlled. In this, some experiments were carried

out that allowed us to obtain information of several cellular

cultures used for parameters adjustment of the mathematical

nonlinear model proposed [2]. We deal with a nonlinear

model approximation we use four state variables in order to

analyze the basic cellular behavior and the capability of the

CPWL model to approximate better this behavior, compared

with a simpler approximation of PWL based on a partition

form of the state space.

For the macrocospic development scale we use the mass

balance equations; with the dynamic general model for a

biological reactor developed in [1].

A. Nonlinear Model

The biological system class of nonlinear dynamics may be

described using the following model [8]:

{

·

x(t) = f(x(t)) + B(x(t))u(t)y(t) = Cx(t)

(1)

where x(t) ∈ Rn is a vector of states at time t, f : R

n →R

n are a nonlinear vector-valued functions, B is a state-

dependent n × m input matrix, u : R → Rm is an input

signal, C is an n × k output matrix and y : R → Rk is the

output signal.

Where u is the control variable, the dilution rate, we can

see that the control variable has a direct relation with state

variables, therefore we have a highly nonlinear behavior.

We present a description including four state variables;

cellular concentration, glucose, dissolved oxygen, and a

waste product acetate, it is the most relevant waste compo-

nent produced in the cellular growth process. In addition,

this model facilitates the reactor star-up and steady-state

operation conditions since the presence of less-desired steady

states at the same inlet conditions makes this bioreactor a

challenging problem for control, how we will show in sim-

ulation section, the transient analysis evidence this behavior.

·

x1

= µm

(

x2

x22

ki+x2+ks

)

(

x3

x3+ka

)

x1 −

−(

kdmkds

(µm−kdmx4)(kds+x2)

)

x1 − x1u.

·

x2 = −k1µm

(

x2

x22

ki+x2+ks

)

(

x3

x3+ka

)

x1 + (S1i − x2)u

·

x3 = k3µm

(

x2

x22

ki+x2+ks

)

(

x3

x3+ka

)

x1 − x3u

·

x4= Kla(C∗ − x4) − k1µm

(

x2

x22

ki+x2+ks

)

(

x3

x3+ka

)

x1

−x4u(2)

wwhere x1 g/L is the cellular concentration, x2 g/L is

the glucose concentration, µm 1/h is the maximum cellular

growth rate, ks g/L is the glucose saturation parameter, x3 is

the glutamine concentration, x4 is the lactate concentration,

ka = 0.06 g/L is the glutamine saturation parameter,

k1,2,3 are yield coefficients, 1.46, 0.9, and 0.06 respectively.

S1i = 1.5, is the initial glucose concentration in the feeding

medium, kdm = 0.012 is the maximum cellular death rate,

and kds = 0.08 is the death saturation parameter. The

volumetric oxygen transfer coefficient, Kla is a function

of stirred speed but is also affected by the air flow rate

and factors like viscosity, and foaming. C ∗ is the dissolved

oxygen concentration in equilibrium with the oxygen in gas

bubble.

III. BIOLOGICAL CPWL MODEL

The approximation deals with the PWL approximation

technique developed in the past few years by Julian et

al. [4],[5] and later improved by Storace and Defeo [6],

by analysis the qualitative behaviors of dynamical systems

characterized by PWL flows that approximate the flows of

some academic examples. To guarantee that the dynamical

behavior of the PWL-approximate flow will be faithful to

that of the original system for any values of some significant

parameters, we use the input variable u like a parameter, and

simulate the PWL-flow.

It has been shown in [2] that the cellular growth has several

growth phases: a first phase, after inoculation, is called (i)

Latency, an (ii) acceleration state, when cells begins the

growth process, an (iii) exponential state, in this phase cells

reach a constant growth with the major growth rate, a (iv)

deceleration state, cells reduced the constant growth for

absence of substrate and accumulation of toxic substrates

and finally a (v) stationary state, when the cellular growth

reaches a constant value, since substrate is over or it is in

a low constant value.We give first some definitions about

CPWL (Canonical Piecewise-linear) based in the previous

work in [4], [5] for the modeling and in [6] for analysis of

this PWL systems.

A. Orthonormal Canonical Piecewise Linear Functions

The representation proposed in [4] is the first PWL

expression able to represent PWL mappings in arbitrary

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dimensional domains [9].The orthonormal definition of the

PWL functions given before in [5] is used in this paper

to represent the nonlinear static mapping of cellular growth

process in bioreator of 4th order (2). With this in mind, a

brief description of this representation and its most important

characteristics are given next, for further details refer to [4],

[5]. For this approximation we have that given a nonlinear

system (3) the following steps are necessary to obtain a

CPWL approximation:

Step 1. Given a compact hyperrectangle grid domain,

order all vertices of that grid,

Step 2. Group the vertices into simplicial cells,

Step 3. Find CPWL basis functions,

Step 4. Find the CPWL model approximation.

In order to perform these steps we present the following

definitions and methods:

Definition 1. Let x0,x1, ...,xn be n + 1 points

in the n−dimensional space. A simplex (or polytope)

△(x0,x1, ...,xn) is defined by

△(x0,x1, ...,xn) ={

x : x =∑n

i=0µix

n}

where 0 ≤ µi ≤ 1, i ∈ {0, 1, ..., n} and∑n

i=0 µi=1. A

simplex is said to be proper if and only if it can not be

contained in a (n − 1) dimensional hyperplane.

The representation proposed in [4] requires the definition

of a rectangular compact domain of the form

S = {y ∈ Rm : 0 ≤ xi ≤ niδ, i = 1, 2, ..., m} . (3)

where δ is the grid size and ni ∈ Z+, being Z+ the

set of positive integers. Then this domain is subdivided

using a simplicial boundary configuration H , a simplicial

boundary configuration is characterized by the property that

it produces a division of the domain into simplices. If S is

defined as in (4), the space of all continuous PWL mappings

defined over the domain S partitioned with a simplicial

boundary configuration H is denoted by PWL[SH ] and it

is a linear vector space with the sum and multiplication of

functions by a scalar defined as usual. At this point it is

convenient to introduce the set VS of all simplex vertices

contained in S. Depending on the number of coordinates

different from zero, these verticies are organized in classes;

for instance, all the vertices in S that have k(k ≤ m)coordinates different from zero are called class k vertices.

The set VS plays an important role due to the fact that

the function values at the vertices of VS constitute all the

information that is needed to fully characterize any PWL

function fp : S �−→ R1. Moreover, the space PWL[SH ] of

all the PWL mappings defined over the set S, partitioned by

the boundary configuration H , is a linear vector space.

There are many possible choices for the PWL basis

functions, each of which is made up of N (linearly inde-

pendent) functions belonging to PWL N−dimensional linear

space. We have that any basis can be expressed as a linear

combination of the elements of the β basis, which is defined

by recursively applying the following generating function

γ(u, v) [4],[5]:

γ(x1, x2) = {||x1| + x2| − ||x2| − x1| +

+ |x1| + |x2| − |x2 − x1|}/4 (4)

Using (4), the functions γ0(x1) = x1, γ1(x1) =γ(x1, x1), γ2(x1, x2) = γ(x1, x2), and in general

γk(x1, ..., xk) = γ(x1, γk−1(x2, ..., xk)) (5)

are defined. A distinctive property of (5) is that it possesses

k nesting of absolute value functions, and accordingly it is

said to have nesting level (n.l) equal to k.The first element of the basis is a constant term γ 0(1). the

remaining elements are formed by the composition of the

function γk(·, ..., ·), k = 1, 2, ..., m, with the linear functions

πk,jk(x) =xk − jkδ

k = 1, 2, ..., m, jk = 0, 1, ..., nk − 1. Note that the

hyperplanes associated with these linear functions, namely

{x ∈ Rn : πk,jk

(x) = 0}, are the responsible for subdivision

of S into hypercubes. As a result, the basis can be expressed

in vector form, ordered according to its n.l. as

Λ =[

Λ0T

, Λ1T

, ..., ΛmT]

(6)

where Λi is the vector containing the generating functions

defined in [4] with i nesting levels. Accordingly, any fp ∈PWL[SH ] can be written as

fp(x) = cTΛ(x) (7)

where c =[

c0T

, c1T

, ..., cmT

]T

, and every vector ci is a

parameter vector associated with the vector function Λ i.In order to obtain an orthonormal basis, it is necessary

to define an inner product on PWL[SH ]. If VS is the set of

vertices of S and f, g belong to PWL[SH ],the functional

〈; 〉 �−→ R given by

〈f, g〉 =∑

vi∈S

f(vi)g(vi) (8)

defines an inner product and with this inner product the

space PWL[SH ] becomes a Hilbert space with the norm de-

fined by (8). The new basis elements are linear combination

of (6), that is Υ(x) =TΛ(x), and the matrix T may be

obtained by two different methodologies. In this work we

use the one built using the Gran-schmidt procedure as given

in [5].

Also, the CPWL functions of this class can uniformly

approximate any continuous function g : S �−→ R1. For

finding the required approximation, we use a routine of [5]

that finds a vector of parameters c that is the solution of the

least square problem minx ‖Ax − b‖2, being A = ΥT (X),X the input matrix and b the output to be approximated in

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sparse format. In accordance with [5], the CPWL approxi-

mation of the nonlinear function g is defined as the function

fp ∈PWL[SH ] satisfying

fCPWL = Ac. (9)

B. Cellular Growth CPWL Model

The PWL approximation based in a fourth Order nonlinear

model is obtained over the domain

S ={

y ∈ R5 : a1 ≤ x1 ≤ b1, a2 ≤ x2 ≤ b2,

a3 ≤ x3 ≤ b3, a4 ≤ x4 ≤ b4, a5 ≤ u ≤ b5.}

with

a1 = 0.0 b1 = 4

a2 = 0.0 b2 = 4.5

a3 = 0.0 b3 = 4

a4 = 0.0 b4 = 2

a5 = 0.0 b5 = 0.3

The domain S is partitioned by performing m1,m2, m3, m4 subdivisions along the state components x1,x2, x3, x4 respectively, and m5, subdivisions along the

parameter component u, as the control variable. The basis

were derived from a set of samples of f corresponding

to a regular grid of n1 × n2 × n3 × n4 × n5 point over

domain S. In particular, we will consider the following PWL

approximation fCPWL of f : m1 = m2 = m3 = m4 = 6,

m5 = 3, n1 = n2 = n3 = n4 = 15, n5 = 10. Through

several values of subdivisions we find that the least values

are performing the systems well from both qualitative and

quantitative points of view, we have to point out that the

algorithm using orthonormal basis allow a lager number of

inputs due to the ability of handling sparse matrices. Thus,

it is possible to detect the simplices that contribute to the

approximation, and it gives a measure of the nonlinearity of

the function. It is possible to take smaller values of m in

order to reduce the number of parameter but the quality of

the approximation become worse.

C. Model Simulation Results - Nonlinear vs CPWL

In order to analyze the system behavior to different input

values corresponding with the dilution rate u, we simulate the

original nonlinear system with CPWL approximation for u =0, it is when there is not feeding, batch operation mode, it is

the natural system behavior. Focus now on the simulation of

the more realistic case, a fourth order nonlinear model. The

simulation provides a plausible explanation of the different

steady state. First, as in the previous section we simulate the

CPWL model without input variable, dilution rate, in order

to obtain a batch mode behavior. All the simulations of the

4th order model with initial value x0 = (0.12, 4.2, 2.6, 0.0)′.Fig. 1 shows comparison between the nonlinear system and

CPWL approximation obtained by numerically integrating

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

Time [h]

x1−Cells[g/L] response for input case u=0.0

0 10 20 30 40 500

1

2

3

4

5

Time [h]

x2−Glucose[g/L] response for input case u=0.0

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

Time [h]

x3−Glutamine[g/L] response for input case u=0.0

0 10 20 30 40 500

0.05

0.1

0.15

0.2

Time [h]

x4−Lactate[g/L] response for input case u=0.0

Nonlinear model

cpwl model

Nonlinear model

cpwl model

Nonlinear model

cpwl model

Nonlinear model

cpwl model

Fig. 1. Comparison of system response computed with 4th order nonlinearand cpwl models to the dilution u = 0.0

the dynamical system having on the right-hand side of

fCPWL for a dilution rate u = 0.

In this case we can see the behavior of the four state

variables, the CPWL model is qualitatively similar to the

original system. For simulation purpose we increased the

dilution rate to the optimal value, and the CPWL presents

very good agreement. The dilution rate was then increased

to reach a high dilution rate, where normally the system is

not working, and the cellular concentration is too low. It is

of particular interest to show that model presented here is

able to vindicate the multiple state stationaries. Fig. 2 shows

the comparison between the nonlinear system and CPWL

approximation for these case, dilution rate, u = 0.031/h.

IV. PROBING FEED CONTROLLER

We will now desribe briefly the control thecnique devel-

oped in [13] for E. coli cultivations. The key idea is to exploit

the characteristic saturation in the oxygen supply system that

occurs at acetate formation, when the glucose uptake rate

excceds the glucose uptake rate critic. By superimposing a

short pulse on the feed rate and evaluate the response in the

dissolved oxygen signal. If a pulse response is visible, the

feed rate is increased at the end of a pulse. When overfeeding

is detected (no pulse response) the feed rate is decreased.

This is ”optimal” in the sense that it gives the maximum

growth rate without overflow metabolism, thus minimizing

the cultivation time.

A. Feedback algorithm

A simple algorithm that can be interpreted as proportional

incremental controller is used to adjust the feed rate F. At

each cycle of the algorithm a pulse is given and depending on

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0 10 20 30 40 500

0.5

1

1.5

2

2.5

Time [h]

x1−Cells[g/L] response for input case u=0.031

0 10 20 30 40 500

1

2

3

4

5

Time [h]

x2−Glucose[g/L] response for input case u=0.031

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

Time [h]

x3−Glutamine[g/L] response for input case u=0.031

0 10 20 30 40 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time [h]

x4−Lactate[g/L] response for input case u=0.031

Nonlinear model

cpwl model

Nonlinear model

cpwl model

Nonlinear model

cpwl model

Nonlinear model

cpwl model

Fig. 2. Comparison of system response computed with 4th order nonlinearand cpwl models to the dilution u = 0.031

the response the feed rate is adjusted according to the flow.

A reaction to a pulse is said to occur if the amplitude of the

response exceeds a critic oxygen reactive during the pulse

[13]. In this specific case we used a proportional probing

feed controller which change depending if it’s in a pulse up,

the increase in the feed F is decided by

dF (t)

dt= k

Cpulse(t) − Cpref

C∗ − x4F (10)

where Cpulse(t) is the pulse response, Cpref is the desired

pulse response and k the controller gain. In the case for down

pulses we get

dF (t)

dt=

(

kCpulse(t) − Cpref

C∗ − x4− γp

)

F (11)

where γp is the pulse amplitude. The pulse height, Fpulse,

must given an oxygen response that exceeds the reaction

levels, but it must also be ensured that the dissolved oxygen

level does not reach zero during the pulse.

Fpulse = γpF (12)

γp ≈ 4Creac/(C∗ − Csp)

where Creac is the reaction amplitude, it should be chosen

large enough for the algorithm to be unaffected by the

background variability in Cpulse, a value in 3 − 5%, is a

reasonable default choice.

Fig. 3 shows a simulation with a proportional controller

according to the description above, Fig. 4 shows the cellular

growth,

0 2 4 6 8 10 120.01

0.015

0.02

0.025System response for probing control during fed−batch simulation

Feed r

ate

[L/h

]]

0 2 4 6 8 10 120

0.5

1

Cells

X [g/L

]

0 2 4 6 8 10 120.4

0.5

0.6

0.7

Time [h]

Oxy

gen C

[g/L

]

Fig. 3. Systems response for probing control in a part of fed-batchcultivation

V. CONCLUSIONS AND FUTURE WORK

In this paper preliminary results on a Piecewise-linear

approximation of nonlinear cellular growth have been pre-

sented, based on orthonormal CPWL functions. It has also

been proved that this structure allows one to approximate

the dynamical behavior for different initial conditions in the

transient analysis. The main interesting point over this CPWL

model is the fact that we used a points of the nonlinear

system model, then for a real implementation, these points

could be taken from the sensors or data from the process. We

have tested the CPWL model with the probing proportional

controller based on the dissolved oxygen signal, its behavior

exhibited some optimal characteristics. This model although

identified for the BHK experiments seems to be generally

applicable to mammaliam systems other than BHK, such as

Hybridoma and Chinese Hamster Ovary (CHO) cells that

display similar behavior [9].

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