# Pharma 4.0-Artificially Intelligent Digital Twins for Solidified Nanosuspensions

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Process Design

^{®}platform Process Systems Enterprise, gPROMS, (www.psenterprise.com/products/gproms, accessed on 2 August 2022). Before formulation, it is crucial to select the compatible polymer or surfactant to coat the liquid crystals, the one which best enhances the API’s dissolution performance [9]. This pivotal step was included in the process model; the selection was based on the implementation of the computational statistical associating fluid theory criteria (PC-SAFT) to substitute for the experimental trials.

^{−1}), according to Equations (1)–(3) [7].

^{−1}K

^{−1}) is the universal gas constant. In Equation (2) γ (Ν m

^{−1}) is the surface tension, V

_{m}(m

^{3}mol

^{−1}) the API’s molar volume, r (m) the particle’s characteristic size and C (m) a parameter equal to $1.5{\left(\frac{{V}_{m}}{{N}_{A}}\right)}^{\frac{1}{3}}$, while in Equation (3) Δ (m) is the material’s distance between the molecular layers, m

_{stab}(−) the stabilizer’s number of segments per chain based on PC-SAFT theory, and ε (eV) and σ (m) are the depth of pair potential and the segment diameter, respectively. The terms ε

_{stab-API}and σ

_{stab-API}were calculated using the Berthelot-Lorentz combining rules. Stabilizer candidates were chosen, namely Poloxamer-188, Poloxamer-407 and HPC-SL. As far as poloxamers are concerned, since both are copolymers containing ethylene oxide (EO) and propylene oxide (PO) groups in different proportions, the component’s PC-SAFT parameters were calculated using Berthelot-Lorentz rules (see Table 1).

^{®}package of Siemens Process Systems Enterprise (https://www.psenterprise.com/products/gproms/properties, accessed on 2 August 2022). The Saft-γ-Mie equations of state were utilized to calculate the temperature dependence on the density and the heat capacity of the components.

^{−1}) is the breakage rate of a particle of size interval i, E

_{kin}(J) and E

_{fract}(J m

^{−3}) are the kinetic energy of the particles and the fracture energy, respectively, P

_{y}(Pa) is the yield pressure, ρ (kg m

^{−3}) is the particle’s density, V (m

^{3}) is the mill’s chamber volume, H (Pa) is the particle’s hardness, x

_{i}(m) is the particle size i, K

_{1C}(Pa m

^{−1/2}) is the stress intensity factor, W

_{m,kin}the mass specific impact energy (J kg

^{−1}) and b(i,j) (−) is the mass fraction of the product that fell from size interval j to i. In Equation (8), δ is the solubility parameter (Pa

^{1/2}), V

_{i}(m

^{3}) is the particle’s volume, υ (−) is the Poisson’s ratio, Υ (Pa) is the Young’s modulus of elasticity and V

_{0}(m

^{3}) is the unit’s crystal volume. While all the other parameters in Equations (6) to (8) are pre-estimated, the breakage rate parameter c (−) is a tuning parameter, estimated by experimental data. The second breakage function and breakage rate used for the wet milling simulations is proposed by Austin et al. [16] and it is referred to as the Austin model (see Equations (9) and (10)):

^{−1}) are the tuning parameters, while x

_{i}and x

_{j}are the product’s and the post-breakage final particle size accordingly (m), and x

_{crit}is the critical particle size, namely the size after which no breakage occurs. The last breakage function was the one proposed by Kapur et al. [17] and it is referred to as the Kapur model (model Equations (11) and (12)):

^{−1}) are the tuning parameters. Apparently, all three breakage functions and rates encumber tuning and physical parameters respectively, with Austin presenting the highest number of considered tuning parameters and de Vegt the lowest. The parameter estimation was conducted in the Siemens Process Systems Enterprise gPromsFP

^{®}Model Validation platform (https://www.psenterprise.com/products/gproms/modelbuilder, accessed on 2 August 2022), applying the Maximum Likelihood Estimation method. In the spray drying process, the droplets’ hydrodynamic diameter was assumed to obey lognormal distribution and for the drying rate calculation the Oakley’s model was adapted [18]. For each particle size interval i was defined from the particle size distribution and was dispersed in a droplet, which in turn belongs to a size interval z, the local mass and energy balance describing the spray drying model which is described respectively by the Equations (13) and (14).

^{−1}) is the corresponding solids particles flowrate, ${x}_{i,z,j,t}$ (kg kg

^{−1}) is the dry basis moisture content of the liquid specie j, ${\dot{N}}_{i,z,j,t}$ (kg s

^{−2}) is the drying rate time derivative, ${C}_{p,s,i}$ (J kg

^{−1}K

^{−1}) is the solid material’s specific heat capacity and ${C}_{p,j}$ the liquid specie’s corresponding one, ${T}_{i,z}$ (K) is the solid particle’s temperature, h (J m

^{−2}s

^{−1}K

^{−1}) is the heat transfer coefficient, ${A}_{i,z}$ (m

^{2}s

^{−1}) is the shrinking rate of the surface area of the droplet, ${T}_{i,z}$ (K) is the droplet’s temperature, and λ

_{j}(J kg

^{−1}) is the latent heat of vaporization of the liquid specie j. The local vapor phase’s mass balance for the evaporated liquid specie j and the vapor’s phase energy balance is described in Equations (15) and (16), respectively.

^{−1}) are the inlet and outlet mass flowrate of the vapor phase respectively, ${x}_{v,j,in}$ and ${x}_{v,j,out}$ (kg kg

^{−1}) are the inlet and outlet mass fractions of the liquid specie j in the vapor phase respectively, ${H}_{v,in}$ and ${H}_{v,out}$ (J s

^{−1}) are the inlet and outlet enthalpy flowrates of the vapor phase respectively and ${C}_{p,v,j}$ (J kg

^{−1}K

^{−1}) is the specific heat capacity of liquid specie j in the vapor phase, t

_{τ}(s) is the droplets residence time inside the spray dryer’s chamber and T

_{v}(K) is the vapor phase’s temperature. The unhindered drying rate Ν

_{u,z,j}(kg s

^{−1}), i.e., the drying rate of the very same droplets without containing solids described by Equation (17) and is interlinked to the actual drying rate by the relative drying rate f (−) (Equation (18)).

_{wb,j}(K) is the wet bulb temperature of the liquid specie j.

#### 2.2. Experimental Study and Digital Twin Thread Structuring

_{s}, at time t

_{n}, the DAE system transforms into a linear system featuring six equations bearing six unknown variables, each derivative of them being approximated as in Equation (25).

_{l}is the lth polynomial’s level coefficient.

#### 2.3. Integration of Artificial Neural Networks for Parameter Tuning

_{i,j}connecting a layer with n

_{j}neurons with the next layer with n

_{i}neurons, followed the He initialization (Equation (28)).

## 3. Results

#### 3.1. Material Critical Quality Attributes and the Material System’s Interfacial Gibbs Energy Assesment

^{−1}) decreases as well, see Equation (29) [9]. In addition, the zeta-potential bourn within the semi-solid interface that the stabilizer’s addition forms [22], is utilized as indicator of its efficiency (Table 3). A stabilizer causing high absolute values of zeta-potential, creates strong repulsive forces (Coulomb forces) preventing the particles from aggregating.

_{B}(J K

^{−1}) is the Boltzmann constant, Mr (g mol

^{−1}) and ρ (g m

^{−3}) are respectively the molecular weight and the density of the particle, N

_{A}(mol

^{−1}) the Avogadro number and S

_{0}(−) the solubility of the pure API in water in absence of the stabilizer.

#### 3.2. Parameter Fitting

#### 3.3. Sensitivity Analysis

#### 3.3.1. Wet Milling

^{−1}) and d (−) were interrogated, and it was found that by increasing each or both of a and d, the breakage rates of interval sizes i (Equation (10)), and thus the comminution efficiency, increases (Figure 6). The breakage distribution function b(i,j) during the Austin model’s relative sensitivities remained the same, with φ = 0.3, γ = 1.17 and β = 4. In comparison to de Vegt model, a desired D50 size is achieved via various profile paths (Figure 7a). In addition, even when φ = 1 and γ = 1.25, namely when the distribution function b(i,j) is the same with the de Vegt’s one, the existence of the exponent d as a secondary tuning parameter in the S(i) calculation function allows the formation of multiple profiles leading to the same result (Figure 7b). This is the benefit of engaging multiple tuning parameters within the breakage rate function, as they render the digital twin capable of fitting multiple profiles.

#### 3.3.2. Spray Drying

^{−1}). The air water capacity threshold rises proportionally with temperature when the temperature increases the air’s given humidity (g g

^{−1}) covering a lower proportion of the moisture threshold and its relative humidity decreases, allowing humidity absorption [7].

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Digital twin deployment and parameter estimation strategy by steps: (

**a**) real data sampling from the physical object, (

**b**) data multiplication via pattern recognition using artificial intelligence algorithms aiming accuracy enhancement, (

**c**) determination of the proper descriptive physical and empirical laws, and (

**d**) adjustment of tuning parameters to achieve best curve fit.

**Figure 2.**Saft-γ-Mie stabilizer candidates, material properties results (

**a**) Poloxamers’ densities (

**b**) Poloxamers’ heat capacities (

**c**) HPC-SL density (

**d**) HPC-SL heat capacity as a function of temperature.

**Figure 3.**Wet milling dynamic comminution profile of experimental data points, ANN generated and best-fit polynomial curve.

**Figure 4.**Result dynamic profiles for the three breakage functions (

**a**) de Vegt function’s best fit curve generating a 0.89% MSE (

**b**) Kapur function’s best fit curve generating a 0.08% MSE, and (

**c**) the Austin function’s best fit curve generating a 0.02% MSE.

**Figure 6.**D50 time profiles with Austin’s breakage function in various a and d parameter values with γ = 1.17 and φ = 0.3 (

**a**) a = 8 × 10

^{−5}and (

**b**) d = 1.6.

**Figure 7.**Austin’s breakage function for a D50 = 1 μm final size via various time profile paths with (

**a**) γ = 1.17 and φ = 0.3 and (

**b**) with γ = 1.25 and φ = 1 (de Vegt approximation).

**Figure 8.**D50 time profiles with Kapur’s breakage function in various k and A parameter values with e = 6 (

**a**) A = 0.008 and (

**b**) k = 1.2.

**Figure 9.**Kapur’s breakage function for a D50 = 1μm final size via (

**a**) various time profile paths with e = 6 and (

**b**) various profile paths with e = 1.25 (de Vegt approximation).

**Table 1.**PC-SAFT parameters of EO and PO groups for the calculation of the Poloxamers copolymers’ corresponding ones.

Group | m_{seg} (−) | σ_{i} (A) | u_{i}/k (K) | Source |
---|---|---|---|---|

EO | 0.052 ΜW_{total} | 2.89 | 206.74 | [10,11,12,13,14] |

PO | 0.037 MW_{total} | 3.34 | 192.72 |

**Table 2.**Discrete experimental sampling values used as training data for the ANN. These values represent three performed experiments using Itraconazole as API and Poloxamer-188 as stabilizer [19], which proved to be the most suitable after the Gibbs energy analysis.

Time (s) | Experiment 1 D50 (μm) | Experiment 2 D50 (μm) | Experiment 3 D50 (μm) |
---|---|---|---|

360 | 1.39 | 1.31 | 1.37 |

720 | 0.874 | 0.792 | 0.846 |

1440 | 0.784 | 0.742 | 0.783 |

2160 | 0.522 | 0.501 | 0.520 |

2880 | 0.467 | 0.434 | 0.436 |

3600 | 0.305 | 0.297 | 0.301 |

MODEL | PARAMETER | TYPE | VALUE | UNIT |
---|---|---|---|---|

Water quantity | input | 9 | mL | |

API content | input | 0.5 | g | |

Stabilizer content | input | 0.25 | g | |

Mannitol content | input | 1 | g | |

Wet mill | Initial particle size (D50) | input | 1.5 | μm |

Grinding time | input | 1 | h | |

Rotor speed | input | 600 | rpm | |

Rotor diameter | input | 40 | mm | |

Equipment volume | input | 48 | mL | |

D50(t) | output | - | - | |

Air temperature | input | 110 | °C | |

Air flow | input | 800 | L h^{−1} | |

Spray dryer | Air pressure | input | 5 | bar |

Drying chamber volume | input | 5 | L | |

Drying time | input | 1 | h | |

Final product size (D50) | output | 10 | μm |

Stabilizer | ${\mathit{G}}_{\mathit{m}}^{\mathit{i}}$ | Density (kg m^{−3}) | Z-Potential |
---|---|---|---|

HPC-SL | 0.0019 | 1320 | −11.7 |

Poloxamer-407 | 0.0039 | 954 | −13.7 |

Poloxame-188 | 0.0056 | 951 | −17.0 |

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**MDPI and ACS Style**

Davidopoulou, C.; Ouranidis, A. Pharma 4.0-Artificially Intelligent Digital Twins for Solidified Nanosuspensions. *Pharmaceutics* **2022**, *14*, 2113.
https://doi.org/10.3390/pharmaceutics14102113

**AMA Style**

Davidopoulou C, Ouranidis A. Pharma 4.0-Artificially Intelligent Digital Twins for Solidified Nanosuspensions. *Pharmaceutics*. 2022; 14(10):2113.
https://doi.org/10.3390/pharmaceutics14102113

**Chicago/Turabian Style**

Davidopoulou, Christina, and Andreas Ouranidis. 2022. "Pharma 4.0-Artificially Intelligent Digital Twins for Solidified Nanosuspensions" *Pharmaceutics* 14, no. 10: 2113.
https://doi.org/10.3390/pharmaceutics14102113