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Linearized Parameter Estimation Methods for Modeled Crystallization... 69
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primary nuclei-originated. Grown seed crystals, of which the product crystals are
composed in the successful full seeding, are defined as seed-grown crystals. Grown
secondary nuclei originated from seed-grown crystals and their grown descendants,
of which the product crystals may be mainly composed in the partial seeding, are
defined as seed-originated crystals. Grown primary nuclei, grown secondary nuclei
originated from grown primary nuclei, and their grown descendants, of which the
product crystals are completely composed in the internal seeding, are defined as
primary nuclei-originated crystals. Mass fractions from each origin are computed in
Sect. 2.
Consequently, the statistics mentioned above versus seed loading ratio, or ratio of
seed loading mass to theoretical crystal yield, are computed and illustrated several
other materials.
This trend indicates that partial seeding is most effectively performed at the first
local minimum point of the CV. Therefore, the seed loading ratio at which the CV
takes the first local minimum can be regarded as the optimal one under a certain
condition of the seed quality and the cooling method. Then, the optimum seed
loading ratio is estimated for some cooling rates and some seed crystal sizes, and the
resulting relation among optimum seed loading ratio, cooling rate, and seed crystal
size is illustrated in Fig. 7. In addition, the resulting relation among local minimum
CV, cooling rate, and seed crystal size is shown in Fig. 8. Here, the local minimum
CVs correspond to the first local minima, and the optimum seed loading ratios to the
arguments of the first local minima, in the charts of CV versus seed loading ratio.
In Fig. 7, the optimum seed loading ratio is affected both by the seed quality and
by the cooling method. On the other hand, in Fig. 8, the local minimum CV is not
affected by the seed crystal size but by the cooling rate, which might be attributed
to the role of the seed crystals in the partial seeding as the catalysts for secondary
nucleation. In short, the seed quality will not affect the optimal product quality but
the optimal control.
2.3 Process Design
In Sect. 2.3, a simple method for the crystallization process design is developed
for the optimization of partially seeded crystallization of the model substance. As
mentioned in Sect. 2.2, the seed quality will not affect the optimal product quality,
which enables the seed slurry to replace the seed crystals filtrated, dried, milled,
and sieved. The seed slurry can be prepared by recycling the product slurry or in
the other unseeded crystallizer. The crystal quality of the seed slurry cannot be
controlled but monitored with the PAT tools. The acquired data will be utilized for
determining the optimal control procedure.
The seed quality may be determined by the size and the standard deviation
in the size. However, in the simulation, the standard deviation affects neither the
minima nor the arguments and may not matter to the partial seeding. As for the
cooling method, one can change the cooling period and the temperature profile for
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0
,noitairavfotneiciffeoC CV
100
,noitcarfssam w [%]
80
60
40
20
Lp/L
― CV
--- w
-ʀw
-w
100
s
sg
so
po
10
[-]
3,0,s
/L
3,0,p
L
Crystal size ratio,
or
10
1
0
0
-15
10
0.00.00.01.0
10
-10
10
-5
Seed loading ratio, Cs[-]
Fig. 6 Mean volume size ratio of product crystal to seed one, CV, and mass fractions of crystals
from each origin versus ratio of seed loading mass to theoretical crystal yield at the cooling rate of
3.3 K/h and at the seed size of 31.6 μm. Seed, product, seed-grown, seed-originated, and primary
nuclei-originated are denoted by the subscripts s, p, sg, so, and po, respectively. (Reproduced from
Ref. [23])
-2
0.0
10
,oitargnidaoldeesmumitpO
0.0
10
-4
0.556 K/h
0.795 K/h
1.14 K/h
1.63 K/h
-6
0.0
10
[-]
-8
s,opt
0.0
10
C
-10
0.0
10
-12
0.0
10
-6
10
0.0 0.0 0.0 0.0 0.0
Cooling rate,
-5
10
10
R =
-4
Seed mean volume size, L
2.33 K/h
3.33 K/h
4.77 K/h
6.83 K/h
9.77 K/h
14.0 K/h
20.0 K/h
-3
10
s,3,0
[m]
10
-2
Fig. 7 Optimum seed loading ratio for partial seeding versus seed mean volume size and versus
cooling rate. (Reproduced from Ref. [23])
controlling the product quality. For example, in Fig. 6, the cooling period is fixed to
6 h, and the solution is cooled linearly, which means that the decrease in temperature
is proportional to the first power of time. Therefore, the product CVs in Fig. 8 may
be improved by changing the temperature profiles, such as the exponent of time.
In other words, the cooling may be programmed for optimization of the partial
Linearized Parameter Estimation Methods for Modeled Crystallization... 71
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80
[%]
0
Cooling rate,
R =
0.556 K/h
0.795 K/h
1.14 K/h
1.63 K/h
muminimlacoL CV
40
2.33 K/h
3.33 K/h
4.77 K/h
6.83 K/h
9.77 K/h
14.0 K/h
20.0 K/h
20
-6
0.0 0.0 0.0 0.0 0.0
10
Seed mean volume size, L
Fig. 8 Local minimum value of CV versus seed mean volume size and versus cooling rate.
(Reproduced from Ref. [23])
seeding. Then, the exponent of time for temperature profile is optimized for partial
seeding under several conditions of the cooling period, and the optimum exponent
and the resulting minimum CV versus the cooling period are depicted in Fig. 9.
At the same time, the re-optimized seed loading ratio for the programmed cooling
is shown in Fig. 10. Here, typical mean size, or L
and the seed loading ratio need to be optimized simultaneously, and the regression
equations are also shown. As is mentioned in Sect. 2.3, these regression equations
will be utilized for the process design. Nevertheless, it should be noted that there is
room for improvement in the temperature profile in this case, where the decrease in
temperature is set to be proportional to the power function of time.
Before a concrete example is provided, the several conditions of the experimental
procedure are added to those considered in Sect. 2.2 as follows. At first, mass of
solvent is set to 3 kg, which fixes the theoretical crystal yield to 113 g. Next, the
seed slurry, in which the seed mean size is measured at 100 μm, is prepared in the
other unseeded crystallizer and then added to the main crystallizer. Finally, in order
to improve productivity, the product crystals are required to have the CV not more
than 40%. Under these conditions, cooling and seeding methods are optimized to
meet the demand for the product quality and to minimize the cooling period.
This type of optimization involves complex non-linear problems, but they can
be solved as a simplified linear programming problem of the regression equations
mentioned in Sect. 2.2. The regression equations used for the process design are
shown below:
10
-5
10
-4
1,0
-3
10
[m]
s,3,0
, is used, both the exponent
10
-2
CV
/% =−18.5log
0,opt
/s)+ 113 (31)
τ
(
1
10
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60
50
[%]
0,opt
roftnenopxemumitpO
[-]
opt
,gniloocdemmargorp np
1.0
0.8
np
= 0.714 log10(τ1/ s) - 1.88
opt
0.6
40
0.4
cooling, CV
CV
0.2
/ % = -18.5 log10(τ1/ s) + 113
0,opt
30
Minimum CV for programmed
1200 3600 10800
Cooling period, τ1[s]
Fig. 9 Optimum exponent for programmed cooling and resulting minimum product CV versus
cooling period for the partial seeding
-3
0.0
10
Mean size,
[-]
s,opt
,oitar C
0.0
10
C
= 4.12ʹ1012ʹ
s,opt
-6
(L
s,1,0
/ m)
2.95
(τ1/ s)
-1.26
L
s,1,0
3.16 μm
10.0 μm
=
31.6 μm
100 μm
Re- gnidaoldeesdezimitpo
10
0.0
-9
1200 3600 10800
Fig. 10 Re-optimized seed loading ratio for programmed cooling versus cooling period for the
partial seeding
C
s,opt
Here, CV
cooling, C
is local minimum CV, np
0,opt
is optimum seed loading ratio, and τ1is cooling period. Eq. (31)
s,opt
is concerned with the satisfaction of the demand for the quality and with the
Cooling period, τ1[s]
np
= 0.714 log
opt
= 4.12 × 10
10
12
L
s,1,0
opt
/s)− 1.18 (32)
τ
(
1
/m
2.95
−1.26
/s
τ
)
(
1
(33)
is optimum exponent for programmed
Linearized Parameter Estimation Methods for Modeled Crystallization... 73
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minimization of the production time, Eq. (32) with the optimization of the cooling
method, and Eq. (33) with the optimization of the seeding method.
These equations are utilized for the process design as follows. At first, the
demand for the quality is substituted into Eq. (31), and the resulting inequity is
solved to derive the minimum cooling period required to meet the demand. This
method supposes the tendency that higher product quality requires longer cooling
period. Next, the minimum cooling period obtained is substituted into Eq. (32)to
derive the optimum temperature profile. Finally, the minimum cooling period and
the seed mean size measured are substituted into Eq. (33) to derive the optimum
seed loading quantity.
In the case of the model substance, at first, CV
(31) to derive τ
≥ 2.51 h. Then, τ1= 2.51 h into Eq. (32) to derive np
1
≤ 40% is substituted into Eq.
0,opt
opt
= 0.948,
where, like in natural cooling, the cooling is a little faster at an early stage than at
a late stage. In partial seeding, this type of cooling might be work well, because
the faster cooling at an early stage might induce the secondary nuclei, and because
the slower one at the last stage might enlarge the nuclei. Finally, τ
L
= 100 μm into Eq. (33) to derive C
s,1,0
= 6.96 ×10−5, and the optimum seed
s,opt
= 2.51 h and
1
loading ratio multiplied by the theoretical crystal yield is the optimum seed loading
quantity of 7.86 mg. These control variables completely define the optimized
method of cooling and seeding for partially seeded crystallization.
2.4 Case Study: L-Arginine Crystallization
In Sect. 2.2, seeded crystallization in the model system is investigated to characterize partial seeding. In Sect. 2.4, we show the case study of partially seeded
crystallization of Arg, which were originally reported by Unno and Hirasawa [25].
An aqueous solution of Arg is selected as a target substance. The experimental
procedure was as follows. At first, the seed suspension to which the seed crystals had
been sieved and added was prepared and kept at 30
of Arg was dissolved in 300 mL of water at 35
◦
temperature of 30
C. Finally, the solution was linearly cooled down to 20◦C, stirred
at 300 rpm, and monitored with the PAT tools. In cooling, the seed suspension was
added when the solution temperature reached 30
temperature was kept for half an hour. The seed loading ratios were set to around the
optimum ratiopredicted by simulation. As for the simulation, the physical properties
and the kinetic parameters reported by Unno et al. [7] were utilized, and similar
preconditions as in Sect. 2.2 were assumed to be established with such exceptions
as follows: primary nucleation was neglected, the coefficients of each rate varied
with temperature according to the Arrhenius equations, and the growth rate was
expressed as the function of the relative supersaturation.
As a result, simulated and measured charts of CV of the product crystals
versus seed loading ratio are illustrated in Fig. 11.InFig.11, the simulated
optimum seed loading ratio is reasonably close to the measured one. However,
◦
C. Then, a saturating quantity
◦
C, which was higher than saturation
◦
C. After the cooling, the solution
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55
54
[%]
0
53
detalumiS CV
120
105
90
[%]
0
52
75
51
Sim.
Meas.
Obs.
50
-8
10
1.E-08 1.E-06 1.E-04 1.E-02
10
-6
10
-4
10
-2
Measured CV
60
Seed loading ratio [-]
Fig. 11 Simulated and measured relations between seed loading ratio and product CV at the
cooling period of 1.67 h and with the seed crystals sieved to 44–74 μm. The error bar is standard
error of three measurements. (Reproduced from Ref. [7])
the minimum CV is not predicted successfully due to the ignorance of primary
nucleation, agglomeration, and breakage. In addition, the error bars are so long
that one can hardly recognize significant differences among the plots, which might
be attributed to the stochastic nature of nucleation in a laboratory scale and to
the measurement errors of the very small suspension volume added. These results
suggest the difficulties in validation of the optimal partial seeding in laboratories.
2.5 Quality Stability
In Sect. 1.6, we show how to solve the SDEs including stochastic nucleation, and the
experimental results in Sect. 2.4 imply that the stochastic nature of nucleation may
affect the fluctuation of the product quality under a certain controlling condition.
In Sect. 2.5, we discuss the simulation for the effect of stochastic nucleation on the
product quality in seeded crystallization.
As in Sect. 2.2, an aqueous solution of potassium sulfate is selected as a
target substance, and similar preconditions are assumed to be established with
the exception that primary nucleation and secondary nucleation are considered to
be stochastic processes. Then, the same experimental procedure as in Sect. 2.2 is
considered, and the product CV is calculated 50 times per 1 condition.
Consequently, the simulated relation between seed loading ratio and product
CV is illustrated by the box plot in Fig. 12 under an example condition. The
deterministic CVs are also shown in Fig. 12. These results may imply that in
some cases, the fluctuation of the product quality caused by the stochastic nature
of nucleation is not negligible. For example, at the seed loading ratio of 10
−2
in Fig.
Linearized Parameter Estimation Methods for Modeled Crystallization... 75
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― : Median ڧ : Quartile
㺎 :Maximum 㺎 : Minimum
㸩 : Outlier ࠐ : Mean
ڸ : Deterministic
-1010-910-810-710-610-510-410-310-210-1100
10
Fig. 12 Stochastic and deterministic relations simulated between seed loading ratio and product
CV at the cooling period of 6 h, the seed mean size of 316 μm, and the solvent mass of 10 g. In
the box plot, the maximum whisker length is 1.5 times of each box length
12, the box length of the quartile is 5 percentage points despite the median of a little
less than 50%.
In addition, another simulation suggests that primary nucleation may occur
in the range of the seed loading ratio not more than 10
nucleation not more than 10
−1
. Compared to these ranges, Fig. 12 might indicate
−4
and that secondary
that stochastic nucleation, both primary and secondary, causes the fluctuation of
measured values and that stochastic primary nucleation specially causes the error
between the deterministic value and the mean of stochastic ones.
The stochastic behavior of nucleation, which will make it difficult to control
the crystallization process, may be inevitable if nucleation happens. However, the
fluctuation and the error can be reduced by means of nucleation at high speed
or scale-up of the crystallizer. Moreover, one can predict them in the simulation
mentioned above. This prediction will have to be taken into account for the process
design. For instance, the optimal control which can produce the crystals with high
quality but may causethe significant fluctuationwill have to beavoided or improved.
3 Conclusions
At first, the mathematical models for the crystallization phenomena, such as growth,
nucleation, breakage, and agglomeration, are introduced to describe pharmaceutical
crystallization processes, and parameter estimation methods are developed for
the parameters of these models. For the simple parameter estimation, in-line
measurements with PATs, such as FBRM and ATR-FT-IR, may be helpful. By using
these PAT tools, the kinetic parameters of growth and secondary nucleation for Arg
76 I. Hirasawa et al.
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crystallization can be estimated simply and linearly. As for primary nucleation, the
stochastic nature is significant, and hence the SDEs including stochastic primary
nucleation should be solved so many times, as well as repeated measurements, that
the kinetic parameters can be estimated. With the PAT tools, some parameters of
breakage may be estimated simply and linearly, but others may non-linearly. It
is difficult to analyze solely the agglomeration kinetics due to the dependence of
agglomeration on supersaturation, but the kinetic parameters of agglomeration may
be estimated when the parameters of the other phenomena are known.
Then, the developed models are applied to the optimization of pharmaceutical
crystallization processes. Among several seeding policies classified based on seed
loading quantity, partial seeding may be a good choice for pharmaceutical processes.
In partially seeded crystallization of a model substance, the seeding and cooling
methods are simultaneously optimized for the best product quality, which results
in the design formulae of the seeding and cooling conditions. By using the design
formulae, the detailed operating conditions of the crystallization process designed
optimally can be calculated for the model substance. This optimization method
was applied to partially seeded crystallization of Arg. Consequently, the simulated
optimum seed loading ratio was reasonably close to the measured one. However,
the minimum CV was not predicted successfully due to the ignorance of primary
nucleation, agglomeration, and breakage, and the results may have been affected
by the stochastic nature of nucleation. As for the stochastic nature of nucleation,
the simulated result in seeded crystallization of a model substance suggests that
the stochastic behavior of both primary and secondary nucleation may cause the
fluctuation of the product crystal quality. Therefore, the stochastic behavior of
nucleation will have to be considered for the process design.
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L-serine and L-proline oncrystallization
Mathematical Modeling of Different
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Breakage PBE Kernels Using Monte
Carlo Simulation Results
Ashok Das and Jitendra Kumar
1 Introduction
The production of particles with some specific internal and external properties is
crucial in pharmaceutical, chemical, mineral, food processing, and other material
processing industries. Some of the important particulate processes used in these
industries are crystallization, agglomeration, milling, grinding, polymerization, etc.
In these processes, particles change their internal and external properties (e.g., size,
shape, porosity, enthalpy, etc.), and one can observe aggregation, fragmentation,
nucleation, rupture, and growth of particles. The most popular method to track
the macroscopic behavior of the system is to use the population balance equations
(PBEs) [1]. The PBE is an integro-differential equation, which tracks the evolution
of the number density function with respect to time. The PBE uses certain
mathematical kernels to describe the particulate processes, such as aggregation,
breakage, growth, and nucleation. However, the sole focus of this chapter will be
to discuss the PBE kernels corresponding to different types of breakage processes.
Due to the integro-differential nature, PBEs are analytically solvable for some
extremely trivial classes of kernels only. That is why PBEs are often solved
numerically. In the literature, several numerical techniques are available to solve
the PBE, such as sectional method [2–5], method of moments [6, 7], finite element
method [8–10], finite volume technique [11–14], etc. Most of these numerical
techniques discretize the domain of concern to solve the equations. Additionally,
the researchers have also used the stochastic and discrete nature of the Monte Carlo
technique to solve the PBE [15–17]. However, the ubiquitous use of population
balance modeling is hindered in practical situations due to the unavailability of
A. Das · J. Kumar ()
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal,
India
e-mail: jkumar@maths.iitkgp.ac.in
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
A. Fytopoulos et al. (eds.), Optimization of Pharmaceutical Processes, Springer
Optimization and Its Applications 189, https://doi.org/10.1007/978-3-030-90924-6_4
79
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