## Abstrict
The present invention allows biomass and/or biomass-produced components
to be obtained at high yields by executing numeric simulation of
flow to reasonably identify the running conditions suitable for
a fermenter. The control device according to the present invention
comprises an input means for entering measured data from a measurement
means, which measures nutrient components, the concentrations of
oxygen, carbon dioxide, and biomass in a culture medium; a computation
means for calculating nutrient components uptake rate, oxygen uptake
rate and carbon dioxide exhaust rate per unit amount of biomass
from the measured data entered at the aforementioned input means,
as well as volumetric mass transfer coefficient kLa from turbulent
energy k and a turbulent energy dissipation rate e, both of which
are calculated by a transport equation, as well as a diffusion coefficient
D, followed by calculating the concentrations of the nutrient components,
dissolved oxygen, and dissolved carbon dioxide in any area in the
fermenter using an algorithm to numerically integrate a differential
equation describing variations in medium components over time from
the calculated nutrient components uptake rate, the calculated oxygen
uptake rate, the calculated carbon dioxide exhaust rate, and the
calculated volumetric mass transfer coefficient kLa; and a display
means for displaying concentration distributions of the nutrient
components, dissolved oxygen, and dissolved carbon dioxide in the
fermenter based on the concentrations of the nutrient components,
dissolved oxygen, and carbon dioxide in any area of the fermenter
calculated at aforementioned computation means.
## Claims
1. A control device for fermenter comprising: an input means connected
to the fermenter, in which biomass is cultured while oxygen gas
is being blown into a culture medium and the culture medium is being
stirred, for entering measured data output from a measurement means
for measuring nutrient components, concentration of oxygen, concentration
of carbon dioxide, and concentration of biomass in a medium; a computation
means for calculating nutrient components uptake rate, oxygen uptake
rate and carbon dioxide exhaust rate per unit amount of biomass
from the measured data entered at the aforementioned input means,
as well as volumetric mass transfer coefficient kLa from turbulent
energy k and a turbulent energy dissipation rate e, both of which
are calculated by a transport equation, as well as a diffusion coefficient
D, followed by calculating the concentrations of the nutrient components,
dissolved oxygen, and dissolved carbon dioxide in any area in the
fermenter using an algorithm to numerically integrate a differential
equation describing variations in medium components over time from
the calculated nutrient components uptake rate, the calculated oxygen
uptake rate, the calculated carbon dioxide exhaust rate, and the
calculated volumetric mass transfer coefficient kLa; and a display
means for displaying concentration distributions of the nutrient
components, dissolved oxygen, and dissolved carbon dioxide in the
fermenter based on the concentrations of the nutrient components,
dissolved oxygen, and carbon dioxide in any area of the fermenter
calculated at aforementioned computation means.
2. The control device for fermenter according to claim 1, wherein
the aforementioned computation means compares the calculated dissolved
oxygen concentration and a preset targeted value for dissolved oxygen
concentration so as to generate a control signal for controlling
the amount of oxygen gas to be supplied to the fermenter.
3. The control device for fermenter according to claim 1, wherein
the aforementioned computation means calculates a value for an indicator
of biomass death so as to generate a control signal for controlling
a rotation speed of impellers in the fermenter.
4. The control device for fermenter according to claim 1, wherein
the aforementioned differential equation contains experimental constants
and the aforementioned computation means calculates the aforementioned
experimental constants so that the measured data entered at the
aforementioned input means corresponds to data on the measured value
obtained by numerically integrating the aforementioned differential
equation by a least squares method to re-establish the aforementioned
differential equation.
5. The control device for fermenter according to claim 1, wherein
the aforementioned input means enters measured data on biomass concentration,
components having positive effects on biomass concentration, and
components having negative effects on cell growth, wherein when
a formula calculating an yield of a product to be harvested is a
target function, a concentration of the component having negative
effects on cell density and cell growth is a state variable, and
a concentration of the component having positive effects on cell
growth is a control variable, the aforementioned computation means
divides variance ranges of the state variable and the control variable
into a finite number of partial regions, respectively, so as to
generate control signals for time series supply of the components
having positive effects on cell growth such that the target function
can be maximized using dynamic programming.
6. A fermentation unit comprising: a fermenter, in which biomass
is cultured while oxygen gas is being blown into a culture medium
and the culture medium is being stirred; a measurement means connected
to the aforementioned fermenter for measuring nutrient components,
concentration of oxygen, concentration of carbon dioxide, and concentration
of biomass in a medium in the fermenter; an input means for entering
measured data regarding at least nutrient components, concentration
of oxygen, concentration of carbon dioxide, and concentration of
biomass in a medium in the fermenter, which are measured at the
aforementioned measurement means a computation means for calculating
nutrient components uptake rate, oxygen uptake rate and carbon dioxide
exhaust rate per unit amount of biomass from the measured data entered
at the aforementioned input means, as well as volumetric mass transfer
coefficient kLa from turbulent energy k and a turbulent energy dissipation
rate e, both of which are calculated by a transport equation, as
well as a diffusion coefficient D, followed by calculating the concentrations
of the nutrient components, dissolved oxygen, and dissolved carbon
dioxide in any area in the fermenter using an algorithm to numerically
integrate a differential equation describing variations in medium
components over time from the calculated nutrient components uptake
rate, the calculated oxygen uptake rate, the calculated carbon dioxide
exhaust rate, and the calculated volumetric mass transfer coefficient
kLa; and a display means for displaying concentration distributions
of the nutrient components, dissolved oxygen, and dissolved carbon
dioxide in the fermenter based on the concentrations of the nutrient
components, dissolved oxygen, and carbon dioxide in any area of
the fermenter calculated at aforementioned computation means.
7. The fermentation unit according to claim 6, further comprising:
an oxygen gas feed unit for feeding an oxygen gas into the fermenter
based on a control signal, wherein the control signal is calculated
by comparing the dissolved oxygen concentration that are calculated
by the aforementioned computation means and a preset targeted value
for dissolved oxygen concentration at the aforementioned computation
means and the the control signal controls the amount of oxygen gas
to be supplied into the fermenter.
8. The fermentation unit according to claim 6, wherein, the value
for the indicator of biomass death is calculated at the aforementioned
computation means and the rotation speed of the impellers in the
fermenter is controlled based on the control signal generated at
the aforementioned computation means.
9. The fermentation unit according to claim 6, wherein the aforementioned
differential equation contains experimental constants and the aforementioned
computation means calculates the aforementioned experimental constants
so that the measured data entered at the aforementioned input means
corresponds to data on the measured value obtained by numerically
integrating the aforementioned differential equation by a least
squares method to re-establish the aforementioned differential equation.
10. The fermentation unit according to claim 6, further comprising:
a feed unit for supplying the components having positive effects
on cell growth in the fermenter, wherein the aforementioned measurement
means measures a data on biomass concentration, components having
positive effects on cell growth, and components having negative
effects on cell growth, wherein when a formula calculating an yield
of a product to be harvested is a target function, a concentration
of the component having negative effects on cell density and cell
growth is a state variable, and a concentration of the component
having positive effects on cell growth is a control variable, the
aforementioned computation means divides variance ranges of the
state variable and the control variable into a finite number of
partial regions, respectively, so as to generate control signals
for time series supply of the components having positive effects
on cell growth such that the target function can be maximized using
dynamic programming, and wherein the feed unit supplies the components
having positive effects on cell growth in the fermenter based on
the control signals for time series supply.
## Description FIELD OF THE INVENTION
[0001] The present invention relates to a control device controlling
a fermenter for culturing biomass such as mammalian cells or microorganisms.
BACKGROUND OF THE INVENTION
[0002] In an industrial culture, oxygen necessary for biomass to
grow in a culture medium containing nutrient components such as
glucose and glutamine is supplied in a form of a bubble aerating
gas or supplied by aeration onto a liquid free surface while the
culture medium is often frequently stirred to grow the biomass,
harvesting a responsible material produced by the biomass. To successfully
achieve biomass culture, it is required that inner fermenter conditions
suitable for biomass culture be secured. In contrast, the structure
and running conditions of the fermenter contribute to deviation
from these suitable inner fermenter conditions.
[0003] To ensure an increased volumetric mass transfer coefficient
kLa for oxygen in the fermenter, and homogeneous mixture of the
nutrient components and a dissolved oxygen concentration, the culture
medium is stirred by impellers. Is difficult to achieve high kLa
and sufficient homogeneity as for high-viscosity culture media.
Though an attempt has been made to experimentally obtain the volumetric
oxygen transfer coefficient kLa for each type of fermenter, in such
an experimental method (Japanese Patent Publication (Kokai) No.
2001-231544), kLa varies with the tank diameter of the fermenter,
the diameter of bubbles, the diameter of impellers, and the rotation
speed and consistently depends on the type of fermenter. Therefore,
this method lacks generality. In the case where aeration onto the
liquid free surface and aerating into the culture medium are combined,
the ratio between both two depends on the conditions, making it
difficult to obtain general knowledge. To enhance an increase in
kLa and the homogeneity, the rotation speed of the impellers may
be increased, though usually, agitation power constraints it. In
culturing the mammalian cells, such a problem arise that to strong
agitation may bring the cells to death. To avoid experimental constraints,
an attempt has been made to use fluid numeric simulation in designing
the fermenter structure (Japanese Patent Publication (Kokai) No.
2001-75947), which has provided no knowledge of a fluid field suitable
for biomass growth. No method for applying fluid numeric simulation
to control has been established.
[0004] Any starvation of nutrient components in the culture process
inhibits growth of the biomass. In many cases, the components such
a lactate, ammonia, and alcohol produced as by-products suppress
the growth of the biomass, deteriorating the yield of responsible
products. Accordingly, it is always important to measure the components
in the culture medium and to sustain the nutrient components during
fermenter's running. However, no satisfactory technique has been
established for reflecting the measurement results in fermenter
control. An attempt has been made to make a model of a biomass growing
process by a differential equation of reaction dynamics, though
no computer-aided control method, which combines both the techniques
for modeling of the biomass growing process and achieving of hydrodynamic
conditions in the fermenter, has been reported.
SUMMARY OF THE INVENTION
[0005] To solve these problems, an object of the present invention
is to provide the control device for fermenter, which is capable
of producing the biomass and/or components produced by the biomass
at a high yield by applying the fluid numerical simulation technique
to reasonably find the running conditions suitable for the fermenter.
[0006] The present invention, which has attained the abovementioned
object, is composed of a plurality of the following means.
[0007] The control device for fermenter according to the present
invention comprises; an input means connected to the fermenter,
in which biomass is cultured while oxygen gas is being blown into
a culture medium and the culture medium is being stirred, for entering
measured data output from a measurement means for measuring nutrient
components, concentration of oxygen, concentration of carbon dioxide,
and concentration of biomass in a medium; a computation means for
calculating nutrient components uptake rate, oxygen uptake rate
and carbon dioxide exhaust rate per unit amount of biomass from
the measured data entered at the aforementioned input means, as
well as volumetric mass transfer coefficient kLa from turbulent
energy k and a turbulent energy dissipation rate e, both of which
are calculated by a transport equation, as well as a diffusion coefficient
D, followed by calculating the concentrations of the nutrient components,
dissolved oxygen, and dissolved carbon dioxide in any area in the
fermenter using an algorithm to numerically integrate a differential
equation describing variations in medium components over time from
the calculated nutrient components uptake rate, the calculated oxygen
uptake rate, the calculated carbon dioxide exhaust rate, and the
calculated volumetric mass transfer coefficient kLa; and a display
means for displaying concentration distributions of the nutrient
components, dissolved oxygen, and dissolved carbon dioxide in the
fermenter based on the concentrations of the nutrient components,
dissolved oxygen, and carbon dioxide in any area of the fermenter
calculated at aforementioned computation means.
[0008] According to the present invention, the variations of medium
components over time and spatial distributions of the nutrient components
in the fermenter are calculated by combining the differential equation
describing the medium components with equations of fluid dynamics
for numerical, discrete, and algebraic evaluation. This gives information
on how the biomass grows in the any region in the fermenter and
on what kind of fluid field is suitable for biomass growth. Moreover,
the present invention provides the volumetric mass transfer coefficient
kLa as a function k.sub.La (k, .epsilon., D) among k, .epsilon.,
and the diffusion coefficient D based on the result of an experimental
study on the relationship among a mass transfer coefficient for
a turbulent field, and turbulent energy k and turbulent energy dissipation
rate .epsilon.. This allows the volumetric mass transfer coefficient
k.sub.La for the any region in the fermenter of any structure to
be calculated based only on the properties of the turbulent field
and the values for gas properties without depending on the structure
of the fermenter. Therefore, the concentrations of the aforementioned
medium components, as well as of oxygen, carbon dioxide, and so
on, can be calculated in the any point in the fermenter. Also, the
structure of the fermenter suitable for biomass growth can be determined.
Further, the present invention provides the nutrient supply algorithm
based on time differential equations for the medium components by
means of dynamic programming.
[0009] The control device for the fermenter and a fermentation
unit according to the present invention can display the distributions
of dissolved oxygen concentration and dissolved carbon dioxide concentration
on its display screen to facilitate the determination of conditions
suitable for biomass growth in the fermenter.
BRIEF DESCRIPTION OF THE DRAWINGS
[0010] FIG. 1 is a block diagram showing the structures of a control
device for fermenter and a fermenter unit according to the present
invention.
[0011] FIG. 2A and FIG. 2B show mesh data on the inside of the
fermenter generated by a mesh data generator.
[0012] FIG. 3 is a flowchart explaining an algorithm for sequentially,
numerically, discretely, and algebraically integrating the differential
equations for time.
[0013] FIGS. 4-1A, 4-1B, 4-1C, and 4-1D show an example of the
results of computations indicated on a display, in which FIG. 4-1A
is an example of the displayed flow rate vector; FIG. 4-1B is an
example of the displayed gas holdup ratio, FIG. 4-1C is an example
of the displayed distribution of turbulent energy k, and FIG. 4-1D
is an example of the displayed distribution of turbulent energy
dissipation rate .epsilon..
[0014] FIGS. 4-2E, 4-2F, 4-2G, and 4-2H show an example of the
results of computations indicated on the display; in which FIG.
4-2E is an example of the displayed distribution of volumetric oxygen
transfer coefficient KLa, FIG. 4-2F is an example of the displayed
distribution of Kolmogorov scale, FIG. 4-2G is an example of the
displayed image of the oxygen concentration distribution, and FIG.
4-2H is an example of the displayed image of the carbon dioxide
concentration distribution.
[0015] FIG. 5A and FIG. 5B show an example of a graph of the result
of computations indicated on the display; in which FIG. 5A is an
example of the displayed graph of the relationship between the rotation
speed of impellers; and the turbulent energy k and the turbulent
energy dissipation rate .epsilon., and FIG. 5B is an example of
a graph of the relationship between the rotation speed of the impellers;
and the total volumetric oxygen transfer coefficient Kla, the volumetric
liquid-surface oxygen transfer coefficient Kla,s, and the volumetric
bubble-oxygen transfer coefficient Kla,b.
[0016] FIG. 6 is a flowchart explaining a regula falsi algorithm
for obtaining the amount of aerating oxygen.
[0017] FIG. 7 is a Kolmogorov's characteristic diagram being set
the rotation speed as a parameter.
[0018] FIG. 8 is a characteristic diagram showing the dissolved
carbon dioxide concentration in the fermenter calculated by evaluating
the differential equations being set the rotation speed of the impellers
3 as a parameter under the condition where the cell density Xa and
the dissolved oxygen concentration Do are fixed.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0019] Now, by reference to the drawings, the control device for
fermenter according to the present invention is described in detail.
As shown in FIG. 1, the control device for fermenter according to
the present invention controls the fermentation unit having a fermenter
1, which cultures the biomass under given conditions. The fermentation
unit comprises the fermenter 1, a aerating unit 2, which supplies
a oxygen gas into the fermenter 1, impellers 3, which stire the
culture medium in the fermenter 1, a measurement unit 4, which measures
the concentrations of oxygen, nutrients, biomass, or the like in
the culture medium, a nutrient component feed unit 5, which supplies
the nutrient components in the culture medium, and a drive control
unit 6, which supplies drive control signals to the aerating unit
2, the impellers 3, and the nutrient component feed unit 5 in the
fermenter 1.
[0020] As shown in FIG. 1, the control device according to the
present invention comprises a mesh data generator 7, which generates
mesh data for the fermenter 1 based on shape data on the fermenter
1, a processor 8, which executes computations in accordance with
a running control program, memory 9, which stores the running control
program, and an output display 10, which indicates the results of
computations executed at the processor 8. The control device according
to the present invention can be specifically implemented using hardware
resources comprising an interface for entering data output from
the measurement unit 4 of the fermenter 1, an input device for entering
the shape data on the fermenter 1, or the like, the computation
means such as a CPU, a memory means such as hard disk, nonvolatile
memory, and volatile memory, and an displaying means such as a display.
Alternatively, the control device according to the present invention
may be implemented as an integrated type, which has been integrated
in the fermenter unit. This means that the present invention may
implement the fermentation unit having the control device for fermenter.
[0021] The drive control unit 6, which is incorporated a PID (Proportional
Integral Differential) control system, outputs the control signals
based on target control values calculated by the processor 8 using,
for example measured values input from the measurement unit 4 via
the interface.
[0022] The measurement unit 4 may measure, for example the concentration
of the biomass contained in the culture medium sampled by a sampler,
the components having positive effects on cell growth, and the components
having negative effects on cell growth and also measure the concentrations
of dissolved oxygen and dissolved carbon dioxide contained in the
culture medium by a chromatographic system. The term "the components
having positive effects on cell growth" described herein includes
the nutrient components such as glucose and glutamic acid. The term
"the components having negative effects on cell growth"
described herein includes substances such as lactate and carbon
dioxide produced in the culture process.
[0023] The mesh data generator 7 enters data to be calculated by
means of the input means so as to generate mesh data, and sends
it to the processor 8. The term "the input data" described
herein are the shape data such as the diameter and the height of
the fermenter, the diameter width, and number of impellers of the
fermenter; conditions such as an impeller rotation speed and a sparger
gas flow rate; and initial values data such as the initial concentrations
of culture medium components and biomass. The mesh data generated
here may be output to the input data display unit, enabling the
target to be analyzed to be checked as shown in FIGS. 2A and 2B.
[0024] The processor 8 executes computations in accordance with
the program stored in the memory 9. As shown in FIG. 3, the program
comprises the algorithm, which numerically, discretely, and algebraically
integrates the time differential equation. In other words, the program
comprises a step 1, at which t is initialized to t.sub.0; a step
2, at which a time step is updated to tn+1; a step 3, at which A(t.sub.n+1)=A(t.sub.n)+.DELTA.tf(A(t.sub.n))
is calculated; and a step 4, at which n=n.sub.max is determined.
[0025] Here, the differential equation n to be integrated at the
processor 8 herein is an equation of motion (Navie-Stokes equation)
for a flow rate of flow u.sub.f(m/s), which may be represented by:
[ Formula .times. .times. 1 ] .times. .differential. u .fwdarw.
f .differential. t + ( u .fwdarw. f .gradient. ) .times. u .fwdarw.
f = - 1 .rho. .times. .gradient. .times. P + .gradient. ( v t .times.
.gradient. u .fwdarw. f ) + .rho. .times. .times. g .fwdarw. ( 1
) In the formula (1), .rho. is the density (kg/m.sup.3) of a fluid,
P is pressure (Pa), g is acceleration for gravity (m/s.sup.2), and
.nu..sub.t is an eddy viscosity coefficient (m.sup.2/s).
[0026] The differential equations integrated at the processor 8
are the transport equation of turbulent energy k(m.sup.2/s.sup.2)
represented by: [ Formula .times. .times. 2 ] .times. .differential.
k .differential. t + ( u .fwdarw. f .gradient. ) .times. k = .gradient.
( v t .times. .gradient. k ) + P k - ( 2 ) and the transport equation
for the turbulent energy dissipation rate .epsilon. (m.sup.2/s.sup.3)
represented by: [ Formula .times. .times. 3 ] .times. .times. .differential.
.differential. t + ( .times. u .fwdarw. f .gradient. ) .times. =
.gradient. ( v t .times. .gradient. .times. ) + c 1 .times. k .times.
P k - c 2 .times. 2 k , .times. .times. C 1 = 1.44 .times. .times.
.times. C 2 = 1.92 ( 3 ) A turbulent energy production term P.sub.K(m.sup.2/s.sup.3)
is calculated by the following formula (4): [ Formula .times. .times.
4 ] .times. P k = i = 1 3 .times. j = 1 3 .times. 1 2 .times. v
t .function. ( .differential. .times. u _ i .differential. x j +
.differential. .times. u _ j .differential. x i ) 2 ( 4 )
[0027] The eddy dynamic viscosity .nu..sub.t is calculated by the
following formula (5): [ Formula .times. .times. 5 ] .times. v t
= c .mu. .times. k 2 , C .mu. = 0.09 ( 5 ) It should be noted that
in formulae (4) and (5), C.sub.1, C.sub.2, and C.sub..mu. are model
constants of a turbulent model.
[0028] A gas holdup ratio .alpha..sub.b in the culture medium is
calculated by: [ Formula .times. .times. 6 ] .times. .differential.
.alpha. b .differential. t + ( u .fwdarw. g .gradient. ) .times.
.alpha. b = S .alpha. ( 6 ) In the formula (6), bubbles are assumed
to move at a flow rate u.sub.g(m/s) different from the flow rate
of flow u.sub.f(m/s). Where, u.sub.g is obtained by: [Formula 7]{overscore
(u)}.sub.g={overscore (u)}.sub.f+{overscore (u)}.sub.d (7)
[0029] In the formula (7), u.sub.d is a bubble terminal velocity
depending on a bubble diameter, which is often entered in the form
of input data or a function formula being set the bubble diameter
as a parameter. It should be noted that the flow rate of bubbles
u.sub.g may be obtained by the dynamic equation similar to that
in the formula (1); however, when the bubbles have any of smaller
diameters, it is convenient to use the formula (7). The formulae
(1) to (7) may be used to calculate the flow rate and gas holdup
distributions in the fermenter.
[0030] The transport equation of cell density (Xa)(cells/mL) of
the cells transported at a flow rate u.sub.f, which is integrated
at the processor 8, is represented by: [ Formula .times. .times.
8 ] .times. .differential. X a .differential. t + ( u .fwdarw. f
.gradient. ) .times. X a - .gradient. ( v t .times. .gradient. X
a ) = .mu. .times. .times. X a - K d .times. X a ( 8 ) In the formula
(8), .mu. is a specific growth rate (1/s) and Kd is a death rate
(1/s). The transport equation of the medium component transported
at the flow rate u.sub.f, for example glucose (Glc) transported
at the flow rate u.sub.f, which is integrated at the processor 8,
is represented by: [ Formula .times. .times. 9 ] .times. .differential.
Glc .differential. t + ( u .fwdarw. f .gradient. ) .times. Glc -
.gradient. ( v t .times. .gradient. Glc ) = - q Glc .times. X a
( 9 ) Similarly, the transport equation of the medium component,
for example glutamine (Gln) transported at the flow rate u.sub.f
is represented by: [ Formula .times. .times. 10 ] .times. .differential.
Gln .differential. t + ( u .fwdarw. f .gradient. ) .times. Gln -
.gradient. ( v t .times. .gradient. Gln ) = - q Gln .times. X a
- .kappa. gln .times. Gln ( 10 ) The transport equation of the medium
component, for example lactate (Lac) transported at the flow rate
u.sub.f is represented by: [ Formula .times. .times. 11 ] .times.
.differential. Lac .differential. t + ( u .fwdarw. f .gradient.
) .times. Lac - .gradient. ( v t .times. .gradient. Lac ) = q Lac
.times. X a - .kappa. Lac .times. Lac ( 11 ) The transport equation
of the medium component, for example ammonia (Amm) transported at
the flow rate u.sub.f is represented by: [ Formula .times. .times.
.times. 12 ] .times. .times. .differential. Amm .differential. t
+ ( .times. u .times. _ .times. f .gradient. ) .times. Amm - .gradient.
( v .times. t .times. .gradient. Amm ) = q .times. Amm .times. X
.times. a + .kappa. .times. Gln .times. Gln ( 12 )
[0031] It should be noted that the medium components are not limited
to these substances and the transport equation may be described
for any of the biological metabolites involved in mammalian cell
culture.
[0032] In formulae (8) to (12), the right sides are reaction dynamics
model equations, which describe growth, production, and consumption
of the cells (formula (8)) and medium components (formulas (9) to
(12)) through the process of metabolism. These reaction models are
different depending on the type of a stem or cell line to be cultured.
The models applicable to the general processes of mammalian cell
culture are exemplified by the following formulae.
[0033] In the formulae (8) to (12), the reaction equations extracted
only for mammalian cells are represented by: d X a d t = .mu. .times.
.times. X a - k d .times. X a .times. .times. d Glc d t = - q Glc
.times. X a .times. .times. d Gln d t = - q Gln .times. X a - .kappa.
Gln .times. Gln .times. .times. d Lac d t = q Lac .times. X a -
.kappa. Lac .times. Lac .times. .times. d Amm d t = q Amm .times.
X a + .kappa. Gln .times. Gln [ Formula .times. .times. 13 ] Where,
.mu. = .mu. max .times. Glc Glc + K Glc Gln Gln + K Gln 1 [ 1 +
Lac 2 K Lac ] 1 [ 1 + Amm 2 K Amm ] [ Formula .times. .times. 14
] and k.sub.d=k.sub.d0e.sup.-.alpha..mu. [Formula 15] are defined
In addition, q Glc = .mu. Y Glc + m Glc + q E , Glc Glc .times.
Glc - Glc 0 Glc ( Glc - Glc 0 Glc ) + K Glc Glc + q E , Glc Gln
.times. Gln - Gln 0 Glc ( Gln - Gln 0 Gln ) + K Gln Glc .times.
.times. q Gln = .mu. Y Gln + m Gln + q E , Gln Gln .times. Gln -
Gln 0 Gln ( Gln - Gln 0 Gln ) + K Gln Gln .times. .times. q Lac
= .mu. Y Lac + m Lac + q E , Lac Glc .times. Glc Glc + K Glc Lac
.times. .times. q Amm = .mu. Y Amm + m Amm + q E , Amm Glc .times.
Glc Glc + K Glc Amm + q E , Amm Gln .times. Gln Gln + K Gln Amm
[ Formula .times. .times. 16 ] are defined.
[0034] The transport equation for the dissolved oxygen calculated
at the processor 8 is represented by: [ Formula .times. .times.
17 ] .times. .times. .differential. D O .differential. t + ( u _
f .gradient. ) .times. D O - .gradient. ( v t .times. .gradient.
D O ) = - q O .times. .times. 2 .times. X o + K L , O , b .times.
a b .function. ( D O , b eq - D O ) + K L , O , s .times. a s .function.
( D O , s eq - D O ) ( 13 ) Similarly, the transport equation for
dissolved carbon dioxide calculated at the processor 8 is represented
by: [ Formula .times. .times. 18 ] .times. .times. .differential.
D CO 2 .differential. t + ( u _ f .gradient. ) .times. D CO 2 -
.gradient. ( v t .times. .gradient. D CO 2 ) = q CO .times. .times.
2 .times. X a + K L , CO 2 , b .times. a b .times. ( D CO 2 , b
eq - D CO 2 ) + K L , CO 2 , s .times. a s .function. ( D CO 2 ,
s eq - D CO 2 ) ( 14 ) In both formulae (13) and (14), the left
sides indicate a transport term in a flow field and the right sides
indicate terms for carbon dioxide production and consumption caused
by mass transfer at a gas-liquid interface and metabolism. In the
formulae, q.sub.02 is the oxygen uptake rate (mg/cells) per unit
cell and q.sub.C02 is a carbon dioxide exhaust rate (mg/cells) per
unit cell. For mass transfer at the gas-liquid interface, the volumetric
mass transfer coefficient k.sub.La is obtained by multiplying a
mass transfer coefficient velocity K.sub.L at the gas-liquid interface
and a specific surface area a, which comprises of those at a bubble-liquid
interface and at a gas-culture medium interface on the liquid free
surface. A subscript b is used for that of the bubble-liquid interface
and a subscript S is used for that of the liquid free surface. The
specific surface area a.sub.S (1/m) is obtained by dividing a liquid
free surface area S.sub.surf (m.sup.2) by the volume of the fermenter
Vol (m.sup.3) (a.sub.S=S.sub.surf/Vol). The specific surface area
ab is obtained by a.sub.b=6.alpha./Dp using a gas holdup ratio number
density .alpha..sub.b and the bubble diameter Dp(m). The gas holdup
ratio number density .alpha..sub.b is obtained by the aforementioned
formula (6). S.sub..alpha. is bubbles holdup blown from the sparger
at an interval of unit time.
[0035] In the aforementioned formula, .mu..sub.max, k.sub.d0,.alpha.,K.sub.i,K.sup.i.sub.j,
Y.sub.i,m.sub.i, q.sub.Ei.sup.j,.kappa..sub.j(i,j=Glc, Gln,Lac,Amm)
[Formula 19] are experimental constants used in the reaction dynamics
model, respectively, and D.sub.O,b.sup.eq D.sub.O,S.sup.eq, D.sub.CO2b.sup.eq,
D.sub.cO2,S.sup.eq [Formula 20] are the concentration of dissolved
oxygen, which comes to balance a partial pressure of oxygen in the
bubbles, the concentration of dissolved oxygen, which comes to balance
a partial pressure of oxygen on the liquid surface, the concentration
of dissolved carbon dioxide, which comes to balance a partial pressure
of carbon dioxide in the bubbles, and the concentration of dissolved
carbon dioxide, which comes to balance a partial pressure of carbon
dioxide on the liquid surface, respectively. In the formula (13),
the oxygen gas in the bubbles and on the liquid free surface dissolves
into the liquid under the volumetric oxygen transfer coefficient
K.sub.L,o,ba.sub.b or K.sub.L,o,sa.sub.s, and oxygen of q.sub.o2
per cell unit and per time unit are consumed by a cell. In the formula
(14), carbon dioxide of q.sub.co2 per cell unit and per time unit
are exhausted from a cell, and transferred in the form of gas from
the liquid into the bubbles and on the liquid free surface under
the volumetric carbon dioxide transfer coefficient K.sub.L,co2,ba.sub.b
and K.sub.L,co2,sa.sub.s and finally purged out from the fermenter
1.
[0036] Each of mass transfer velocities K.sub.L,i,j (i=O, CO.sub.2,
j=b, S) described herein is given as a function of only the turbulent
energy k, the turbulent energy dissipation rate .epsilon., and the
diffusion coefficient Di (i=O, CO.sub.2) that is a property value,
and does not depend directly on the size of the fermenter 1, the
bubble diameter, and so on. The turbulent energy k and the turbulent
energy dissipation rate .epsilon. are calculated by the formulae
(2) and (3) in the algorithm and thereby, they may be obtained at
any point in any fermenter 1 only by means of computations with
no experiment.
[0037] According to the present invention, the use of these volumetric
mass transfer coefficients allows the fluid state in any shape of
fermenter, as well as all the concentration distributions of the
biomass metabolite, aerating gas, and dissolved gas to be obtained
by means of computations and therefore, the shape and culture conditions
of the fermenter 1 suitable for biomass culture may be known.
[0038] The results of computations executed at the processor 8
are sent to the display unit 10 and displayed. The display unit
10 may display the results of computations in the form of distribution
images and/or graphs. The examples of distribution images are shown
in FIGS. 4-1A to 4-1D and FIGS. 4-2A to 4-2H. FIG. 4-1A shows an
example of the displayed image of a flow rate vector, FIG. 4-1B
shows an example of the displayed image of the gas holdup ratio,
FIG. 4-1C shows an example of the displayed image of the distribution
of the turbulent energy k, FIG. 4-1D shows an example of the displayed
image of the distribution of the energy dissipation rate .epsilon.,
FIG. 4-2E shows the displayed image of the distribution of the volumetric
oxygen transfer coefficient Kla, FIG. 4-2F shows the displayed image
of the Kolmogorov scale distribution, FIG. 4-2G shows an example
of the displayed image of the oxygen concentration distribution,
and FIG. 4-2H shows an example of the displayed image of the carbon
dioxide concentration distribution.
[0039] Examples of the displayed graphs are shown in FIGS. 5A and
5B. FIG. 5A shows an example of the displayed graph of the relationships
between the impeller rotation speed, and turbulent energy k and
the turbulent energy dissipation rate .epsilon.. FIG. 5B shows an
example of the displayed graph of the relationship between the impeller
rotation speed, and the total volumetric oxygen transfer coefficient
Kla, the volumetric liquid-surface oxygen transfer coefficient Kla,s,
and the volumetric bubble-oxygen transfer coefficient Kla,b.
[0040] Thus, according to the control device for fermenter of the
present invention, displaying the results of computation executed
at the processor 8 may give quantitative information on the condition
inside the fermenter 1. For example, as shown in FIGS. 4-1 and 4-2,
it is clear that according to the control device for fermenter of
the present invention, the turbulent energy k and the turbulent
energy dissipation rate .epsilon. take relatively larger values
around the impellers and the volumetric oxygen transfer coefficient
K.sub.L,o,b,a.sub.b also takes a larger value as a function of these
factors around the impellers. As known from FIG. 5A, as the impeller
rotation speed increases, the flow rate in the fermenter becomes
higher and k and .epsilon. increase accordingly, while the turbulent
energy k increases in proportion to the square value for the rotation
speed and the turbulent energy dissipation rate .epsilon. increases
in proportion to the cubic value for the rotation speed. As shown
in FIG. 5B, regarding the volumetric oxygen transfer coefficient,
the dependency of K.sub.L,o,b,a.sub.b for bubbles and K.sub.L,o,S,a.sub.s
for the liquid surface on the rotation speed is different. This
is because the dependency of average k and .epsilon. values in the
fermenter 1 on the rotation speed is different from the dependency
of k and .epsilon. values in the vicinity of the liquid free surface
on the rotation speed.
[0041] According to the control device for fermenter of the present
invention, it is preferable that the differential equations describing
the variations of the medium components over time in the aforementioned
formulae (8) to (12) are modified to the differential equations
suitable for different culture conditions. This means that it is
preferable that the variations of the medium components over time
depend on the cell line being cultured and the culture environment
and thereby, the experimental constants are determined so that suitable
differential equations may be established.
[0042] To achieve this, the processor 8 processes time-series measured
data sent from the measurement unit 4 equipped in the fermenter
1 to determine the experimental constants for the differential equations,
which best reflect the time-series measured data. In this case,
it is assumed that no spatial distribution is included in the experimental
constants for the differential equations. The component concentrations
measured in the actual fermenter 1 generally can give only information
on the averages of the components in the fermenter 1 and thereby,
to obtain the detail spatial distributions, fluid dynamics calculations
in the aforementioned formulae (1) to (7) are requisite. Accordingly,
the differential equations given herein are assumed to describe
the variations in spatial average concentrations of the medium components.
[0043] Specifically, the processor 8 takes differences between
the calculated values for a historical curve of component concentration,
which are obtained by numerically integrating the differential equations
of the medium components and the values for a historical curve measured
to determine the experimental constants, for which the differences
between these values take the minimum values. For example, assuming
that a set of components in the aforementioned formulae (8) to (12)
is Xi (I=cell density, glucose, glutamine, lactate, ammonia) and
a set of experimental constants is Kj (j=1, m), a set of the differential
equations may be formally represented by: d x i d t = f i .function.
( x 1 , .times. .times. x n , K 1 .times. .times. .times. K m )
[ Formula .times. .times. 21 ] Defining that the observed data is
Xi.sup.obs (t) and the calculated value is Xi.sup.cal (t), the calculated
value is obtained by: x.sub.i(t)=x.sub.i(0).intg..sub.0f.sub.i(x.sub.1,
. . . x.sub.n, K.sub.1, . . . K.sub.m)d.tau. [Formula 22] Taking
the square value of the difference between the measured value and
the calculated value and representing a function G by: G .function.
( K 1 .times. .times. .times. K m ) = .intg. 0 T .times. i = 1 n
.times. [ x i .function. ( 0 ) + .intg. 0 t .times. f i .function.
( x 1 , .times. , x n , K 1 , .times. .times. K m ) .times. d .tau.
- x i obs .function. ( t ) ] 2 .times. d t , [ Formula .times. .times.
23 ] the function G is represented as a function of the experimental
constant Kj(j=1, m). Accordingly, the model constant Kj(j=1, m),
for which the function G takes the minimum value, may be obtained
by the extreme-value search theory, for example, the sequential
quadratic programming or conjugate gradient method. This method
may be applicable provided that the differential equation has been
formulated with several experimental constants included and is not
affected by the actual differential equation. Alternatively, a method,
by which a Kalman filter is structured, may be applied in sequentially
obtaining the experimental constants. Using the experimental constants
obtained in this way in executing the computations based on the
aforementioned formulae (1) to (12), the results of computations
approximate to the actual fermenter conditions may be obtained.
This is useful in improving the accuracy of control of the fermenter.
[0044] It is preferable that in the control device for fermenter
according to the present invention, the processor 8 calculates the
target value for control and the drive control unit 6 generates
the drive control signals based on the target value for control.
The drive control signals generated at the control device for fermenter
according to the present invention control the aerating unit 2,
the impeller 3, and the nutrient component feed unit 5 in the fermenter
1 to establish desired conditions in the fermenter 1.
[0045] The amount of oxygen gas aerated into the fermenter 1 by
controlling the aerating unit 2 is determined in the process described
below.
[0046] First of all, the target value for control is calculated
in terms of the preset target value for a concentration of dissolved
oxygen in the following manner. When the cell density Xa and agitation
rotation speed rpm to increase the amount of aerating oxygen So
are increased and the dissolved oxygen concentration Do that is
obtained by integrating the differential equations, is a monotone
increasing function of the amount of aerating oxygen So. Accordingly,
using, for example, the regula falsi algorithm shown in FIG. 6,
the amount of aerating oxygen So may be obtained. This means that
the operator: enters the amount of biomass (X), the rotation speed
(rpm) of the impellers 3, and the target value for dissolved oxygen
concentration (Do) at the input unit (step 10); reads the preset
the maximum amount of aerating gas S.sub.max (e.g., S.sub.max=10.sup.18)
and specifies the minimum amount of aerating gas S.sub.min (e.g.,
S.sub.min=0) (step 11); and sets the amount of aerating gas So (e.g.,
So=(S.sub.max+S.sub.min)/2) (step 12). Next, the dissolved oxygen
concentration Do in the fermenter 1 is calculated based on the differential
equations in the formula (13) at the processor 8 (step 13). Next,
The calculated dissolved oxygen concentration Do and the target
value entered at the step 10 are compared (step 14) and if the calculated
dissolved oxygen concentration agrees with the target value, the
processing is terminated. If the calculated dissolved oxygen concentration
does not agree with the target value and exceeds it (step 15), the
amount of aerating gas So is re-set to S.sub.max (step 16) and if
the calculated dissolved oxygen concentration does not exceed the
target value (step 15), the amount of aerating gas So is re-set
to S.sub.min (step 16). Finally, the step 11 and subsequent steps
are repeated until the measured dissolved oxygen concentration Do
agrees with the target value entered in the step 10.
[0047] For the agitation rotation speed of the impellers 3, a value
suitable for the biomass to be cultured may be set. Generally, an
increase in rotation speed of the impellers 3 brings an increase
in turbulent energy k and turbulent energy dissipation rate .epsilon.,
which in turn, makes the volumetric oxygen transfer coefficient
larger to facilitate dissolution of the oxygen gas into the culture
medium. Accordingly, for example, in culturing microorganisms, the
target value for control over the agitation rotation speed may be
set to the maximum rotation speed under the constraint of an agitation
power.
[0048] On the other hand, in culturing mammalian cells, the target
value for control over the agitation rotation speed should be determined
based on two factors, namely cell death due to shear stress and
accumulation of dissolved carbon dioxide. When an increase in agitation
rotation speed results in the increased flow rate in the fermenter
1, some of the cells come to death due to hydrodynamic shear stress.
Therefore, running the impellers 3 at too high rotation speeds is
not preferable in culturing the mammalian cells. The shear rate
y of the flow represented by; .gamma. = i = 1 .times. j = 1 .times.
( .differential. u _ .times. i .differential. xj + .differential.
u _ .times. j .differential. xi ) 2 [ Formula .times. .times. 24
] and the Kolmogorov's eddy length scale .eta. represented by: .eta.
= ( v 3 ) 1 4 [ Formula .times. .times. 25 ] are known to be indicators
for the occurrence of cell death. If .gamma. exceeds a given value
.gamma..sub.max in the flow rate field in the fermenter, it is determined
that the cell death occurs. Similarly, if the Kolmogorov's eddy
length scale .eta. is smaller than cell's size scale .eta.c, it
is determined that the cell death occurs. In the present invention,
all the flow rate gradients .differential.ui/.differential.xj and
the turbulent energy dissipation rates .epsilon. may be numerically
obtained and therefore, either value of .gamma. or .eta. may be
calculated at any point in the fermenter. By reference to the value,
the probability of cell death may be estimated.
[0049] The example of the displayed image of the Kolmogorov scale
distribution calculated at the control device according to the present
invention is shown in FIG. 4-2F. An example of the displayed graph
of the Kolmogorov scale using the rotation speed as a parameter
is shown in FIG. 7. With the control device according to the present
invention, based on the graph shown in FIG. 7, the possibility of
cell death depending on the agitation rotation speed of the impellers
3 may be estimated and the agitation rotation speed of the impellers
3 may be optimized.
[0050] The agitation rotation speed of the impellers 3 affects
the dissolved carbon dioxide concentration in the culture medium.
If the dissolved carbon dioxide concentration in the culture medium
increases, the growth rate of cells deteriorates. FIG. 8 shows the
dissolved carbon dioxide concentration in the fermenter 1 obtained
by solving the differential equations using the rotation speed of
the impellers 3 as a parameter under the fixed conditions of cell
density Xa and the dissolved oxygen concentration Do. As known from
FIG. 8, when the rotation speed of the impellers 3 increases, the
dissolved carbon dioxide concentration in the fermenter 1 reduces.
This is because that with the increased rotation speed, the volumetric
carbon dioxide transfer coefficient in the bubbles or on the liquid
free surface becomes large and thereby, the speeds of transforming
the dissolved carbon dioxide from the biomass transforms into the
liquid phase, and purging the dissolved carbon dioxide from the
fermenter 1 through the liquid free surface increase. The control
device of the present invention allows the possibility of cell death
depending on the agitation rotation speed of the impellers 3 to
be estimated based on the graph shown in FIG. 8 and the agitation
rotation speed of the impellers 3 to be optimized.
[0051] Based on the graph illustrating the Kolmogorov scale shown
in FIG. 7 and the graph illustrating the dissolved carbon dioxide
concentration shown in FIG. 8, the rotation speed of the impellers
3, at which the dissolved carbon dioxide concentration is minimized
under such a condition that cell death does not occur, may be determined.
[0052] In addition, the processor 8 according to the present invention
determines the amount of aerating gas supplied into the fermenter
1 and the target value for control over the rotation speed of the
impellers 3, as well as the target value for control over nutrient
supply. Under control over nutrient supply, the cells are continuously
cultivated while the nutrient components consumed in the culture
process are additionally supplied together with the fresh culture
medium, which is called the Fed Batch method. In the Fed Batch method,
the medium is continuously added with no medium extracted and thereby,
the amount of liquid increases as the culture process advances.
In culturing the mammalian cells, such nutrient components (the
components positively involved in cell growth) as glucose, glutamine,
amino acid, and serum are supplied. No medium is extracted and therefore,
lactate and ammonia accumulate with no reduction in their amounts.
[0053] Focusing on the differential equations describing the variations
in the medium components over time shown in the aforementioned formulae
(9) to (12), the following quantitative information may be given.
Decreases in glucose and glutamine reduce the cell growth rate,
inhibiting an increase in cell count. Assuming that the yield of
the product is proportional to the cell count, it is considered
that supplying sufficient amounts of glucose and glutamine increases
the yield of the product. On the other hand, focusing on the differential
equations describing the variations in lactate and ammonia over
time, it is considered that increases in glucose and glutamine make
the lactate and ammonia concentrations higher, while increases in
lactate and ammonia (the components negatively involved in cell
growth) inhibit the cell growth rate. Accordingly, it is possible
that at any optimal concentrations of glutamine and glucose, the
cell count and the yield of the product are maximized. It is preferable
that the control device according to the present invention optimally
controls the supply of the medium components such as glucose and
glutamine. Specifically, the processor 8 calculates the target value
for control by means of dynamic programming shown in the following
procedure using the differential equations in the aforementioned
formulae (9) to (12). The calculated target value for control is
input into the drive control unit 6. The drive control unit 6 generates
the drive control signals based on the entered target value for
control in order to control the supply of the medium components
by the feed unit 5.
[0054] First of all, assuming that a set of cell density, lactate
and ammonia (the components negatively involved in cell growth)
is a state variable when the differential equations in the formulae
(9) to (12) are given, x=(X.sub..alpha.,Lac,Amm) [Formula 26] is
established. Assuming that glucose and glutamine (the components
positively involved in cell growth) are supplied from the nutrient
feed unit and the concentrations of glucose and glutamine are control
variables, v=(Glc, Gin) [Formula 27] is established. Then, a set
of differential equations in the formulae (9) to (12) may be formally
described by: d x d t = f .function. ( x , v ) [ Formula .times.
.times. 28 ] If a time axis is divided into N portions in the chronological
order, the state vector at a time n+1 is obtained using the value
at a time n by: x.sub.n+1=x.sub.n+f(x.sub.n,v.sub.n).DELTA.t.ident.h(x.sub.n,v.sub.n)
[Formula 29] The yield of the product is considered to be proportional
to the cell density and therefore, assuming that a target function
is described as G = .intg. 0 T .times. X a .times. .times. d t .apprxeq.
n = 0 N .times. g .function. ( x n , v n ) [ Formula .times. .times.
30 ] so as to obtain the target value for maximizing G, v.sub.n=v(x.sub.n)
[n=0, 1, . . . , N]. Where, g(x.sub.n,v.sub.n) is an integral value
for a discrete minute area represented by: g .function. ( x n ,
v n ) = .intg. n t n + 1 .times. X a .times. .times. d t = X a .function.
( t n ) .times. .DELTA. .times. .times. t + ( d X a d t ) t n .times.
.DELTA. .times. .times. t 2 2 [ Formula .times. .times. 31 ] The
expression giving the maximum value for G, F N .function. ( x 0
) = Max v 0 .times. .times. .times. .times. v N .function. [ g .function.
( x 0 , v 0 ) + g .function. ( x 1 , v 1 ) + + g .function. ( x
N , v N ) ] [ Formula .times. .times. 32 ] is defined. Where, x.sub.0
is an initial value for the state variable. In this case, F n .function.
( x N - n ) = .times. Max v N - n [ g .times. ( x N - n , v N -
n ) + .times. F n - 1 .function. ( h .function. ( x N - n , v N
- n ) ) ] = .times. Max v 0 .function. [ g .function. ( x 0 , v
0 ) + F N - 1 .function. ( h .function. ( x 0 , v 0 ) ) ] [ Formula
.times. .times. 33 ] are established. These relational expressions
deduce a recursive expression F N .function. ( x 0 ) = Max v 0 .function.
[ g .function. ( x 0 , v 0 ) + Max v 1 .times. .times. .times. .times.
v N .function. [ g .function. ( x 1 , v 1 ) + + g .function. ( x
N , v N ) ] ] [ Formula .times. .times. 34 ] at any time n with
no loss of generality.
[0055] When the aforementioned recursive expression is used, an
arithmetic algorithm for dynamic programming behaves as described
below. It is assumed that a domain, in which the state variable
X may take a value, is x .epsilon. .OMEGA..sup.3 and a domain, in
which the control variable V may take a value, is v .epsilon. .LAMBDA..sup.2.
To execute numeric calculations, the domain .OMEGA..sup.3 for the
state variable and the domain .LAMBDA..sup.2 for the control variable
are made discrete as long as the memory capacity of a computer accepts.
Specifically, this means that for example, the cell density is divided
into 10 domain elements from 10.sup.6 cells/mL to 10.sup.7 cells/mL
in increments of 10.sup.6 cells/mL, the concentration of lactate
is divided into 10 domain elements from 0 to 1000 mg/L in increments
of 100 mg/L, and the concentration of ammonia is divided into 10
domain elements from 0 to 100 mg/L in increments of 10 mg/L. Accordingly,
the domain .OMEGA..sup.3 is divided into 1000 domain elements .OMEGA..sup.3i
(I=1, . . . , 1000). Similarly, assuming that for example, the concentration
of glucose is divided into 10 domain elements from 0 to 2000 mg/L
and the concentration of glutamine is divided into 10 domain elements
from 0 to 1000 mg/L, the domain .LAMBDA..sup.2 for the control variable
is divided into 100 domain elements .LAMBDA..sup.2j(j=1, . . . ,
100).
[0056] First of all, x .epsilon. .OMEGA..sup.3.sub.i for each of
domain elements is obtained by: F 0 .function. ( x N ) = Max v N
.di-elect cons. .LAMBDA. j 2 .function. [ g .function. ( x N , v
N ) ] [ Formula .times. .times. 35 ] In this case, v.sub.N .epsilon.
.LAMBDA..sup.2.sub.j, which gives the maximum value, is different
for each x.sub.N .epsilon. .OMEGA..sup.3.sub.i and thereby, v.sub.N,
which gives the maximum value, is v.sub.N=v.sub.N(x.sub.N), a function
of x.sub.N. This is stored in memory.
[0057] Next, for each x.sub.N-1 .epsilon. .OMEGA..sup.3.sub.i,
F 1 .function. ( x N - 1 ) = Max v N - 1 .function. [ g .function.
( x N - 1 , v N - 1 ) + F 0 .function. ( h .function. ( x N - 1
, v N - 1 ) ) ] [ Formula .times. .times. 36 ] is calculated. In
this case, the second term in the right side has been already determined
because of the following; F.sub.0(h(x.sub.N-1,v.sub.N-1))=F.sub.0(x.sub.N)
[Formula 37] The v.sub.N-1=v.sub.N-1(x.sub.N-1), which gives the
maximum value for each x.sub.N-1 .epsilon. .OMEGA..sup.3.sub.i,
is stored. Subsequently, in the same manner as that mentioned above,
F n .function. ( x N - n ) = Max v N - n .function. [ g .function.
( x N - n , v N - n ) + F n - 1 .function. ( h .function. ( x N
- n , v N - n ) ) ] [ Formula .times. .times. 38 ] and v.sub.N-n=v.sub.N-n(x.sub.N-n),
which gives the maximum value, are calculated. Finally, F N .function.
( x 0 ) = Max v 0 .function. [ g .function. ( x 0 , v 0 ) + F N
- 1 .function. ( h .function. ( x 0 , v 0 ) ) ] [ Formula .times.
.times. 39 ] and v.sub.0=v.sub.0(x.sub.0) are calculated to complete
a series of computations.
[0058] v.sub.n=v(x.sub.n) [n=0, 1, . . . , N] stored in memory
through the aforementioned computation process is the target control,
x.sub.n [n=0, 1, . . . , N] is a trajectory drawn by the state variable,
and F.sub.N(x.sub.0) is the maximum variable for the target function.
In culturing cells, if a culture period is approximately 10 days
and 24 hours/day are divided into 10 time elements, the time axis
division number N is 100 and if a domain .OMEGA..sup.3 for the state
variable is divided into 1000 domain elements, 10.sup.5 array variables
are required to store each v.sub.n=v(x.sub.n) [n=1, . . . , N],
which is the target control in the aforementioned computation process.
This requires a huge amount of memory capacity, however, the current
computer sufficiently addresses this requirement of memory capacity.
[0059] At the control device according to the present invention,
the target value for control used in controlling the amount of supplied
medium components such as glucose and glutamine may be calculated
by executing the aforementioned computation process at the processor
8. The target value for control calculated at the processor 8 is
input into the drive control unit 6. At the drive control unit 6,
the drive control signals are generated based on the input target
value for control and the measured value read from the measurement
unit 4 by for example, the PID control system. The drive control
signals are output into the nutrient component feed unit 6 so as
to control the amount of nutrient components supplied by the nutrient
component feed unit 6. Accordingly, with the control device of the
present invention, the fermenter may be driven under the condition
where the biomass growth rate is maximized. |