J Pharm Innov DOI 10.1007/s12247-008-9036-0
RESEARCH ARTICLE
Using Compartment Modeling to Investigate Mixing Behavior of a Continuous Mixer Patricia M. Portillo & Fernando J. Muzzio & Marianthi G. Ierapetritou
# International Society for Pharmaceutical Engineering 2008
Abstract In this paper, the development of a compartment model to simulate mixing within a continuous blender is reported. The main benefit of the method is that it can generate extensive modeling predictions in very short computational time. The model can also be used to explore the effect of sampling parameters on estimated mixing performance, a topic that has been central to pharmaceutical manufacturing for the past 15 years and that remains a central issue in the PAT initiative. However, this method requires more input than conventional particle dynamics methods. Thus, we investigate the effects of modeling parameters on mixing performance to develop general guidance needed to adapt this modeling framework to any continuous process. An experimental technique based on longitudinal sampling is used to examine the content uniformity of the blend along the continuous mixer. The model compares favorably with continuous mixing experiments, capture the effects of feeding rate variability, active product ingredient concentration, and blender processing angle, while effectively capturing and making explicit the effect of sampling parameters such as number of samples and sample size. The modeling approach provides a convenient tool for process design. Keywords Compartment modeling . Continuous powder mixing
Introduction A number of approaches exist to model powder mixing processes, including discrete element methods (DEM), P. M. Portillo : F. J. Muzzio : M. G. Ierapetritou (*) Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ 08854, USA e-mail:
[email protected]
continuum models, and Markov chains. However, due to computational boundaries, uncertainty in material properties, and limitations in terms of process representation, these approaches are applicable only to simplified cases. For example, mixing simulations using DEM [7, 29, 21, 34] are common, yet the largest number of particles used within these simulations was 250,000 for a period of 120 s [24]. Under these conditions, the CPU time on a Beowulf cluster (performed in parallel using up to 128, 3.6 GHz processors of a 1152 processor) was about 5 to 8 h per second of simulation. This computational expense limits the number of particles that can be modeled, the details of the system geometry [13], and the shape of the particles considered. Moreover, a model that takes longer than the process itself is poorly suited for process control purposes. Thus, the main purpose of this paper is to introduce an alternative modeling approach, which will require more input than particle dynamics methods, but is also much faster from a computational viewpoint. The method is a feasible route for the development of models useful for real-time closed-loop control of manufacturing processes. Consistent with the nature of pharmaceutical manufacturing, which proceeds largely in batch mode, a majority of the modeled mixing processes confine the particles within a vessel. In recent years, however, a keen interest in continuous processes has arisen. In such systems, timedependent inflow and outflow of materials results in fluctuations in total mass and species concentration within the system at a given time, causing unsteady flow and mixing performance. Successful modeling of such systems has been limited. Unsteady state flows have been examined for hoppers using particle dynamics/discrete element modeling (DEM) [40]. The advantages of DEM approaches are that the models use parameters with clear physical meaning (friction, adhesion, and restitution), although the relatively
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small number of large particles used by these models negates some of its physical appeal. Nonetheless, DEM models can be used to consider mechanistic effects, such as the effects of geometry or flow rate on stress and shear along the flow. Their disadvantages, as mentioned above, are the excessive computational requirements and the fact that they are largely limited to spherical particles. The approach used in this study, compartment modeling, has been used to model fluid-based systems for several decades but has been seldom applied to particulate systems. The main advantage of compartment models, which is the main motivation for our interest in them, is the low computational time required to simulate motion and mixing of millions of particles, providing calculated results in a time frame that enables the method to be used for process control [32]. The essence of compartment modeling is to divide the mixer into multiple zones, simulating motion and mixing (and stagnation and segregation) by prescribing fluxes of species between zones. Although stresses and strains on trajectories and wall collisions are not calculated explicitly, particles are still considered as discrete entities. Because these models do not attempt to capture the actual physics of motion but only motion itself, they are not limited by particle size or geometry, provided that fluxes can be measured or calculated (much in the same way as population balances and other statistically formulated models can make extensive predictions once aggregation and breakup kernels are formulated) [25]. Once flux parameters are selected, the simulation can be used to provide extensive information about the dynamics and the outcome of the process; for example, it can be used to predict mixing performance (in terms of a relative standard deviation of sample composition) along the axial length or at the discharge. More importantly, the model is valuable for predicting response dynamics after an atypical event such as a burst of a given ingredient at the entrance of the mixer, thus providing a starting point for the development of control laws. As mentioned, compartment models require substantial input (from experiments or from particle dynamics calculations) to provide realistic predictions. Unfortunately, the literature reveals little experimental work on continuous mixing, particularly for pharmaceutical applications. Kehlenbeck and Sommer [22] examined a CaCO3–maize starch mixture. In the authors’ previous work [32], an acetaminophen (APAP) formulation was examined, showing that mixing performance was strongly affected by blender operating parameters such as angle of incline and impeller speed. Some evidence exists to indicate that a well-controlled continuous mixing process can enhance productivity significantly [26, 30]. However, to develop an effective control law, the first requirement is the development of efficient predictive models that can be used for
process optimization and control. Moreover, the use of accurate models can reduce the need for sampling, which is difficult in continuous operation, always inconvenient, and an additional source of uncertainty often leading to type I errors. In this paper, compartment modeling is used to simulate continuous mixers based on currently available experimental information. The structure of the paper follows: In the next section (“Modeling Methodology”), the proposed compartment modeling is presented, followed by the experimental setup used to examine the powder flow (“Equipment and Materials”). Experimental measurements are described in “Homogeneity Measurements.” The method used to determine the fluxes from experimental work is explained in “Powder Fluxes.” The effects of modeling parameters are explored in “Compartment Modeling Parameters,” and the effects of sampling parameters (sample size, number of samples) are explored in “Sampling.” Predictions regarding effects of processing conditions are experimentally validated in “Experimental Validation.” “Summary and Conclusion” includes a summary and conclusion.
Experimental Modeling Methodology Several limitations hinder the application of existing modeling approaches [37, 23] for real time optimization and control. To overcome these problems, we propose a method that predicts product homogeneity using the basic ideas of compartment modeling [33]. The main idea behind compartment modeling is that it discretizes the space of the mixing vessel into locally homogeneous compartments. Particles move from compartment to compartment to account for both convection and dispersion. In this paper, a three-dimensional compartment modeling approach is utilized. As shown in Figs. 1 and 2, the mixer is discretized in both the axial and radial dimensions. The axial compartments are denoted by the index i, whereas the radial dimension is represented by index j, identifying each compartment as Sij. The total number of radial compartments (at a given axial position) is represented as Nr and the number of axial discretizations as Na. The total number of particles in the vessel (Nt) is the sum of particles in all of the compartments; each compartment Nr P Na P contains Nij particles so Nt ¼ Nij . The intensity of i¼1 j¼1
fluxes within the mixer depends on the number of compartments and the way that the compartments are connected. For example, in a binary system, two different types of particles are represented (Aij and Bij) and Nij =Aij +Bij. The effects of other particle properties (size, morphology, etc.) are captured by their fluxes.
J Pharm Innov Fig. 1 Radial and axial view of horizontal cylindrical mixing vessel investigated
One of the most important parameters in setting up the compartment model is the determination of the flux, which is a function of the number of particles exchanged between compartments and number of compartments. The fluxes between compartments are fine-tuned to account for the varying powder flow rates and stagnant mixing regions. The higher the rate, the larger the number of particles exchanged between compartments. The number of particles exchanged between two compartments Sij ! Si0 j 0 is denoted by lij!i0 j0 and represents a randomly selected number of particles that are transferred from compartment Sij to compartment Si′j′. The particles are selected randomly because we consider a perfectly random mixture; this results with the same probability of finding a particle of a constituent of the mixture for all points in the mixture as stated by Fan et al. [15]. It should be noted that this flow is between two compartments, and compartments exist in a three-dimensional space, thus capturing a bi-directional flow. The flux, Fluxij!i0 j0 , is defined as the ratio of the number of particles transferred to a neighbor compartment (Si′j′) with respect to the total number of particles within the compartment. The total number of particles within each compartment may remain constant (ΔNij =0) or may change (ΔNij ≠0).
The compositional distribution of particles within a compartment may also remain constant. This would require that, at time point k and a subsequent time point k+1, the ratios between each species (for a binary system A and B) also A B remain constant, i.e., $k!kþ1 Nijij ¼ 0; $k!kþ1 Nijij ¼ 0. This is statistically very improbable [34]; a more probable scenario is one where the distribution changes and the component A B distribution changes $k!kþ1 Nijij ¼ 6 0; $k!kþ1 Nijij ¼ 6 0. In this paper, the total number of particles within each compartment remains constant, but the component ratio within each compartment changes. Two important comments need to be made explicit at this point. The first major comment concerns the estimation of fluxes. As mentioned previously, this is a critical input for the model. In principle, flux information can be obtained in several ways: 1. From comparisons with samples taken along the axis of the mixer 2. From comparisons with samples taken at the discharge 3. From positron emission particle tracking or particle image velocimetry measurements 4. From DEM models run for short periods of time
Fig. 2 Schematic of a compartment diagram used to model the cylinder-mixing vessel
j=1
si+1, j
j=Na
si, j
si+2, j
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si+3, j s…, j Radial (i-component)
Axial (j-component)
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Over the long term, the authors plan to evaluate all of these sources of information to develop a robust approach for compartment modeling. In this article, however, the goal is to introduce the approach, provide preliminary validation, and demonstrate its use for evaluating sources of error associated with sampling. Only the first two sources of information are considered for this work. The second comment is that we assume all radial compartments to be equally filled. In reality, the mixer fill level is half empty; however, during operation, particles moved quickly along the radial direction. Although the mixer theoretically may have a low fill level, the material is largely uniformly distributed in the radial direction. For the mixer examined here, radial heterogeneity is not a concern, and transport in the radial direction is largely a minor concern. Radial compartments are retained as part of the model only to lay the foundation for more complex situations (to be examined in future papers), where mixing in the radial direction could be an important issue. Considering the specific geometry of the continuous mixing system examined in this paper, uniform fluxes between radial compartments are used because the convective blades rotate radially. The axial fluxes are uniform along the horizontal length of the mixer because an equally distributed number of blades are located along the axis (as shown in Fig. 1). Independent of the mixer inclination, due to mass conservation, powder flow is assumed to be constant along the axial direction. A weir is used to maintain a specified fill level. Once the number of particles that are interchanged between compartments is determined, the actual particles exchanged are randomly chosen among all particles within the compartment.
Equipment and Materials The continuous mixer modeled in this paper is manufactured by Buck Systems in Birmingham, England. The mixer is 0.31 m long, and the radius is 0.025 m. The mixer’s rotation rate ranges from 17 to 340 rpm, impeller speed was a constant 50 rpm for this experiment. An adjustable number of triangular flat blades are placed within the horizontal mixer. The feeding system used in these experiments is composed of two vibratory mechanisms (Eriez, Erie, PA, USA). The variability in the flow rate depends on powder properties such as particle size, surface roughness, electrostatic charge, and other properties that affect cohesion [8]. The feeding flow rate is dependent on the powder’s bulk density and the vibration speed. The average bulk density of lactose 125 and acetaminophen/ lactose 125 blends are shown in Table 1. The error in the estimation of the density is measured as the standard
Table 1 Density of powder formulations Active % Lactose 125 M Lactose 125 M and 8% 30-µm milled APAP Lactose 125 M and 9% 30-µm milled APAP Lactose 125 M and 10% 30-µm milled APAP Lactose 125 M and 12% 30-µm milled APAP Lactose 125 M and 15% 30-µm milled APAP Lactose 125 M and 16% 30-µm milled APAP 100% 30-µm milled APAP
Density (g/ml) 0.785±0.009 0.939±0.009 0.882±0.013 0.928±0.013 0.919±0.018 0.92±0.014 0.934±0.025 0.320±0.030
deviation of the ten samples; the lack of linearity corresponds with the work of others [41, 42]. The average flow rate and its variability are calculated by weighing the mass of powder discharged in 1 s interval. The deviation from the mean flow rate will affect the overall amount of each ingredient fed to the system. In the flow studies conducted for this paper, the authors found higher variability in the flow rate of the pre-blend composed of acetaminophen and lactose 125 compared to pure lactose 125. This is possibly due to electrostatic agglomeration of acetaminophen in the vibratory feeder, as well as the increased cohesion of the blend containing acetaminophen [28]. The lactose feed mass flow rate was determined three times; outflow variability was found to be 12.54±1.80 gs−1, 13.65± 2.70 gs−1, and 10.29±2.90 gs−1, with an overall average of 12.16±2.46 gs−1. The pre-blend component flow rate was also determined three times, generating outflow values of 6.56±1.08 gs−1, 7.42±1.44 gs−1, and 5.65±0.96 gs−1, with an overall average of 6.55±1.16 gs−1. Experimental measurements of the fluxes between compartments is a complicated task. For batch mixers, fluxes can be estimated using specialized experimental procedures such as solidification [38] and image analysis of a discretized mixer [27]. However, these approaches, which require sacrificing the mixing vessel, are difficult to implement for continuous mixing experiments. Therefore, the authors resorted to sampling from five carefully selected sites on the mixer. Samples were approximately 1.6 g each. The mixing vessel was constructed with five sample ports on the top, allowing for axial sampling (Fig. 3). The five sampling ports were chosen to maximize the number of axial areas examined while maintaining an adequate port diameter to allow sample collection. From each opening, five samples were obtained using a probe. Experimentally, samples were not removed from the inflow to avoid changing the target concentration, and because materials have yet to be mixed at the inflow positions. To measure mixture homogeneity as a function of axial length, retrieved samples were analyzed using near infrared spectroscopy (NIR). The NIR system used to analyze the experimental
J Pharm Innov Fig. 3 Axial sampling ports of the horizontal cylindrical vessel
Calculate Tracer %
samples collected for this work was the Nicolet Antaris, Near-IR Analyzer (Thermo Electron Corporation). A calibration set of 90 samples was created with APAP/ lactose concentration varying over the APAP concentration range from 0% to 9%. Sample size was approximately 5 g. These samples were then scanned over the wavelength range from 10,000–4,000 cm using OMNIC software (Thermo Electron Corporation) for operating the instrument. Two spectral regions were important: 4,728–4,613 and 4,972– 4,887 cm. Using TQ Analyst (Thermo Electron Corporation), a partial least-squares regression model was created for APAP concentration, using mean-centered second-derivative spectra (no spectral smoothing was applied). The performance of the model was then assessed using a set of validation samples (samples of known concentration not used in the calibration set). The calibration curve for the acetaminophen/lactose 125 blend is illustrated in Fig. 4. The RMSEP is 0.23, and the RMSEC is 0.352 (R2 =0.9933). One of the prerequisites for spectroscopic analysis involves determination of the amount of scan averaging required to achieve adequate signal for the required degree of sensitivity. Based on the data in Fig. 5, where samples containing 0.6%, 3.5%, and 5% acetaminiophen were analyzed with increasing numbers of spectral scans, the authors determined that 38 scans per sample spectrum were required to achieve the necessary sensitivity. Using the Antaris instrument, the authors chose to perform invasive sampling and offline analysis for this work rather than online analysis because of
10 9 8 7 6 5 4 3 2 1 0
the challenges of achieving adequate scan averaging with online analysis. Additionally, using an offline method simplifies the analytical method validation effort.
Results and Discussion Homogeneity Measurements The homogeneity of a powder mixture may, in some cases, be evaluated by calculating the variability of the retrieved samples. For batch mixing vessels, the samples are typically retrieved throughout the powder bed [31], whereas for continuous mixing, the powder can be analyzed both at the outlet [32] and along the axis (as presented in this paper). For binary mixtures, measuring one of the components is sufficient to determine the distribution within each sample. However, for a multi-component mixture composed of r species, r−1 measurements per sample are required to determine the component distribution. This is important in pharmaceutical applications because both the amounts of active and excipients must be consistent to achieve uniformity of dosage and dissolution [12]. Consider a set of n samples retrieved from a binary mixture where the amount of a minor component (the tracer) within the kth sample is represented by Xk. The average tracer concentration of all the n samples is defined as X. The homogeneity of the mixture is represented by the relative standard deviation (RSD) of the tracer concentration of all n samples. Xk ¼fc1 ; . . . cm1 ;cm g
R2 = 0.9933 RMSEP=.230 RMSEC=.352
X¼
0
2
4
6
8
Actual Tracer % Calibration
Validation
Fig. 4 NIR calibration curve (partial least-squares regression, secondderivative spectra)
X1 þ . . . . . .Xn n
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n X Xk X 2 1u t RSD ¼ n1 X k¼1 Each sample Xk is composed of “m” number of particles, where the “lth” particle is represented by cl. The greater the number of particles, the larger the sample size. In addition,
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a
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.6 % Active
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Fig. 5 Determination of required scan averaging. Prediction of three samples, including a 0.6%, b 3.5%, and c 5% of tracer (APAP) concentration, analyzed with six to 68 scans
the greater the number of samples, the higher the number of Xnth terms. Within each compartment, a number of samples are retrieved; the RSD of the ith radial compartment and jth axial compartment is denoted as RSDij. The variability within all the radial compartments at a fixed axial position “j” is denoted as RSDj. This measurement is calculated by determining the relative standard deviation between all samples retrieved within all radial compartments at a fixed axial position j. Sijk represents the tracer concentration of the kth sample from the ith radial and jth axial compartment. The total number of samples taken within the compartment is n. The average of the n samples retrieved within all Nr radial compartments at a fixed axial (jth) position is represented by Xj. Xj ¼
Nr n 1 X 1X Xijk Nr i¼1 n k¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nr P n 2 P 1 Xijk Xj Nr n RSDj ¼
i¼1 k¼1
Xj
Lower RSD values mean less variability between samples, which implies better mixing. To obtain the RSD within the axial section, the RSD within all the radial sectors that lie within the axial compartment are measured. Samples are retrieved from each of the radial slices, and the RSD of all the samples within that axial portion are used to determine sample homogeneity.
Powder Fluxes Powder mixing occurs by three mechanisms, including shear, dispersion, and convection. One method to charac-
terize the rate of powder movement is to isolate a finite area of the vessel and perform a mass balance, as is often done in the characterization of convective–dispersive fluid flow problems. The convective–dispersion equation has been applied to describe blending of the particulate material [20] using the following expression: @C ðx; rÞ @C ðx; rÞ @C ðx; rÞ þ Vx ðx; rÞ þ V ðx; rÞr @t @x @r 2 2 @ C ðx; rÞ @ C ðx; rÞ ¼ Dðx; rÞ þ @x2 @r2 where C represents the concentration of one component, Vx refers to the axial transport velocity, Vr refers to the radial transport velocity, and D is the dispersion coefficient, which is hereby assumed to be isotropic along the radial (r) and axial (x) directions. Berthiaux and coworkers [5] stated that the velocity in the radial direction could be neglected because they assumed it is orders of magnitude lower than that of the axial component, suggesting that the dynamic change in concentration occurs due to the rate of dispersion in the radial and axial direction, as well as the axial transport in the axial direction. The mass balance resulting from this equation can be discretized and solved numerically. A general schematic of a discretized area within the vessel that can be represented by the previous expression is shown in Fig. 6. A simple mass balance of one component within a finite area of the vessel can be applied, resulting in the following expression: @C ¼ Ff Ci1;j Ci;j þ Fb Ciþ1;j Ci;j @t þ Fr Ci;jþ1 þ Ci;j1 Ci;j where Fr, Ff, and Fb represent the radial, forward, and backward fluxes, respectively; Cij is the concentration of each finite area where i and j represents the axial and radial
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Fr Fr Ff Fb
per compartment. These parameters affect both the quality of the results and the computational time. Other parameters examined that are difficult to characterize experimentally but that are known to affect homogeneity estimates are the number of samples retrieved and the sample size. These parameters are examined in “Sampling.” Compartment Fluxes
xi Fig. 6 General scheme of radial and axial convective dispersion model
position, respectively. To define the dynamic change in concentration, the flow rates must be known (Fr, Ff, Fb) and can be calculated based on the bulk particle flow rate. While the fluxes between compartments should also be defined to implement the proposed compartment model, one clear difference of the compartment model approach is that the fluxes represent a finite number of discrete particles interchanged between compartments. The fluxes are not considered to be a continuous property but a number of discrete particles that are stochastically chosen. Thus, the concentration of the outflow stream will not necessarily be the same as the concentration in the compartment the stream is leaving; deviations can occur due to the stochastic nature of the particle selection process. It is important to mention that samples were not taken from the outflow stream until a steady state was reached. In this paper, the fluxes are found using the experimental measurements along the axial length for the experimental vessel (Fig. 3). The variability between the samples retrieved from one spatial area was calculated using the measurements described in the previous section. The slope of the change in RSD versus axial length, m, can be found using linear regression, as shown in Fig. 7, where m ¼ $RSD ‘ . In cases were the RSD function is nonlinear (e.g., an exponential trend), it is recommended that the fluxes chosen use the slopes corresponding to the appropriate axial position. In this scenario, the slope, m, will not @ ðRSDÞ be considered a constant, m ¼ @‘‘ 6¼ C, but a function of ‘. The following section will focus more on using the slope to adapt it to the compartment methodology.
Effective use of the proposed compartment modeling approach discussed in “Modeling Methodology” requires that the fluxes between the compartments must be determined. This is accomplished in the following way: Samples are retrieved at the designated axial locations shown in Fig. 3, and the slope of the sample variability along the axial length of the mixer is calculated. Due to the symmetry of the convective motion and short axial distance ð$‘k Þ, the change in variability between the designated locations is assumed to be linear. As the axial length ð$‘k Þ is represented by NAk compartments, the change in content uniformity along these compartments is represented as N$‘Akk . This describes a scale effect where NAk compartments are used to model the $‘k axial length of the vessel. Although the change in concentration that occurs along the axis is monitored, the content uniformity of the samples retrieved at a fixed axial position measure the radial variability. In the existing model, the vessel diameter is discretized into (NR) radial slices, and the particles fluxes are between NR −1 compartments. Mixing occurs by the convective motion of the blades rotating 360° at a constant rate. As a result, the rate of the number of particles exchanged within the radial compartments is uniformly expressed among the radial m $‘
NA compartments as = ¼ NR 1 . The flux depends on the number of particles exchanged between compartments relative to the total number of particles. Thus, the radial flux, Fluxij→i′j′, represents the fraction of the total number of particles exchanged, from one compartments (ij) to another (i′j′), to the total number of particles in compartment ij. In the current model, the rate of mixing depends on the number of the particles exchanged between the compartments.
0.3 0.25 0.2
RSD
rj
y = -0.1493x + 0.206 R2 = 0.9206
0.15 0.1
Compartment Modeling Parameters
0.05 n=3
0 0
The modeling parameters examined in this section consist of the fluxes between the compartments, the number of radial and axial compartments, and the number of particles
0.2
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Fig. 7 Experimental RSD obtained at no inclination for n=3 experiments at a speed of 50 rpm vs. axial length with a linear slope of 0.15
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i0 ¼1 j0 ¼1 i¼1 j¼1
defined, considering the total number of particles within the vessel and the rate of interchange between all the compartments in the vessel. As Fig. 8 illustrates, increasing the flux results in lower RSD curves that in effect means faster mixing. It is important to mention that in the authors’ previous work [32], it was illustrated that increasing the rotation rate within the mixer did not improve the overall mixing performance. This occurs because, at higher rotation rates, the particles also tend to spend less time in the mixer, leading to a lower total mixing rate for higher rotation rates. Radial Compartments The number of radial compartments has an effect on the estimate of radial variability of the flowing powder at or near the inflow region, in particular. The greater the number of radial compartments, the higher the precision with which the radial component of the compositional variance can be determined. The number of compartments, however, has an impact on the computational requirements: The larger the number of compartments, the greater the number of fluxes between the compartments that must be defined. Statistically, the lower the number of compartments, the lower the initial variability because higher uniformity exists between a smaller number of larger compartments (averaging effect).
0.12 0.1
RSD
Increasing the number of particles exchanged between compartments results in higher fluxes and a faster mixing rate, which ultimately means that the mixed state is reached faster. The particles selected to transition to other compartments are random. The fluxes are defined as the ratio between the number of particles interchanged within two compartments (Sij, Si′j′) relative to the total number of particles within each 1 0 j0 compartment, Fluxij!i0 j0 ¼ ij!i Nij . The non-dimensional flux can range between 0 and 1, depending on the level of exchange, from no mixing, where no particles are exchanged, to a high degree of mixing, where all particles are exchanged. NR P NA P NR P NA P A total mixing rate, MRtotal ¼ Fluxij!i0 j0 , can be
0.08 0.06 0.04 0.02 0 0
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5 Radial Compartments 15 Radial Compartments
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Fig. 8 Effect of flux (0.1, 0.18, and 0.22) between compartments; all the other modeling parameters are kept constant
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Consider a binary formulation, where some of the compartments are loaded with one component, and all other compartments are loaded with the other component. Figure 9 illustrates the effect of loading the exact percentage of tracer in each example but varying the number of radial compartments. In the case of an example with five radial compartments, one is loaded with the tracer and the other four with non-tracer; in the case of ten radial compartments, two are loaded with tracer and the other eight with non-tracer. The flux is adjusted to account for the change in the number of fluxes as well as for the variation in the flux magnitude. This keeps the percentage of the tracer constant to keep the total number of particles dispersed between compartments constant. For example, if a finite number of particles denoted as Ptotal exist and they are divided among a number of radial compartments, Nradial, the flux is expected to change along with the number of radial compartments. Thus, the greater the number of radial compartments, Nradial, the lower the flux between the compartments, given a greater number of fluxes between compartments. At a fixed axial position, the mixing rate between the radial compartments can be found using the following equation: NR P NR P 0
0.06
0.6
Fig. 9 Effects of changing the number of radial partitions, while the number of axial partitions, total mixing rate, and the total number of particles are kept constant
Fluxijnew !i0 j0new ¼ i ¼1 0.07
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i¼1
Fluxij!i0 j Nij
Nij NRnew
8i
In this case study, the relative standard deviation (RSD) between the radial compartments represents the radial degree of homogeneity of the mixture, where the number of samples is the number of compartments and the sample size is the number of particles per compartment. The Central Limit Theorem states that, for independent samples, sample variance is inversely proportional to sample size. In agreement with the theorem, the case study with the lowest number of radial compartments contained the largest number of particles per radial compartments and resulted in the lowest variability. This, in fact, introduces a grid
J Pharm Innov 0.07 0.06 0.05
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effect in the estimate of homogeneity. To avoid this grid dependence problem, sample size should be consistent (i.e., should contain the same number of particles), regardless of the total number of compartments or the number of particles per radial compartment.
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In this section, the effect of varying the number of axial compartments is examined, while the number of radial compartments, the total dispersion, and the total number of particles are kept constant. Ten, 20, and 30 axial compartments are used to track the relative standard deviation along the axial trajectory. At the horizontal mixer setting, the axial flux is the dominant axial rate of transport (i.e., there is no back flow), and there is basically a uniform residence time. To compare the results using a larger number of axial compartments, the flux between the compartments was reduced to maintain the same overall axial mixing rate. When adjusting the number of axial compartments, the total number of compartments changes, affecting the total mixing rate. To keep the total mixing rate constant (MRtotal =C2), when the number of axial compartments, NA, changes, the fluxes must be adjusted to keep the sum of all the fluxes constant: C2 ¼
NR X NA X NR X NA X i0 ¼1 j0 ¼1 i¼1 j¼1
Fluxij!i0 j0 ¼
NR NX NR NX Anew X Anew X
Fluxij!i0 j0
i0 ¼1 j0 ¼1 i¼1 j¼1
Counter-intuitively, the reduction in the fluxes when the number of axial compartments changed from 10, 20, and 30 result in a reduction of the CPU time required, even though a greater number of compartments were used. All simulations were conducted using a Sun Sparc 900 MHz Processor 2 GB. It is important to mention that the advantage of the model is that it is not limited in the number of axial measurements it can obtain; however, experimentally, we are often limited by sampling constraints. Thus, the model’s benefit is that it can be used to examine the effect of sampling location. The authors have discussed the effect of sampling location in their previous work [33]. In summary, decreasing the number of axial compartments will decrease the accuracy of the model’s content uniformity predictions, as illustrated in Fig. 10.
10 Axial Compartments 30 Axial Compartments
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Fig. 10 Effects of changing the number of axial partitions from ten to 30, keeping the radial partitions and total number of particles constant
experimental conditions. In reality, particle size of common excipients, such as lactose, may be in the micron range [9]. Calcium carbonate particles are found in the range of 5–214 µm [19]. Therefore, the actual number of particles involved is several orders of magnitude larger than that considered in traditional particle dynamics simulations. The question that remains, however, is what is the minimum number of particles required in the simulation to represent the physics realistically. This is an open question that depends, at least in part, on flow properties such as cohesion. In some rotating drum case studies, decreasing the number of particles and increasing the particle size will still result in the same dilation effect or powder flow properties [16]. However, in mixing, it is important to use a large enough number of particles to minimize effects of sampling parameters on the estimate on mixing performance. In the modeling approach explored in this work, the authors specified (a) the number of particles within each compartment; (b) the number of particles in the entire vessel, which reflects the sum of particles in each compartment; and (c) the fluxes between compartments. Figure 11 illustrates the effect of changing the number of particles in each compartment by a factor of 10× while keeping the number of compartments constant. The overall effect of increasing the number of particles is not substantial in terms of the RSD curve trend and slope. However, the CPU time required for the larger number of particles was increased from 51 CPU seconds to 8,005 CPU seconds, a factor of 157.
Sampling Number of Particles Most of the existing simulation approaches suffer from limitations in the number of particles they can handle. For example, simulation involving from 4,000 to 30,000 particles in high-shear mixers that range in particle diameter from 8 to 200 mm [7, 14, 36] is far from the practical
Sampling is very important in determining the quality of the pharmaceutical product. The Food and Drug Administration’s Guidance [17] on powder blends and finished dosage units (http://www.fda.gov/cder/guidance/5831dft.htm) focuses on the use of stratified sampling of blend and dosage units to demonstrate adequacy of mix for powder blends
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Fig. 11 Effect of the number of particles per compartment on RSD profile considering 250 compartments in both cases
processed in batch processing equipment. However, at this time, the FDA has yet to establish any guidelines on sampling the continuous mixing process. Many of the existing sampling issues mentioned in this paper have not yet been addressed, either in regulations or in the literature, to any significant degree. Thus, it is deemed important here to begin examining the sampling issues related to continuous mixing. Sampling questions that arise is whether to increase the number of samples taken throughout the vessel or to increase the sample size. A small number of powder samples results in greater variability (and lower statistical power) for the homogeneity index. The effect of the sample size and the number of samples can be accurately predicted only for random mixtures (i.e., mixtures that are statistically homogeneous). While the random mixture model is a useful limiting case, real systems, and, in particular, systems that show mixing problems, deviate substantially from the random model because the main sources of heterogeneity (agglomeration, segregation, and stagnation) all cause nonnormality in composition distributions. For such systems, which are precisely the systems that sampling is (or should be) designed to “catch,” the effect of sampling parameters cannot be easily predicted and must be laboriously determined. Fortunately, compartment modeling is an excellent tool for simulating sampling of non-homogeneous systems, nicely adding an arrow to our QbD quiver. Compartment modeling is useful for predicting powder behavior (during mixing) as a function of sample size and for evaluating sampling protocols.
Sample Size Sample size is defined as the mass of powder within one sample. The limitations that exist using a large sample size are that dead or segregated zones that are smaller than the sample space can remain hidden. Decreasing sample size
may result in higher measurement errors relative to the true blend homogeneity. However, it is important to keep in mind that the scale of scrutiny is based on the sample size of the target delivery system. Experimentally, these effects can be noticed using advanced sampling techniques, including solidification of the powder bed [10] or vessel partitions [27]. However, such experiments are somewhat impractical, as described previously. Minimizing disturbance of the powder bed by optimizing sampling is a priority. Online non-invasive sampling would be ideal, but method validation is challenging because the sample is not captured and cannot be analyzed by a corroborating method. Online sampling requires both rapid data acquisition and an accurate/fast algorithm to accomplish prediction in a short sampling time window. The small sample size associated with some online reflectance measurements [2] has led us to examine small sample sizes. To the best of our knowledge, NIR has not been successfully integrated online for continuous powder mixing processes. In this work, the effect of variable sample size and number of samples on the axial measurements is examined both experimentally and in silico. To determine the effect of sample size, the authors examined 250 compartments containing a total of 10,000 particles. Samples were taken from the radial compartments and used to calculate the RSD of the tracer concentration at a fixed axial position. The total number of samples used was constant, but the number of particles in each sample was varied from 50 to 1,000. Figure 12 illustrates the unique RSD curves for each sample size. Clearly, the variability is higher for the small sample size. In fact, the average variability between the unique sample sizes and the result for all particles was 59% for 50 particles, followed by 17.1% for 500 particles and 12.5% for 1,000 particles. Computationally increasing the number of particles in the sample results in a very small increase in computational time: about 51 CPUs for 50 particles, 56 CPUs for 500 particles, and 61 CPUs for 1,000 particles.
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J Pharm Innov Table 2 Effect of the sample size using two focal diameters under NIR absorbance
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0.15 0.156 0.22 0.229 0.183 0.195
0.086 0.176 0.171 0.199 0.092 0.087
1.457 2.566 2.438 1.841 2.215 2.256
1.364 2.434 2.246 1.706 2.278 2.446
The effect of increasing sample size on the content uniformity measurements was examined experimentally for continuous mixing experiments where the percentage of APAP in the inflow varied. The samples were analyzed using NIR with 10.8- and 30.8-mm diameter windows. The larger window resulted in a larger sample size. The results shown in Table 2 illustrate that increasing the sample size for six different continuous mixing experiments with varying APAP target mean concentrations resulted in a lower sample-to-sample variability as well as a change in the average concentration of APAP measured. The experiments illustrate that increasing sample size generally reduced the variability between samples, providing evidence that compartment modeling could be a valuable tool to quantitatively estimate the effect of sample size on measurement uncertainty. Number of Samples Increasing the number of samples expands the sampling space, reducing the probability of overlooking stagnant regions within the blender. Increasing the number of samples also increases the statistical power of the sample population estimates. A disadvantage of the invasive 0.25
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sampling is that it has the potential to alter the powder distribution. Thus, determining the optimal number of samples is an important goal, requiring a compromise between experimental limitations and providing reassurance that no unmixed zones are overlooked within the vessel. It is important that the variability in population estimates are sufficiently low to give a high-quality assessment of blend properties. Experimentally, the authors studied the number of samples retrieved across the axial length of the mixer. As described in the experimental section, the mixing system was rotated at an impeller speed of 50 rpm in the horizontal position. The results shown in Fig. 13 indicate that, by varying the number of ~1.6-g samples retrieved at each spatial area (from three to five samples), experimentally and computationally, the variability in the content uniformity decreased as number of samples increased. The model showed lower RSD values than the experiments as expected, as invasive sampling is inherently a highly variable process. Questions that arise, then, are (1) “What is the optimum number of samples?,” and (2) “How accurately can the model capture the improving RSD profile as the number of samples increases?” The RSD behavior for increasing numbers of samples is shown in Fig. 14, where the number of samples is varied while keeping fixed the number of particles per sample (100). With a constant 100 particles per sample, RSD was not linearly related to the number of samples. It should also be
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Fig. 13 Compartment model and experimental results RSD measurements for three and five samples (model, 100 particles per sample; experimental, 30-mm diameter) taken throughout the axial length; the error bars are obtained from replicating the experiment three times
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Fig. 15 Relative standard deviation as a function of axial length within the mixer; the error bars are obtained from replicating the experiment three times
mentioned that the computational time was trivially increased, from 51.8 CPUs for 25 samples to 52.3 CPUs for 50 samples and 55.5 CPUs for 200 samples. The RSD trend might show more sensitivity to the number of samples with a larger number of particles per sample.
Experimental Validation Experimental work provided the data for validation of the modeling results. Computational data was derived by selecting five samples of 100 particles from radial and axial compartments. Computationally, the mixing rate was chosen to match the outflow variability. Figure 15 compares the computational results to the experimental RSD data. While the computational model provides results that are, on average, similar to the experimental data, the data derived from the model does not illustrate the variability found in the experimental results. The variability in experimental data may be due to sampling, analytical error, or true mixture variability.
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Fig. 17 Experimental and computational RSD results obtained from acetaminophen concentration. The computational results are shown as a function of four different sample sizes 20 particles, 40 particles, 400 particles, and all of the particles (concentration of the entire output)
Processing Angle The authors have previously illustrated the impact of processing angle on the relative standard deviation of the outflow content uniformity [32], where increasing processing angle was demonstrated to enhance mixing due to increased residence time. Here, we examine the effect of using an upward positive processing inclination of +17° while maintaining the impeller speed at 50 rpm. Six samples were retrieved for each spatial area. The experimental results and computational output utilizing compartment modeling are both shown in Fig. 16, where the data correlates well with the authors’ previous work. RSD values are consistently lower and more stable than in the horizontal mixer. As suggested by Berthiaux et al. [6], a more homogenous blend, as witnessed in the inclined blender, may be affected less by sampling, thereby reducing the sampling error.
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Fig. 16 Experimental results obtained from changing the processing angle and compartment modeling results at the upward processing angle
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Fig. 18 Compartment model results showing the effect of variability in the active mass fraction of the inflow and the RSD of the outflow homogeneity of the powder
J Pharm Innov
Product Formulation A large array of publications exists regarding the different physical properties affecting powder flow. These properties include particle size [18], particle morphology, powder density, and surface structure [1], mixing [11], and diagnostics [4]. Lacking “mixing rules,” systematic characterization of the properties of all relevant powders that could be used in a continuous mixing process is an immense task. In this work, the authors have demonstrated a compartment modeling approach to continuous powder blending, demonstrating its capability for mixtures with variable active-ingredient concentrations. Experiments were conducted at various APAP concentrations using two sample sizes in the NIR analysis of the experimental data. The results showed that increasing APAP concentration decreased the sample-to-sample variance in the outflow, as illustrated in Fig. 17. In addition, the larger sample size resulted in an improved RSD as anticipated. Compartment modeling also indicated that outflow variability was a function of sample size (Fig. 17). The effects of inflow APAP fluctuations have also been examined. Continuous mixing performance is sometimes assessed by computing the variance reduction ratio (VRR), which accounts for the inflow fluctuations that arise due to the feeding mechanism. Utilizing vibratory feeders, we have demonstrated that feeding variations are present, limiting the flow-rate capacity. Predicting the effect of inflow variability on outflow has sparked interest in the pharmaceutical community. The variability of the active ingredient mass is modeled with a pseudo-random flow rate calculated from the active experimental variability. Using the methodology proposed in this paper, the effect of the input variability on the outflow was calculated (Fig. 18). The top curve represents the mass fraction of active ingredient at a test point, while the bottom curve is the corresponding outflow. The variance of the fluctuating inflow in Fig. 18 is 1.82E-04. After mixing, the resulting RSD profile has a variance of 3.02E-05 with a resulting VRR, defined as the inflow variance over the outflow variance of 16. The VRR that is high implies good mixing as discussed by Williams and Rahman [39], and Beaudry [3].
Summary and Conclusion The focus of this paper is to introduce a compartment modeling method that provides a fast and convenient alternative to particle dynamics modeling methods. The method is suitable for gaining understanding of the effects of sampling parameters for process characterization and, potentially, for control purposes. Given the current interest
in continuous powder blending, the case study selected in this paper focuses on the axial mixing behavior within a continuous mixer as a function of processing parameters. The paper outlined the suitability and flexibility of compartment modeling to capture the dynamics of continuous powder blending. The advantages of this approach are (a) the sampling flexibility, (b) the short computational time, and (c) the ability to predict the axial and outflow variability given inflow fluctuations (which will ultimately be useful for online control and optimization). The modeling parameters that were examined include the number of axial and radial compartments, the number of particles, and particle fluxes between compartments. Results proved that the compartment modeling approach is both feasible and relatively convenient. Effects of sample size and number of samples were clearly captured by the model. Fluxes were easily adjusted to account for differences in mixing performance resulting from changes in processing parameters. While having been extensively used for fluid processes, the compartment modeling approach is in its infancy when it comes to powder processes. Much work remains to be done for the method to reach full blossom. To wit, multiple sources of information regarding fluxes will be tested and incorporated into a general framework. Methods for accounting for particle agglomeration and attrition can be included. The model can be coupled with process simulators for control processes. These and other expansions of the present work will be addressed in future publications. Acknowledgment The authors would like to thank the National Science Foundation for their financial support through grants NSF0504497 and NSF-ECC 0540855 both to Fernando J. Muzzio as well as the Nanopharmaceutical IGERT Fellowship to Patricia M. Portillo.
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