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Nonsmooth Modeling for Simulation and Optimization of Continuous... 245
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relaxation of the steady-state constraint, making it superior to the traditional direct
steady-state optimization approach [16]. The formulation is:
Y(p,tf) (23a)
max
p∈P
s.t. model (1), ∀t ∈[0,t
q(p,t) ≤ 0, ∀t ∈[t
Pr
P − M
on −spec
l
≤ p ≤ pu, (23d)
p
n
p
where p ∈ R
t
, with lower and upper bounds, pland pu, respectively, and q are the quality
off
is the vector of decision (optimization) variables, including tonand
(p,tf) ≤ 0, (23c)
],
f
], (23b)
on,toff
constraints defining an on-spec product. Overall, the campaign duration will be
discretized into n
time intervals. In general the decision variables are the valve
t
positions at the discretized time intervals. In this example there are 5 valves, thus
the total number of decision variables will be (5 + 1) × n
. For more details on the
t
normalized and discretized formulation see [23].
The overall yield is optimized in this formulation. It is defined by:
If M
mass of on-spec product
Y =
mass of raw material fed
rm
is the total mass of the raw material fed to the process. It can be calculated
, ∀t ∈ (0,t
]. (24)
f
from the following ODE:
rm
dM
dt
M
(p,t) = F
rm
(0) = 0.
rm
(p,t), ∀t ∈[0,tf], (25)
on-spec
If M
is the total on-spec product produced in the campaign, it can be
calculated from:
on-spec
dM
M
where F
pr
is the mass flow rate of the product. Notice that product accumulation
is only considered during the on-spec production epoch, but the consumption of
the main reactant is considered over the entire campaign. In general, other endpoint constraints can be added, for example, to enforce shutdown specifications,
according to the specific requirements of the campaign. Nonetheless, as we show
dt
on-spec
(p,t) =
(0) = 0,
⎧
0,t∈[0,t
⎨
pr
(p,t), t ∈[ton,t
F
⎩
0,t∈ (t
on
off,tf
off
),
],
(26)
],
246 M. Patrascu and P. I. Barton
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later, optimizing over the yield (instead of the total production) results in optimal
shutdown procedures without the need to add explicit constraints.
The quality constraints, q, of the on-spec product are path constraints, which
are handled by introducing auxiliary variables, x
, transforming them to end-
α
point equality constraints in terms of the auxiliary variables, using the hybrid and
nonsmooth formulation equation:
dx
dt
α
(p,t) =
x
(0) = 0,
α
⎧
⎨
max(0,q
⎩
0,t∈[0,t
(p,t)),t∈[ton,t
α
0,t∈ (t
off,tf
),
on
α ∈{a, b, c, d}, (27)
],
off
],
where q
is the original path constraint. The new end-point constraints are:
α
(p,tf) = 0,α∈{a, b, c, d}. (28)
x
α
4.2 Optimal Dynamic Operation
Here, we look at an example problem which was optimized by the above approach.
An end-to-end continuous manufacturing plant, as depicted in Fig. 5, was modeled
by a nonsmooth DAE system (composed of 2132 equations), and the yield of the
production campaign was optimized, given a limited campaign duration [23, 24].
Consider the solution of Formulation (23) with P = 120 kg for a campaign time
of 200 h (t
first off-spec epoch, two for the on-spec epoch and another one for the last off-spec
epoch, such that n
time horizon based on this discretization. Integration and sensitivity analysis was
performed by DAEPACK [38]. The local optimization was performed by IPOPT
with C++ interface [40].
We examine the results in terms of the objective function (the yield) and the
constraints (overall on-spec productivity and impurity levels). Figure 6a presents the
yield and the productivity, and Fig. 6b depicts the final product impurity profiles. A
few interesting details are revealed by inspecting the optimal solution:
1. The overall productivity for the campaign is exactly 120kg (i.e. Constraint (23c)
is active).
2. The impurity levels were pushed to the maximum specified by the quality
path constraints (23b), and were not violated during the entire campaign. Thus,
essentially all the product produced is on-spec, and no product is wasted.
3. It takes about 17 h for an on-spec product to flow and to obtain a positive yield,
which then increase monotonically throughout the campaign time. However,
= 200). The time horizon is discretized into 4 sub-epochs; one for the
f
= 4. We assume piecewise constant decision variables over the
t
Nonsmooth Modeling for Simulation and Optimization of Continuous... 247
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140
(a)
120
100
80
60
40
20
Productivity [kg] / Yield [%]
0
0 50 100 150 200
On-spec product
Yield
Time [h]
0.6
(b)
0.5
0.4
0.3
0.2
Impurities [%w]
0.1
0
0 50 100 150 200
Time [h]
I1+I
I
2
2
Fig. 6 (a) On-spec productivity and yield. (b) The impurity mass fractions in the final product.
The specification levels are indicated by thin dashed lines
1
(a)
0.9
0.8
0.7
0.6
0.5
0.4
Controls
0.3
0.2
0.1
0
Feed flow rate [kg/h]
0 50 100 150 200
Time [h]
Cr
1
u
Cr
2
u
Cr
3
u
Cr
4
u
35
(b)
30
25
20
15
Volumes [L]
10
5
0
0 50 100 150 200
Cr
V
Cr
V
Cr
V
Cr
V
Time [h]
1
2
3
4
Fig. 7 Solution of the optimization problem (23)withP = 120 kg and campaign time of 200 h.
(a) Optimal control profiles. (b) Holdup volumes in the crystallizers
some process attributes reach a steady-state much later (e.g. the holdup volume
of Cr4 reaches steady-state after more than 75 h, see Fig. 7b).
4. A jump in the slope of the yield is apparent at t = 186.1 h, together with some
change of the impurity levels of the product. The final (optimal) yield obtained
is 63 %.
The optimal control trajectories and the resulting holdup volumes in the crystal-
lizers are depicted in Fig.7. The most important decision variablesare the sub-epoch
durations, τ
τ
= 0.0h. The final off-spec epoch, therefore, has been reduced to its lower bound,
4
practically zero. The on-spec epoch is t ∈[16.8, 200] ([τ
, with optimal values: τ1= 16.8h, τ2= 167.5h, τ3= 15.7 h and
k
%
3
,
1
τi]). The relatively
i=1
short initial off-spec epoch, t ∈[0, 16.8), is necessary because of the time required
for the appearance of the API at the required concentration in the product stream.
The optimal start-up procedure is characterized by a relatively short time to produce
the API at the required concentration in the final product, while keeping the impurity
mass fraction peaks below the specification levels.
248 M. Patrascu and P. I. Barton
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More interesting details are revealed by looking at Fig. 7, which shows the values
of the dynamic decision variables for the optimal solution. The feed flow rate of
the reactant C
the end of the time horizon. In practice this is a shutdown procedure. Shutting
down while maintaining on-spec production was enabled by discretizing the onspec epoch by more than just one sub-epoch, and optimizing their duration. The
resulting shutdown procedure is a consequence of maximizing the yield; there is
no justification to continue feeding reactants to the system if there will be no
time to process them. The crystallizers are completely depleted by the end of the
campaign (by appropriately adjusting their valve positions), while satisfying the
product quality constraints. The optimizer finds the optimal time towards the end of
the campaign to make this switch. It is important to emphasize that vessel depletion
was not enforced by any end-point constraint. It is a property of the optimal
solution when optimization of the yield is considered. The fact that a “shutdown”
procedure was achieved while satisfying the on-spec quality path constraints shows
the importance of setting all of the sub-epoch durations as decision variables, as
opposed to only the main epochs (on/off-spec) [25].
is reduced to its lower bound and the vessels are depleted towards
1
4.3 Multi-Objective Optimization
The performance characteristics of the process depend on the design and the
dynamic operating procedures. Often, more than one objective is of interest to the
decision maker, such as yield and productivity. In many processes there is a trade-off
between the optimal productivity and the optimal yield. In suchcases, results may be
presented as Pareto curves (or surfaces)[1, 8, 27], representing a set of non-inferior
solutions.
Pareto curves may be generated by varying P in Formulation (23) and solving
the optimization problem for each value. This is called the -constraint method [8].
After obtaining a solution to one problem we solve a similar problem, changing the
value of P slightly, using the solution from the previous iteration as initial guess
for the decision variables and the multipliers (see the IPOPT documentation). The
results for campaigns of 150–250 h are presented in Fig. 8. The maximum yield
that can be achieved for various campaign durations is comparable, around 67%,
although this decreases slightly as the time horizon decreases. The optimal steadystate yield is shown for comparison as well. The campaign yield gets closer to
this optimal value as the productivity constraint is decreased, and as the campaign
duration is increased. The on-spec productivity, however, changes significantly with
the time horizon. Better results are obtained in terms of yield and productivity for
longer campaigns, as expected.
Although here we focused on the yield and the productivity as the performance
objectives, other performance characteristics can be evaluated and optimized by
introducing appropriate quality constraints or objective functions, such as the
environmental factor (E-factor) [30] or the total energy consumption. Furthermore,
Nonsmooth Modeling for Simulation and Optimization of Continuous... 249
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70
65
60
55
Yield [%]
=150 h
t
50
f
tf =200 h
tf =250 h
45
60 80 100 120 140 160 180 200
Productivity [kg]
Fig. 8 Pareto curves of optimal solutions of Formulation (23) for various campaign durations of
the end-to-end continuous pilot plant
the minimization of the campaign time could be performed readily by defining tfas
the objective function. This information should be available to the decision makers
when deciding on the desired performance and appropriate procedures.
5 Conclusions and Outlook
The nonsmooth formulation approach demonstrated here is capable of simulating
a wide range of dynamic phenomena such as switching between different physical
regimes (e.g., between thermodynamic phases or kinetic regimes), control valve
limitations, and nonsmooth changes in stream compositions. The nonsmooth DAEs
framework guarantees the existence of meaningful sensitivity information, which
can be computed by appropriate sensitivity analysis, and used by optimization
algorithms to find optimal dynamic procedures, as exemplified here even for endto-end continuous manufacturing plants.
Optimization problems for nonsmooth systems are traditionally handled by
introducing cumbersome, unphysical reformulations in the frameworks of mixedinteger and complementarity system approaches. These methods lead to additional
(probably unnecessary) parameters, artificial variables, binary variables,constraints,
and considerations. Consequently, mixed-integer reformulations of nonsmooth
problems may be inaccurate. Furthermore, the introduction of artificial numerical
parameters/binary variables may significantly increase the solver’s running time.
Now, thanks to the developments presented in this chapter, these can be formulated
250 M. Patrascu and P. I. Barton
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and solved in a more straightforward approach that is mathematically sound.
Moreover, local-optimization solvers for nonsmooth problems exist and provide
an approach to find local solutions of larger problems that are outside the scope
of global nonconvex mixed-integer solvers. Extension of deterministic global
optimization methods to nonsmooth DAEs should be immediate since relaxations of
nonsmooth functions can be computed using McCormick’s framework [13, 29, 39].
For dynamic simulation of nonsmooth DAEs, process simulators need to be
extended to include a library of nonsmooth elemental functions (e.g. the Euclidean
norm, min, max, mid, mid with n arguments). The Jacobian software [11], used
for the simulations presented in Sect. 3, already supports most of these nonsmooth
elemental functions. Now, there exists a potential of implementing such nonsmooth
dynamic formulations in online economic optimization and control systems, known
as nonlinear model predictive control [20]. In such systems, an updated optimal
control profile will be calculated based on real-time measurements. These systems
should also be robust and take parametric uncertainty into account.
The dynamic optimization formulation presented in Sect. 4.1 outperforms the
traditional steady-state optimization approach. This is accomplished by enforcing
the quality constraints on an internal epoch and optimizing its duration, allowing
an overall transient behavior (relaxing the steady-state constraint). Importantly, by
allowing the switching times of the controls to vary by the optimizer it is possible
to find optimal start-up and shutdown procedures. Here, we showed that the optimal
solution completely eliminated off-spec production, achieving high levels of yield
for campaigns of 150–250 h. Performance maps of the manufacturing process may
be expressed by Pareto curves, where optimal yield and optimal productivity are
considered as opposing objectives.
Traditionally, the pharmaceutical industry has adopted batch processes as the
main manufacturing approach. These unit operations involve unsteady flow, timedependent conditions and are disconnected from the rest of the plant. Continuous
flow processes allow for higher level of process integration, control, and efficiency.
Sometimes these processes are (wrongly) defined as steady-state processes. However, this is not necessarily the case, and an optimal continuous operation may
exhibit significant transients, as demonstrated here. In this sense, the pharmaceutical
industry may evolve to operate manufacturing processes which hold both batch and
continuous type properties (continuous flow with significant transients).
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Integrated Synthesis, Crystallization,
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Filtration, and Drying of Active
Pharmaceutical Ingredients: A
Model-Based Digital Design Framework
for Process Optimization and Control
Daniel J. Laky, Daniel Casas-Orozco, Francesco Destro, Massimiliano Barolo,
Gintaras V. Reklaitis, and Zoltan K. Nagy
1 Introduction
Over the past two decades, initiatives such as quality-by-design (QbD) [1] and
quality-by-control (QbC) [2] have accelerated a modernization in pharmaceutical
manufacturing. These paradigms require quantitative interpretation of an operating
region of the process, often branded as the design space.Thedesign space has
been previously characterized as “the multidimensional combination and interaction of input variables and process parameters that have been demonstrated
to provide assurance of quality” [3]. Naturally, incorporation of Industry 4.0
standards [4] accompanies modernization via digitalization and computerization of
manufacturing, especially during identification/quantification, and maintenance of
a robust operating region through process design and online process management,
respectively.
Standard computational methods for design space identification typically fall
into three categories: (1) data-driven sampling [5, 6], (2) fully mechanistic, direct
optimization [7], and (3) data-driven modeling with optimization [8, 9], or a
combination of such methods [10, 11]. Each technique requires high-quality
mechanistic models, quantified uncertainty of model parameters, and high-quality
data for meaningful justification of process digitalization.
Design space identification through mechanistic modeling may be applied to
a single unit operation or larger pieces of a manufacturing process. Certain
key operations in pharmaceutical manufacturing, for instance, the crystallization-
D. J. Laky · D. Casas-Orozco · G. V. Reklaitis · Z. K. Nagy ()
Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN, USA
e-mail: znagy@purdue.edu
F. Destro · M. Barolo
CAPE-Lab—Computer-Aided Process Engineering Laboratory, University of Padova, Padova
(PD), Italy
© 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_10
253
254 D. J. Laky et al.
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filtration-drying steps, are highly coupled subprocesses and should be considered
simultaneously whileidentifying optimal processdesign. Achieving such an optimal
operation while considering the interaction of these process steps will be at least as
good as considering the units separately, of which the latter often requires some
heuristic blending of qualitative and quantitative conclusions.
Even with implementations achieving computational tractability with quality
design space identification, there is still much work to be done with regard to
the accessibility and standardization of traditional pharmaceutical unit operations.
Currently, one may use a commercial software, such as gPROMS FormulatedProducts [12], to analyze integrated design and control of pharmaceutical processes.
However, limitations exist on user flexibility for automated simulation and analysis
techniques, the availability of models in the provided model library, the ease of
implementing custom models, and robustness with respect to hybrid modelling (i.e.,
batch, semibatch, and continuous manufacturing steps in the same process).
Given this drawback, when developing intensified or novel processing steps,
collaboration with commercial developers or usage of a more accessible coding
framework (i.e., Python or MATLAB) is often required. For this reason, an opensource pharmaceutical manufacturing package, PharmaPy [13], has been developed
to supplement other (commercial) software packages, as a tool for simulation and
optimization of pharmaceutical processes, focusing on allowing process design and
analysis through automated simulation, custom modeling, and hybrid modeling
capabilities.
Creating and maintaining a process digital twin is a key step while analyzing
and optimizing a given process through digital design. In fact, once a digital twin
is realized, one may perform digital tests of active control loops for model-based
control applications. This paradigm begins pushing the QbD approach to an online
QbC approach, employing the same or similar models and modeling techniques.
In Yu et al. [14], this move has been defined by the FDA as the final step in the
modernization of pharmaceutical manufacturing. Even yet, both approaches provide
the ability for online analysis, requiring user scrutiny when deciding under which
paradigm process-specific design space identification and maintenance fall.
In this work, we present mathematical models relevant to the synthesis, crystallization, filtration, and drying steps of an active pharmaceutical ingredient (API).
Each step encourages QbD by utilizing an integrated simulation framework for
process analysis and optimization. The rest of the chapter is organized as follows. In
Sect. 2, we present the relevant model implementations utilized in the case studies.
In Sect. 3, we present a case study for synthesis-crystallization of paracetamol using
experimental data available from existing literature. Then, a case study showcasing
an intensification of filtration-drying steps within an integrated carousel is presented
as well. Also, to encourage the movement toward QbC, we test an active control
strategy on the digital twin of the integrated filtration-drying carousel.
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