# Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model

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## Abstract

**:**

_{gen}and the apparent overall heat transfer coefficient times surface area UA, were estimated. For the test runs, these parameters were predicted as a function of the process parameters via a power law (PL) model and machine learning (ML) model. The LPM augmented with the PL and ML models was used to predict the temperature evolution in the test runs. The LPM predictions were also compared with those of an enthalpy balance model (EBM) developed recently. The LPM had a fitting capability with a root-mean-squared error (RMSE) lower than 0.9 °C, and a prediction capability, when augmented with the PL and ML models, with an RMSE lower than 4.1 and 2.1 °C, respectively. Overall, the LPM augmented with the PL model had both good descriptive and predictive capability, whereas the one with the ML model had a comparable predictive capability. Despite being simple, with two parameters and obviating the need for sophisticated numerical techniques for its solution, the semi-theoretical LPM generally predicts the temperature evolution similarly or slightly better than the EBM. Hence, this study has provided a validated, simple model for pharmaceutical engineers to simulate the temperature evolution during the nanomilling process, which will help to set proper process controls for thermally labile drugs.

## 1. Introduction

_{gen}and the apparent overall heat transfer coefficient times surface area UA, were obtained by direct fitting of the LPM to the experimentally measured temperature profiles via SigmaPlot. The parameters estimated for the 27 process runs enabled us to train the power law (PL) and the machine learning (ML) model. By using the trained models, we predicted Q

_{gen}and UA of the five test runs. Minitab was used for the PL predictions, whereas Google Colab was used for the ML predictions. By inserting the predicted Q

_{gen}and UA values in the LPM, the temperature profiles of the test runs were simulated. The advantages and disadvantages of the LPM and the EBM as well as the limitations of the LPM were discussed. Not only will this study reveal the fitting capability of the LPM as compared with the EBM, but it will also enable us to assess their comparative predictive capabilities and usefulness for process development and understanding. Overall, this study provides pharmaceutical engineers with a validated, simple model (LPM), which simulates the temperature evolution during the production of drug nanosuspensions and predicts the impact of process parameters, thereby eventually helping engineers to control and optimize the process.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Methods

#### 2.2.1. Wet Stirred Media Milling

#### 2.2.2. Formulation of the Lumped-Parameter Model (LPM)

_{p}is the specific heat capacity, T is the temperature at the mill outlet, Q

_{gen}is the apparent heat generation rate during milling, UA is the apparent overall heat transfer coefficient times surface area, and T

_{ch}is the chiller temperature. Strictly speaking, Equation (1) represents a transient enthalpy balance for a perfectly mixed batch process. The perfect mixing implies that the mill outlet temperature is equal to the temperature of the suspension in the mill chamber. The well-mixedness in the milling chamber has been established as a valid approximation to the residence time distribution in small mills (small length-to-diameter ratio) [64]. Hence, for a recirculation mill operating with a fixed batch size and recirculation rate, Q

_{gen}and UA may only represent the heat generation rate and overall heat transfer coefficient times surface area, in some approximate, apparent, and statistical manner because they are obtained by fitting to experimental data directly. Although UA can be estimated based on heat transfer correlations for the internal and external convective heat transfer coefficients [33], such correlations are approximate, and none exists for the specific stirrer–mill chamber geometry. We also assumed time-invariant, constant Q

_{gen}, C

_{p}, UA, and T

_{ch}(6.1 °C).

_{p}were determined considering the materials in the mill chamber: the beads (zirconia, C

_{p}= 0.46 J/g °C [65]), the suspension (10% FNB with respect to water, C

_{p}= 3.93 J/g °C), and the stirrer element (zirconia, C

_{p}= 0.46 J/g °C). Whereas the stirrer element mass was constant, the bead and suspension mass varied when bead loading was changed in various runs (refer to Table 1). The C

_{p}was calculated as the weighted average of the C

_{p}of individual materials and the mC

_{p}was found to be 465.6, 455.2, 444.8, 460.4, 450.0, and 470.9 J/°C for 0.4, 0.5, 0.6, 0.45, 0.55, and 0.35 bead loadings, respectively.

#### 2.2.3. Fits by the LPM and Predictions by the LPM Augmented with the PL and ML Models

_{gen}and UA were estimated. Then, these parameters were mathematically expressed as a function of the process parameters for the 27 training runs and predicted as a function of the process parameters for the 5 test runs using a power law (PL) model and a machine learning (ML) model. Minitab was used for the PL predictions, whereas Google Colab was used for the ML predictions. Among the several applied machine learning approaches using Google Colab, as shown in Table S1 of Supplementary Materials, k-nearest neighborhood (KNN) [66] with k = 5 was selected because of its low mean squared error (MSE) and mean absolute error (MAE) compared to other methods for the test runs. Therefore, ML refers to KNN (k = 5) for the rest of this study.

## 3. Results and Discussion

#### 3.1. Properties of the Milled Suspensions and Particles

#### 3.2. Fitted LPM Parameters and the Origin of Temperature Rise during the Milling

_{gen}is linearly and strongly correlated with the average mechanical power consumption P (R

^{2}= 0.97). The value of the constant slope of the linear correlation in Figure 1 indicates that about 64% of the power consumption (rate of shaft work) dissipates as heat. This is not surprising at all: only a small fraction of the mechanical energy spent on mixing the suspension–bead mixture is used to deform the particles [31]. Most is converted into heat through dissipative processes such as viscous losses, inelastic bead–bead and bead–wall collisions, etc. [67]. Some of the shaft work is also spent on generating new particle surfaces (surface energy), sound, and the elastic parts of bead–bead and bead–wall collisions [33].

_{rise}in the mill at 6 min was affected by the apparent heat generation rate Q

_{gen}parameter of the LPM. Based on Equation (1), we expect that Q

_{gen}is the driving force for the temperature rise, which was illustrated in Figure 2. Overall, the temperature rise was more pronounced for higher Q

_{gen}or higher P, in view of Figure 1. The Gompertz growth function in Equation (3) fitted the temperature rise well (R

^{2}= 0.95). As Q

_{gen}approaches zero, it predicts a negligibly small temperature rise (~0.7 °C).

_{gen}values do not reflect the actual heat generation rate; by and large, Q

_{gen}is a fitting parameter affected by the accuracy of the experimental measurements and the assumptions made in the model development. On the other hand, Figure 1 and Figure 2 and the correlations therein strongly associate Q

_{gen}with the underlying physics of the conversion of shaft work (power consumption) into heat and ensuing temperature rise. The upshot of these findings is that the LPM differs from a purely empirical model. The latter would fit temperature evolution as a function of time with parameters that have no connection to the physics of the heat generation–transfer phenomena.

#### 3.3. LPM-Fitted Temperature Profiles and LPM–PL/LPM-ML Predictions in the Training Runs

_{gen}and UA in Runs 1–27, which were conducted at various stirrer speeds ω, bead loadings c, and the bead sizes D

_{b}. For the PL model training, Minitab was used, and Equations (4) and (5) were obtained via fitting.

_{gen}and UA, whereas the bead size impact was much weaker. The relative impact of these parameters can be rank-ordered as follows: stirrer speed > bead loading >> bead size. It is well-known that an increase in the stirrer speed and bead loading increases the power consumption in a wet stirred media mill [31,70], which in turn leads to higher Q

_{gen}. Similarly, it is well-known that an increase in stirrer speed and bead loading also leads to an increase in the internal convective heat transfer coefficient in the mill chamber, which could lead to a higher UA [71,72].

#### 3.4. Comparative Analysis of LPM and EBM Fits and Their Predictions for the Test Runs

_{gen}and UA (see the RMSE in Table 3), and illustrated the fitted profiles in Figure 6. The EBM data were retrieved from Guner et al. [33] for comparison. Figure 6 and Table 3 data suggest that the LPM fitted the experimental temperature profiles slightly better than the EBM. The average ± standard deviation of the RMSEs are 0.50 ± 0.12 °C for the LPM and 0.96 ± 0.43 °C for the EBM. Moreover, LPM has a lower deviation from the experimental data and is more consistent, according to the lower standard deviation.

#### 3.5. The LPM and the EBM Comparison and the Limitations of the LPM

_{m}(refer to [33] for details of the EBM). The EBM is so versatile that it can be used to investigate the impacts of different coolant types and flow rates as well as the material of construction of the mill on the cooling rate and temperature evolution. Despite all these capabilities and higher fidelity to the actual milling process, its use entails more time, effort, and accurate numerical methods for the simulations/parameter estimation. Moreover, one must obtain appropriate data and correlations for the physical–thermal–heat transfer properties. The EBM consists of five ordinary differential equations (ODEs) with the fraction ξ of the power consumption P that is converted into heat being the sole parameter of the EBM. The estimation of ξ entails using a sophisticated optimizer coupled to the ODE solver. Hence, for facile modeling of the temperature profiles as well as effective control and optimization of the WSMM process, development of simpler, low-fidelity models such as the LPM may be warranted.

_{ch}+ Q

_{gen}/UA. Although the EBM has higher fidelity to the real WSMM operation, its predictions could be worse, and some predictions exhibited a maximum and ensuing drop in temperature instead of a monotonic approach to a steady-state in the temperature profiles (Figure 7). In general, the temperature drop from the maximum is within a couple of degrees Celsius, and this error was acceptable. Note that even the EBM has its own assumptions and sources of modeling errors; refer to Guner et al. [33] for a detailed discussion of the modeling errors. For example, in the UA

_{m}calculations, the mixture correlations for the physical properties and the internal/external convective heat transfer coefficients have some errors. The LPM is free of that source of modeling error because UA was used as a fitting parameter; therefore, having two fitting parameters, as opposed to the one fitting parameter of the EBM, enhanced LPM’s fitting and prediction capability.

_{gen}and UA are fitting parameters that are not equal to the true heat generation rate and the product of the overall heat transfer coefficient and the heat transfer surface area. In fact, the actual heat generation rate and even the overall heat transfer coefficient vary with time and temperature [33,34]. However, as established in this study, Q

_{gen}is strongly and positively correlated with the power consumption and is the driver for temperature rise, whereas the UA correlation with the process parameters revealed a similar qualitative dependence of the convective heat transfer coefficients on the process parameters in their correlations (refer to such correlations in [33]).

#### 3.6. A holistic Perspective on the Impact of Process Parameters

_{50}and d

_{90}, whereas an increase in the bead size led to bigger d

_{50}and d

_{90}. As the increase in the stirrer speed and bead loading also leads to higher heat generation rate and temperature rise, existence of an optimal set of process conditions is anticipated. Clearly, the milling conditions that are conducive to higher milling efficiency also cause higher heat generation and temperature rise. This is not surprising as both of them are largely determined by the mechanical power consumption during milling. Guner et al. [34] considered a desirable product specification of d

_{10}< 150 nm, d

_{50}< 200 nm, and d

_{90}< 250 nm and max. temperature below 37 °C. Their holistic consideration of a newly-defined thermal desirability score, power (energy) consumption, and total cycle time suggested that Run 13 or 14 had the optimal set of milling conditions: 3000 rpm with 50% loading of 200 or 400 µm beads. Obviously, in general, the optimal conditions depend on the specific pharmaceutical application of the drug nanosuspension with desired particle size specifications as well as the sensitivity of the drug to temperature and stressing in terms of physical stability and chemical degradation.

## 4. Conclusions and Future Outlook

## Supplementary Materials

_{gen}and UA in the training and test tests; Table S2: Particle size statistics for the milled suspensions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | heat transfer surface area, m^{2} |

A_{m} | heat transfer surface area of the mill chamber, m^{2} |

c | bead loading or fractional volumetric concentration of the beads, – |

C_{p} | specific heat capacity, J/g °C |

EBM | enthalpy balance model |

D_{b} | bead size, µm |

LPM | lumped-parameter model |

m | mass in the mill chamber, g |

ML | machine learning |

PL | power law |

P | power consumption, J/min |

Pξ | heat generation rate during milling, retrieved from EBM study, J/min |

Q_{gen} | apparent heat generation during milling, J/min |

RMSE | root-mean-squared error, °C |

T | temperature at the mill outlet, °C |

T_{ch} | chiller temperature, °C |

T_{rise} | temperature rise at 6 min of milling, °C |

T_{0} | initial temperature at the mill outlet, °C |

t | milling time, min |

U | overall heat transfer coefficient, W/m^{2} °C |

UA | apparent overall heat transfer coefficient times surface area, J/min °C |

UA_{m} | apparent overall heat transfer coefficient times surface area in the mill chamber, J/min °C |

ξ | fraction of mechanical power converted into heat, – |

ω | stirrer (rotational) speed, rpm |

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**Figure 3.**Experimental temperature profiles, direct fits by the lumped parameter model (LPM), and predictions by the LPM coupled with a power law (PL) model and a machine learning (ML) model. (

**Left**)-to-(

**right**): increasing bead size, (

**top**)-to-(

**bottom**): increasing bead loading, stirrer speed: 2000 rpm.

**Figure 4.**Experimental temperature profiles, direct fits by the lumped parameter model (LPM), and predictions by the LPM coupled with a power law (PL) model and a machine learning (ML) model. (

**Left**)-to-(

**right**): increasing bead size, (

**top**)-to-(

**bottom**): increasing bead loading, stirrer speed: 3000 rpm.

**Figure 5.**Experimental temperature profiles, direct fits by the lumped parameter model (LPM), and predictions by the LPM coupled with a power law (PL) model and a machine learning (ML) model. (

**Left**)-to-(

**right**): increasing bead size, (

**top**)-to-(

**bottom**): increasing bead loading, stirrer speed: 4000 rpm.

**Figure 6.**Direct fitting of the experimental temperature profiles via the lumped parameter model (LPM) and the enthalpy balance model (EBM) for the test runs (Runs 28–32). The EBM fits were taken from Guner et al. (2022) [33].

**Figure 7.**Predictions by power law (PL) and machine learning (ML) models coupled with the lumped parameter model (LPM) and the enthalpy balance model (EBM) for the test runs (Runs 28–32). The EBM predictions were taken from Guner et al. (2022) [33].

**Figure 8.**Comparison of the root-mean-squared errors of the direct fits and the predictions made by the power law (PL) and machine learning (ML) models coupled with the lumped parameter model (LPM) and the enthalpy balance model (EBM). The EBM predictions were taken from Guner et al. (2022) [33].

Run No. | Stirrer Speed, ω (rpm) | Bead Loading, c (-) | Bead Size, D_{b} (µm) |
---|---|---|---|

1 ^{1} | 2000 | 0.4 | 200 |

2 ^{1} | 2000 | 0.4 | 400 |

3 ^{1} | 2000 | 0.4 | 800 |

4 ^{1} | 2000 | 0.5 | 200 |

5 ^{1} | 2000 | 0.5 | 400 |

6 ^{1} | 2000 | 0.5 | 800 |

7 ^{1} | 2000 | 0.6 | 200 |

8 ^{1} | 2000 | 0.6 | 400 |

9 ^{1} | 2000 | 0.6 | 800 |

10 ^{1} | 3000 | 0.4 | 200 |

11 ^{1} | 3000 | 0.4 | 400 |

12 ^{1} | 3000 | 0.4 | 800 |

13 ^{1} | 3000 | 0.5 | 200 |

14 ^{1} | 3000 | 0.5 | 400 |

15 ^{1} | 3000 | 0.5 | 800 |

16 ^{1} | 3000 | 0.6 | 200 |

17 ^{1} | 3000 | 0.6 | 400 |

18 ^{1} | 3000 | 0.6 | 800 |

19 ^{1} | 4000 | 0.4 | 200 |

20 ^{1} | 4000 | 0.4 | 400 |

21 ^{1} | 4000 | 0.4 | 800 |

22 ^{1} | 4000 | 0.5 | 200 |

23 ^{1} | 4000 | 0.5 | 400 |

24 ^{1} | 4000 | 0.5 | 800 |

25 ^{1} | 4000 | 0.6 | 200 |

26 ^{1} | 4000 | 0.6 | 400 |

27 ^{1} | 4000 | 0.6 | 800 |

28 ^{2} | 2500 | 0.45 | 400 |

29 ^{2} | 2500 | 0.55 | 400 |

30 ^{2} | 3500 | 0.45 | 400 |

31 ^{2} | 3500 | 0.55 | 400 |

32 ^{2} | 4000 | 0.35 | 100 |

^{1}Runs that were used in the training set,

^{2}Runs that were used in the test set.

Run No: Identifier | Q_{gen} (J/min) | UA (J/min °C) | RMSE (°C) |
---|---|---|---|

1: 2000 0.4 200 | 755.2 | 47.79 | 0.40 |

2: 2000 0.4 400 | 1616 | 103.5 | 0.46 |

3: 2000 0.4 800 | 787.4 | 48.08 | 0.56 |

4: 2000 0.5 200 | 441.4 | 25.82 | 0.32 |

5: 2000 0.5 400 | 720.8 | 40.18 | 0.33 |

6: 2000 0.5 800 | 1296 | 72.88 | 0.39 |

7: 2000 0.6 200 | 1343 | 68.63 | 0.44 |

8: 2000 0.6 400 | 2625 | 131.9 | 0.15 |

9: 2000 0.6 800 | 1822 | 89.95 | 0.50 |

10: 3000 0.4 200 | 1402 | 53.92 | 0.55 |

11: 3000 0.4 400 | 2938 | 107.0 | 0.57 |

12: 3000 0.4 800 | 2798 | 101.0 | 0.66 |

13: 3000 0.5 200 | 1837 | 61.49 | 0.40 |

14: 3000 0.5 400 | 3243 | 107.3 | 0.72 |

15: 3000 0.5 800 | 3307 | 107.4 | 0.83 |

16: 3000 0.6 200 | 2624 | 94.08 | 0.43 |

17: 3000 0.6 400 | 4598 | 128.2 | 0.76 |

18: 3000 0.6 800 | 4542 | 124.4 | 0.80 |

19: 4000 0.4 200 | 5075 | 135.1 | 0.42 |

20: 4000 0.4 400 | 6266 | 162.7 | 0.41 |

21: 4000 0.4 800 | 6245 | 162.5 | 0.51 |

22: 4000 0.5 200 | 6116 | 162.7 | 0.90 |

23: 4000 0.5 400 | 8490 | 220.0 | 0.30 |

24: 4000 0.5 800 | 9359 | 238.1 | 0.60 |

25: 4000 0.6 200 | 8383 | 208.9 | 0.62 |

26: 4000 0.6 400 | 10,600 | 261.7 | 0.28 |

27: 4000 0.6 800 | 10,740 | 171.9 | 0.34 |

**Table 3.**Parameters of the LPM estimated by direct fitting as well as predicted using the PL and ML models coupled to the LPM along with the associated statistics.

Runs | Direct Fitting | PL Prediction | ML Prediction | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Q_{gen} (J/min) | UA (J/min °C) | LPM RMSE (°C) | EBM RMSE (°C) ^{1} | Q_{gen} (J/min) | UA (J/min °C) | LPM RMSE (°C) | EBM RMSE (°C) ^{1} | Q_{gen} (J/min) | UA (J/min °C) | LPM RMSE (°C) | EBM RMSE (°C) ^{1} | |

2500 0.45 400 | 1481 | 68 | 0.35 | 0.51 | 1634 | 78 | 0.74 | 0.93 | 1792 | 77 | 1.51 | 1.02 |

2500 0.55 400 | 2495 | 101 | 0.42 | 0.59 | 2118 | 91 | 1.49 | 1.02 | 2506 | 95 | 1.44 | 1.00 |

3500 0.45 400 | 4084 | 118 | 0.50 | 1.56 | 4519 | 137 | 1.56 | 2.22 | 4555 | 132 | 0.59 | 2.17 |

2500 0.55 400 | 5567 | 147 | 0.56 | 1.01 | 5856 | 160 | 1.02 | 1.17 | 5911 | 162 | 1.19 | 1.29 |

4200 0.35 100 | 3185 | 84 | 0.66 | 1.13 | 4180 | 129 | 4.13 | 4.17 | 4359 | 124 | 2.08 | 1.63 |

^{1}RMSE data were taken from Guner et al. 2022 [33] for comparison with the LPM.

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## Share and Cite

**MDPI and ACS Style**

Guner, G.; Yilmaz, D.; Yao, H.F.; Clancy, D.J.; Bilgili, E. Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model. *Pharmaceutics* **2022**, *14*, 2840.
https://doi.org/10.3390/pharmaceutics14122840

**AMA Style**

Guner G, Yilmaz D, Yao HF, Clancy DJ, Bilgili E. Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model. *Pharmaceutics*. 2022; 14(12):2840.
https://doi.org/10.3390/pharmaceutics14122840

**Chicago/Turabian Style**

Guner, Gulenay, Dogacan Yilmaz, Helen F. Yao, Donald J. Clancy, and Ecevit Bilgili. 2022. "Predicting the Temperature Evolution during Nanomilling of Drug Suspensions via a Semi-Theoretical Lumped-Parameter Model" *Pharmaceutics* 14, no. 12: 2840.
https://doi.org/10.3390/pharmaceutics14122840