Process Development in the QbD Paradigm: Mechanistic Modeling of

Article Cite This: Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Process Development in the QbD Paradigm: Mechanistic Modeling of Antisolvent Crystallization for Production of Pharmaceuticals Manu Garg, Milan Roy, Paresh Chokshi, and Anurag S. Rathore* Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India ABSTRACT: Thorough process understanding is a prerequisite for implementing quality by design during development of a pharmaceutical crystallization process. Identification of the critical process parameters and raw material attributes and creation of mechanistic models that can correlate these to the product quality attributes are the first steps in this approach. In this paper, a mechanistic model of antisolvent crystallization has been proposed. The model considers dependence of crystal growth rate on crystal mean size. A risk assessment was performed using Failure Mode and Effects Analysis to identify critical process parameters for designing experimentation. Crystal morphological data required for development of this model have been obtained using focused beam reflectance measurement. MATLAB has been used to identify optimal growth and nucleation parameters. The proposed model compares favorably to other similar models with respect to the accuracy of prediction of crystal size, surface area, and volume, even at varying feedrate profiles of antisolvent during crystallization. The average residual value obtained in given model is of the order of 1/10th of the previous models. The superlative performance likely originates from the fact that most models ignore the size dependence of crystal growth rate. We expect the proposed model to be a useful tool in the arsenal of those involved in development of pharmaceutical crystallization.

1. INTRODUCTION Crystallization is a separation and purification technique that is commonly used in the pharmaceutical industry for manufacturing of pharmaceutical products. It involves a phase change such that a crystalline product is obtained from the solution. Supersaturation, a zone where concentration of the compound is above the saturation concentration, is the driving force for crystallization. There are four ways of achieving supersaturation, namely, temperature change, solvent evaporation, chemical reaction, and salting out. Of these four, cooling and solvent evaporation have been the method of choice for crystallization for production of pharmaceutical products. However, the last couple of decades have witnessed a rising interest in salting out (antisolvent crystallization). This shift has been fueled by the advantages that antisolvent crystallization offers in cases when the solute is highly soluble or unstable at high temperature. Many researchers have explored the nucleation and growth kinetics of antisolvent crystallization. Mydlarz et al. (1993) proposed a three-parameter exponential size-dependent crystal growth rate function.1 Correlation between rates of nucleation and supersaturation has also been proposed by Takiyama et al. (1998) based on analysis of the effect of different concentrations of aqueous and antisolvent solutions on final crystal shape and distribution.2 A comparison between model-based and direct design approach for controlling antisolvent and cooling crystallization was provided by Nagy et al. (2007).3 Nowee et al. (2008) proposed a model for antisolvent crystallization where crystal growth rate was assumed to be independent of crystal size and agglomeration, and attrition phenomena were neglected.4 In a © XXXX American Chemical Society

separate publication, Nowee et al. (2008) presented a model for seeded cooling crystallization, identified the kinetic parameters, and validated the model using experiments.5 Gimbun et al. (2009) proposed a differential algebraic equation framework for solving a general population balance equation.6 Samad et al. (2011) provided a generic multidimensional model-based system for batch cooling crystallization processes.7 Use of focused beam reflectance measurement (FBRM) for optimizing crystallization of needle shaped particles that can deliver a product of required quality attributes has been presented by Leyssens et al. (2011).8 Recent advances include a model-based systematic design and analysis approach for unseeded combined cooling and antisolvent crystallization (CCAC) systems by Yang et al. (2014).9 Apart from these, other researchers have attempted to solve the population balance equation or relate growth function to crystal size.10−18 However, to the best of our knowledge, a comprehensive model for antisolvent crystallization incorporating mass balance, volume conservation, saturation, nucleation, and size dependent crystal growth rate is yet to be proposed. In this work, we propose a model for antisolvent crystallization in which the crystal growth rate is correlated to crystal size. The parameters in the model have been obtained from the experiments, and the proposed model is used to predict the mean crystal size, area, and volume during antisolvent crystallization. Optimization of the model has been performed Received: January 11, 2018 Revised: April 2, 2018 Published: April 25, 2018 A

DOI: 10.1021/acs.cgd.8b00055 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

Article

dm 2 −αρV L dμ3 = dt (1 − αμ3 ) dt

using MATLAB to calculate the growth and nucleation rate parameters. The proposed model has been validated experimentally. It has been demonstrated that the proposed model offers a significantly more accurate prediction when compared with other models that ignore crystal size dependence of crystal growth rate. We expect the proposed model to be a useful tool in the arsenal of those involved in development of pharmaceutical crystallization.

where α is the crystal volume shape factor coefficient, ρ is the crystal density, and VL is the liquid phase volume of the suspension. The value of α has been taken as 10 for needle-like crystals.21 The suspension volume V is given by4 dV dV L dV S = + dt dt dt

2. MATHEMATICAL MODEL The population balance equation (PBE) for the transient of crystal size distribution (CSD) in the absence of crystal breakage and agglomeration can be written as19 ∂(G(L)n(L , t )) ∂n(L , t ) + δ(L0 , L)B = ∂L ∂t

∫0

∞

n(L , t )Lk dL

V S = V × α × μ3

dt

=

∫0

∞

kLk − 1G(L)n(L , t ) dL + δ(0, k)B

The nucleation rate used in this study is given by B = Kb(ΔC)b

(2)

dt

i=1

(10)

⎛ 1 ⎞⎟ G = G0 × ⎜1 − ⎝ mL ⎠

(11)

where m is a constant and G0 captures the dependence of the growth rate on the concentration of the antisolvent as given by the expression:4 ⎛ ΔC ⎞(g0 + g1z3) ⎟ G0 = (k 0 + k1z 3 + k 2z 32)*⎜ ⎝ C* ⎠

(12)

where C* is the equilibrium solubility of solute in mixture, k0, k1, k2, g0, and g1 are growth constants, and z3 is mass fraction of antisolvent in solute free mixture. The above-mentioned equations are solved using ODE45 solver of MATLAB and optimization of parameters is achieved by making an objective function and solving it using the FMINSEARCH function of MATLAB. The parameter matrix was estimated as [kb, b, k0, k1, k2, g0, g1, m]. The objective function, which is minimized by FMINSEARCH, is represented by the following equation

(3)

N k−1 G(Li) + δ(0, k)B = k ∑ wL i i

(9) 22

Here, Kb and b are the nucleation parameters, and ΔC denotes the degree of supersaturation. The crystal growth rate dependent on crystal size as proposed in this study is given by

It is not possible to solve the above equations since the growth term integral cannot be expressed in terms of moments. Hence, the quadrature method of moments (QMOM) was applied to eq 3 by approximating the moments as a set of abscissas and weights, resulting in the following equation: dμk

(8)

Concentration of solute is given as m C = L2 V

(1)

The first four moments have physical significance. While the μ0 represents the total number of particles, μ1 represents the total length of the crystals, μ2 represents total crystal surface area, and μ3 represents total volume occupied by the crystals. These moments are based on per unit volume of total suspension. The moment transformation of eq 1 yields the following governing equations for the moments of the CSD: dμk

(7)

where VS is the solid phase suspension volume and is given by

where n(L,t) is the number density of crystals of size L at any time t, G(L) is the size dependent crystal growth rate, and B is the nucleation rate. Here, δ is the Dirac delta function and L0 is the critical nucleus size. The moment transformation can be applied to the population balance equation for simplification. The kth moment of the crystal size distribution is given by μk =

(6)

(4)

N

Here, wi are the weights, Li are the abscissas, and N denotes the number of quadrature points. The governing equations for the moments (2−4) have been adopted from Marchisio et al. (2003) after neglecting agglomeration and attrition.19 The governing equations for the moments can now be solved using the standard PD algorithm.19,20 The mass fraction of the components is defined as mi xi = ∑m (5)

ε=

∑ (Fmi − Fei)2 i=1

(13)

Fim

where is the model predicted value of a measurable quantity, e.g., average crystal size at a particular time, and Fie is the experimentally obtained value of the same quantity at the same time. An initial guess is given for which the model is solved, and the predicted values are compared with experimental values by evaluating the square of the total error ε (objective function). FMINSEARCH returns an optimum set of parameters for which the objective function is minimized. The model for which the crystal growth rate is given by eq 11 is termed the dependent model. Further, for comparison purpose we also examine the kinetics of crystallization in which the crystal growth rate is considered independent of the crystal size. Such a model is termed independent model. The equations for mass balance of solute, concentration of solute, liquid and solid phase of suspension

where i represents the component in the crystallization system, and m1, m2, and m3 represent the mass of solvent, solute, and antisolvent in the solution, respectively. Similarly, x1, x2, and x3 represent the mass fraction of solvent, solute and antisolvent, respectively. The mass balance for the dissolved solute in terms of third moment can be derived from mass balance equation for solute accumulation,4 volume conservation equation and moment definition, and can be stated as B



Article

Table 1. Risk Assessment (FMEA Approach) potential failure mode

potential effect

severity

detection

RPN

wrong quantity of solvent

impact on API quality/yield impact on API quality/yield impact on API quality partial dissolution

8

improper measuring

1

calibrated dip rod

6

48

8

improper measuring

1

calibrated dip rod

6

48

3

improper weighing

1

calibrated weighing balance

6

18

8

3

digital temperature indicator

5

120

API degradation impact on API quality delay in batch time cycle impact on yield

8 6

3 8

digital temperature indicator valve controlled manually

5 10

120 480

1

improper utility applied/ utility not working improper utility applied no calibration on addition vessel/human error human error

8

6

48

3

human error

8

6

144

impact on API quality

3

VFD not working

1

manual entries in production record with centralized digital clock manual entries in production record with centralized digital clock VFD functional

5

15

wrong quantity of antisolvent wrong quantity of ammonium carbonate low reactor temperature high reactor temperature wrong rate of antisolvent addition wrong stirring time before AS addition wrong stirring time after AS addition inappropriate Kla

causes

occurrence

current process controls

volume and nucleation remain same as that of the dependent model. The growth rate for the size independent model is given by

G = G0

(14)

where G0 is same as that used in the dependent model. For the independent model the use of QMOM and PD algorithm is not required as eq 3 can be written in the form of moments in the following manner:19

dμ0 dt dμ1 dt

dμ2 dt dμ3 dt

=B

(15)

= Gμ0

(16)

= Gμ1

(17)

= Gμ2

Figure 1. Schematic diagram of the experimental setup (A: automated lab reactor, B: temperature control system, C: Pt1000 temperature sensor, D: FBRM PC, E: dosing pump, F: flask for antisolvent, G: FBRM probe).

(18)

Equations 15−18, along with the governing equations for mass, volume, and concentration are solved using ODE45 solver of MATLAB, and parameter optimization is achieved following the same approach as in the dependent model. Parameter matrix in this case changes to [kb, b, k0, k1, k2, g0, g1].

help of a calibrated digital dosing pump. Anchor agitator was used to stir the slurry at a constant 300 rpm. A temperature control system consisting of a Pt1000 temperature sensor and heating/cooling circulator (Huber, Germany) was used to maintain the temperature at 25 °C. A focused beam reflectance measurement (FBRM) probe (Mettler Toledo Lasentec Products, USA) was used to track particle chord length every 10 s. The FBRM signal was converted to digital signal using a PCB. This was then connected to the FBRM PC for data monitoring. Experiment 1 was performed to obtain data required for parameter estimation for both dependent and independent models. Experiments 2− 4 were performed at different flow rates of antisolvent in order to analyze its effect on characteristics of crystallization. These data are then used for model validation and for comparison between the dependent model and independent model. The experimentation details are mentioned in Table 2.

3. EXPERIMENTAL SECTION A quality risk assessment was performed using Failure Mode and Effects Analysis (FMEA) of the given process parameters labeled as potential failure modes (Table 1). The values for severity, occurrence, and detection were chosen using a 10 point scale, and general process controls at plant scale were taken into consideration. The RPN number for “antisolvent flow rate” was found to be 4 times more than the next lower RPN number. Hence, the “antisolvent flow rate” was chosen as the sole variable in the experimentation design. For estimation of kinetic and growth parameters, experiments were performed at different antisolvent flow rates to generate moments data using FBRM.23 Figure 1 shows the schematic of apparatus and instrumentation used in the study. A ternary system was used for performing antisolvent crystallization with the components being dexlansoprazole as solute, n-heptane as antisolvent, and methyl−ethyl ketone as solvent. Next, n-heptane was added to a 25% concentrated solution of dexlansoprazole in methyl−ethyl ketone in the automated lab reactor (ALR). A linear profile for antisolvent addition was followed with the

4. RESULTS AND DISCUSSION Optimization of parameters was performed using the conditions of experiment 1. Table 3 contains the crystallization parameters estimated for the dependent model. C



Article

Table 2. Listing of Experimental Conditions for Experiments 1−4 Parameter

Experiment 1

Experiment 2

Experiment 3

Experiment 4

solute (API) solvent quantity antisolvent quantity antisolvent flow rate (mL/h) addition temperature (°C) agitator type agitator RPM

1T 4T 10 T 500 25−30 anchor 300

1T 4T 10 T 125 25−30 anchor 300

1T 4T 10 T 250 25−30 anchor 300

1T 4T 10 T 166.67 25−30 anchor 300

Table 3. Optimized Growth and Nucleation Parameters for Size Dependent Model

a

parameter

value

unit

kb k0 k1 k2 b g0 g1 m

11.7 0.044 1.8 × 10−4 −1.4 × 10−4 0.86 8.9 × 10−5 −7 × 10−6 20.9

Noa (g/mL)−1 (mL3)−1 μm/s μm/s μm/s dimensionless dimensionless dimensionless (μm)−1

No refers to number of particles.

A similar process of optimization was followed using the conditions of experiment 1 for the independent model, and the resulting parameters are listed in Table 4. Table 4. Optimized Growth and Nucleation Parameters for Size Independent Model parameter

value

unit

Kb k0 k1 k2 b g0 g1

4.8 0.11 −0.001 −0.004 0.47 0.0044 0.0011

No (g/mL)−1 (mL3 s)−1 μm/s μm/s μm/s dimensionless dimensionless dimensionless

Figure 2. Mean crystal size (experiment 1).

The model estimates the transients of the moments of the crystal size distribution from which the average properties of the distribution can be obtained. In particular, the average size of the crystal, defined as μ1/μ0, the mean area of the crystals, defined as μ2/μ0, and the mean volume of the crystals, defined as μ3/μ0 were calculated. These average quantities were selected as these are readily obtained from the experimental data for comparison. The time evolution of these average properties of the CSD is shown in Figures 2, 3, and 4 for Experiment 1. Experimental and simulation results for both models are plotted together for comparison. It is evident from Figure 2 that both the models overpredict the mean crystal size. The mean size increases until a certain time after which it decreases and becomes constant. Since the amount of antisolvent added is quite significant in Experiment 1, the rate of crystal growth is very high. The fast crystallization in a short time tends to decrease the solute concentration in the slurry leading to a decrease in the degree of supersaturation, ΔC. The reduction in driving force for crystallization suppresses the growth; however the nuclei continues to form. Therefore, the peak in mean crystal dimension is followed by a drop. The time for achieving the peak value is predicted

Figure 3. Mean crystal area (experiment 1).

correctly by both the models. While the dependent model predicts the final mean size very accurately, the independent model fails to do so. It can be inferred from Figure 3 that the nature of the curve for the dependent model is very similar to the experimental data. Not only is the peak shape quite similar, but the final value of mean area in crystal size distribution is also correctly predicted. In the case of mean size, the predicted peak is above the actual peak, whereas the predicted peak is lower than the actual peak D



Article

Figure 5. Mean crystal size (experiment 2). Figure 4. Mean crystal volume (experiment 1).

in the case of mean area. This is likely because finer particles have a larger surface area. In the case of mean area, the independent model again fails to match the experimental data as depicted in Figure 3. Figure 4 represents the mean crystal volume and further confirms our findings. The dependent model predicts the actual data accurately including the peak and the final mean crystal volume. As in the case of “mean crystal area”, the independent model severely underpredicts the peak volume. It is evident from Figures 2−4 that while the peak value of number-average crystal size predicted by the dependent model is higher than that found in the experiment, the model prediction of the peak values of the mean crystal surface area and volume are lower than those in the experiment. This observation suggests that the crystal size distribution (CSD) in the experiment is relatively broader than that predicted by the model. Also, the total number of crystals and hence the rate of nucleation are believed to be higher in the experimental crystallization than that predicted by the optimized model parameters. Similar comparison can be made for the results from the dependent and independent models. The higher mean crystal size and lower mean crystal volume predicted by the independent model in comparison to that predicted by the dependent model suggest that the CSD in the independent model is narrower than in the dependent model. This is expected as it is well-known that the size dependent growth rate tends to broaden the distribution. The results clearly demonstrate the importance of considering dependence of the growth rate on the crystal size. As soon as nucleation occurs, the crystals attain a certain mean size. Hence, the model receives the initial mean size as an input instead of its starting value of zero. Figures 5, 6, and 7 represent the mean crystal size, area, and volume obtained for the experiment 2 for which case the flow rate is kept one-fourth of that in experiment 1. The three plots have similar characteristics with respect to the nature of curves and the behavior of dependent and independent models. For all three average quantities, the dependent model reaches the actual value after only around 800 s after which the model predictions are in close quantitative agreement with the experimental values. Unlike transients in Experiment 1, the curves for the mean


Figure 7. Mean crystal volume (experiment 2).

quantities in Experiment 2 do not exhibit a peak value. Further, the mean crystal dimension at the end of Experiment 2 is around 25 μm, taking a time of around 3000 s, whereas that in E



Article

the Experiment 1 is above 30 μm, requiring only 1000 s to attain, exhibiting a peak value of around 50 μm. This is attributed to the lower rate of crystallization in Experiment 2 due to the decreased amount of antisolvent present in the system. It is important to note that the independent model overpredicts the final mean size and underpredicts the final mean area and volume. As mentioned earlier, this observation implies that the predicted distribution in the independent model is narrower than that in the dependent model. The size dependent growth model appropriately captures the broadness of CSD as found in the experiment. Figures 8, 9, and 10 represent the mean crystal size, area, and volume obtained for experiment 3, which was performed with





an antisolvent flow rate of 250 mL/h. Figures 11, 12, and 13 represent the mean crystal size, area, and volume obtained for experiment 4, which was performed with an antisolvent flow rate of 166.67 mL/h. It is evident from Figures 8 and 11 that the independent model overpredicts mean size, while the dependent model accurately


F



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various moments of the distribution, in close agreement with the experimental findings. The role of antisolvent concentration in the slurry is examined by varying its flow rate. The increase in antisolvent concentration leads to faster crystallization. The results clearly elucidate the necessity of considering dependence of crystal growth rate on crystal size. The proposed model will be a useful tool to control the crystal size distribution in pharmaceutical crystallization processes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Anurag S. Rathore: 0000-0002-5913-4244 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Experimental work was carried out at Research & Development Center, Sun Pharmaceutical Industries Ltd (Gurugram, India) with the support of Chemical Engineering Department of Indian Institute of Technology, Delhi (New Delhi, India). Authors acknowledge funding from the Department of Biotechnology, Ministry of Science and Technology (Centre of Excellence for Biopharmaceutical Technology Grant BT/ COE/34/SP15097/2015).

predicts not only the final mean size but also the nature of increase of mean size barring the first 5 min. The discrepancy in early crystallization kinetics is due to the unavailability of appropriate initial conditions for the model. Notwithstanding this initial discrepancy, the model predictions for the various mean quantities are in close quantitative agreement with the experimental findings for all flow rates of antisolvent examined. In comparison to experiment 2, the antisolvent flow rate is kept higher in experiments 3 and 4. The higher antisolvent concentration results in a higher growth rate (refer to eq 12). Thus, the mean crystal size attains the peak value of around 20 μm in around 1000 s in experiment 3 and around 1500 s in experiment 4. The slow crystallization due to reduced antisolvent concentration in experiment 2 requires around 3000 s to attain the final mean size of around 25 μm. We can also observe from Figures 10 and 13 that in both the experiments, the independent model underpredicts the mean crystal volume, same as the case with experiment 2. The underprediction of mean size and overprediction of surface area and volume are an indication of narrow crystal size distribution when size independent growth rate is considered. Importantly, the dependent model correctly predicts the mean crystal volume for both the experiments as shown in Figures 10 and 13. Thus, size dependent growth rate model is necessary to describe the broad crystal size distribution in antisolvent crystallization. Hence, the dependent model correctly predicts the mean crystal size, area, and volume. Robustness of the model is thus proved by testing the model at different flow rates and comparing the model predictions with the experimental data. The proposed model performs satisfactorily except in the initial period.

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REFERENCES

(1) Mydlarz, J.; Jones, A. G. On the estimation of size-dependent crystal growth rate functions in MSMPR crystallizers. Chem. Eng. J. and Biochem. Eng. J. 1993, 53, 125−135. (2) Takiyama, H.; Otsuhata, T.; Matsuoka, M. Morphology of NaCl crystals in drowning-out precipitation operation. Chem. Eng. Res. Des. 1998, 76, 809−814. (3) Nagy, Z. K.; Fujiwara, M.; Braatz, R. D. Recent advances in the modelling and control of cooling and antisolvent crystallization of pharmaceuticals. IFAC Proceeding Volumes 2007, 40, 29−38. (4) Nowee, S. M.; Abbas, A.; Romagnoli, J. A. Antisolvent crystallization: Model identification, experimental validation and dynamic simulation. Chem. Eng. Sci. 2008, 63, 5457−5467. (5) Nowee, S. M.; Abbas, A.; Romagnoli, J. A. Optimization in seeded cooling crystallization: A parameter estimation and dynamic optimization study. Chem. Eng. Process. 2007, 46, 1096−1106. (6) Gimbun, J.; Nagy, Z. K.; Rielly, C. D. Simultaneous quadrature method of moments for the solution of population balance equations, using a differential algebraic equation framework. Ind. Eng. Chem. Res. 2009, 48, 7798−7812. (7) Samad, N. A. F. A.; Singh, R.; Sin, G.; Gernaey, K. V.; Gani, R. A generic multi-dimensional model-based system for batch cooling crystallization processes. Comput. Chem. Eng. 2011, 35, 828−843. (8) Leyssens, T.; Baudry, C.; Hernandez, M. L. E. Optimization of a crystallization by online FBRM analysis of needle-shaped crystals. Org. Process Res. Dev. 2011, 15, 413−426. (9) Yang, Y.; Nagy, Z. K. Model-based systematic design and analysis approach for unseeded combined cooling and antisolvent crystallization (CCAC) systems. Cryst. Growth Des. 2014, 14, 687−698. (10) Mydlarz, J.; Jones, A. G. On modelling the size dependent growth rate of potassium sulphate in an MSMPR crystallizer. Chem. Eng. Commun. 1990, 90, 47−56. (11) Rawlings, J. B.; Miller, S. M.; Witkowski, W. R. Model identification and control of solution crystallization processes: a review. Ind. Eng. Chem. Res. 1993, 32, 1275−1296. (12) Miller, S. M.; Rawlings, J. B. Model identification and control strategies for batch cooling crystallizers. AIChE J. 1994, 40, 1312− 1327.

5. CONCLUSIONS A mechanistic model for the kinetics of antisolvent crystallization has been proposed. The model considers the dependence of crystal growth rate on crystal size. It has been demonstrated that the dependent growth model is superior in accuracy compared to the independent growth model with respect to the prediction of the crystal characteristics such as mean crystal size, surface area, and volume. The model has been validated using the experimental findings under different process conditions. The size dependent growth model captures the broadness in crystal size distribution, as inferred from the G



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(13) Hu, Q.; Rohani, S.; Wang, D. X.; Jutan, A. Nonlinear kinetic parameter estimation for batch cooling seeded crystallization. AIChE J. 2004, 50, 1786−1794. (14) Nowee, S. M.; Abbas, A.; Romagnoli, J. A. Model-based optimal strategies for controlling particle size in antisolvent crystallization operations. Cryst. Growth Des. 2008, 8, 2698−2706. (15) Nagy, Z. K.; Fujiwara, M.; Braatz, R. D. Modelling and control of combined cooling and antisolvent crystallization processes. J. Process Control 2008, 18, 856−864. (16) Nagy, Z. K. Model based robust control approach for batch crystallization product design. Comput. Chem. Eng. 2009, 33, 1685− 1691. (17) Lindenberg, C.; Krattli, M.; Cornel, J.; Mazzotti, M.; Brozio, J. Design and optimization of a combined cooling/antisolvent crystallization process. Cryst. Growth Des. 2009, 9, 1124−1136. (18) O'Ciardha, C. T.; Hutton, K. W.; Mitchell, N. A.; Frawley, P. J. Simultaneous parameter estimation and optimization of a seeded antisolvent crystallization. Cryst. Growth Des. 2012, 12, 5247−5261. (19) Marchisio, D. L.; Pikturna, J. T.; Fox, R. O.; Vigil, R. D.; Barresi, A. A. Quadrature method of moments for population balance equations. AIChE J. 2003, 49, 1266−1276. (20) Marchisio, D. L.; Fox, R. O. Solution of population balance equations using the direct quadrature method of moments. J. Aerosol Sci. 2005, 36, 43−73. (21) Myerson, A. S. Handbook of Industrial Crystallization, 2nd ed.; Elsevier: Amsterdam, 2002. (22) Woo, X. Y.; Tan, R. B. H.; Chow, P. S.; Braatz, R. D. Simulation of mixing effects in antisolvent crystallization using a coupled CFDPDF-PBE approach. Cryst. Growth Des. 2006, 6, 1291−1303. (23) Ruf, A.; Worlitschek, J.; Mazzotti, M. Modeling and experimental analysis of PSD measurements through FBRM. Part. Part. Syst. Charact. 2000, 17, 167−179.

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Process Development in the QbD Paradigm: Mechanistic Modeling of

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