# Modeling of Disintegration and Dissolution Behavior of Mefenamic Acid Formulation Using Numeric Solution of Noyes-Whitney Equation with Cellular Automata on Microtomographic and Algorithmically Generated Surfaces

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Methods

#### 2.2.1. Preparation of Tablets

#### 2.2.2. Determination of Tablet Porosity and Tensile Strength

_{tablet}and ρ

_{i}are the true densities (g/cm

^{3}) of the tablet and each raw material in the tablet, respectively, and X

_{i}is the weight fraction of each component.

_{t}(MPa), according to Equation (3).

#### 2.2.3. Measurement of Granule Size Distribution

#### 2.2.4. X-Ray Microtomography

#### 2.2.5. Disintegration Test

#### 2.2.6. Dissolution Test

#### 2.3. Simulation of Drug Release with Cellular Automata

#### 2.3.1. Application of Noyes-Whitney Equation in Numeric Calculation of Drug Dissolution

_{m}/d

_{t}from a voxel element representing a solid drug particle surrounded by solvent voxels under sink conditions can be mathematically described according to Equation (4).

_{m}/d

_{t}= (A × D)/λ × (Cs − C), C→0 (sink condition),

^{2}) is $\mathrm{A}=\left(1/N\right)\pi {\lambda}^{2}$, N is the number of neighbors (N = 26), diffusion coefficient D (cm

^{2}/s) is according to the Stokes–Einstein relationship [34], Equation (5), Cs is the solubility at equilibrium and at experimental temperature, and C is the concentration of the solid in the bulk of the dissolution medium at time t. The diffusion coefficient was calculated according to Equation (5).

^{−5}× 10

^{247.8/((T+273.15)−140)}, k

_{b}= 1.3806488 × 10

^{−16}(cm

^{2}·kg·s

^{−2}·K

^{−1}) is the Boltzmann constant, T (°C) is the temperature, and R (Å) is the Stokes radius.

^{2}, and D

_{S}is the coefficient of self-diffusion.

^{−7}cm

^{2}/s, which is in combination with Equation (4), and the mass of a single drug voxel yields a C

_{1}value of 22,082 (Table 2), i.e., C

_{1}is a voxel mass at time 0 divided by the rate of the mass transfer from 1/26th of the voxel surface. This constant is used for convenience during simulation, and is just a simulation software-compatible way of describing dissolution kinetics.

- As soon as a disintegrant cell is signaled to get in contact with a medium-type voxel, its state is converted to “active”.
- All “active” disintegrant cells mark their direct neighbors for random scattering within the calculation matrix. The labeling depth, i.e., radius around the active disintegrant particles, can be set through the simulations parameter (C
_{2}). - All marked cells are randomly distributed in the surrounding medium to maximize the contact surface to the liquid.
- As soon as the disintegrant cell is “activated”, it loses its action; therefore, the random scattering of its neighborhood can be fired only once.

^{3}elements, including solid and dissolution medium voxel types. This calculation matrix size was kept for both types of simulation matrices, that obtained from microtomography and those algorithmically created.

- Rule: If a cell has three positive neighbors, then, on the next epoch, this cell becomes positive.
- Rule: If a cell has two positive neighbors, then, on the next epoch, this cell remains unchanged.

#### 2.3.2. Matrix Arrangement of Tablets

^{3}elements), equal to the microtomographic resolution with a voxel side length of 6.5 μm.

#### 2.3.3. Comparison of Drug Release Pattern between Experimental and Simulated Profiles

_{2}) between simulated and experimental release profiles, Equation (7) was used [39].

_{t}is the dissolution rate of the experimental tablet at time t, and T

_{t}is the dissolution rate of the simulated tablet at time t. A similarity factor (f

_{2}) greater than 50 indicates a close correlation between simulated and experimental data.

## 3. Results

#### 3.1. In Vitro Evaluation of Drug Release

_{2}) of the drug release among formulations are summarized in Table 5. As can be seen from Figure 3 and Table 5, the similarity factors (f

_{2}) decreased with increased porosity (i.e., formulations A1 to A2 and A3), suggesting that the release rate is influenced by the tablet porosity; however, it is not applicable for all formulations. The uncompressed granules had a very distinctive release profile, quite different from the tablets.

#### 3.2. Granule Size Distribution Experimentally Measured and Designed in Simulation Matrices

#### 3.3. Comparison between In Vitro and In Silico Drug Release Profiles

_{2}) for the dissolution between the X-ray reconstructed tablet and the experimental tablet were 54 and 72 for formulations A2 and A3, respectively. Also, the obtained similarity factors (f

_{2}) for the dissolution between the algorithmically created matrices and the experimental tablet were 68 and 73 for A2 and A3, respectively. As demonstrated by the similarity factors (f

_{2}), the dissolution profiles from the X-ray reconstructed tablets and algorithmically created matrices were like those obtained from the experimental tablets. It is important to keep in mind that those close correlations are not the results of the fitting, but of ab initio calculations. These results suggest that the simulations of disintegration and dissolution behavior with the calculation matrices obtained from X-ray microtomography are in good agreement with the experimental tablets.

#### 3.4. In Silico Evaluation of Drug Release

_{2}) are summarized in Table 6. The similarity factors (f

_{2}) between in vitro and in silico release were 67, 68, 73, and 71 for formulations A1–A4, respectively, suggesting that the algorithmically created matrices provided a similar drug release to that obtained from the experimental tablets. The simulation dissolution rate calculated with Equation (4) was set to 1.39 × 10

^{−14}g/s for a single contact surface of 9.75 × 10

^{−8}cm

^{2}under an assumption of unstirred layer thickness equal to 2.6 mm. The necessity to use such a large value for an unstirred diffusion layer thickness is dictated by the experimental data [22] and a tendency to produce a cone of powder at the bottom of the dissolution vessel, where mass migration processes are solely diffusion-driven. Unlike the suggested paddle rotation speed [22], the dissolution test was carried out at 75 rpm to reduce the cone effect impact on the release rate [40]. Despite this change, the cone was still clearly observable, and a further increase in the rotation rate would introduce a significant deviation from the literature reference profiles. The similarity of the simulated and the experimental results can be seen for calculation with the disintegration model, whereas the release profiles simulated without the disintegration model resulted in very slow release kinetics (shown in Figure S1). Therefore, this result suggested that the in silico simulation with the disintegration model produced a similar release profile to the in vitro evaluation.

## 4. Discussion

_{2}) when compared to the simulation without disintegration, including the calculation results from the reconstructed tablets. This suggests that the proposed in silico disintegration procedure is in a good agreement with the experiment. However, the existing deviation between the in vitro and in silico drug release still suggests that there are more subtle mechanisms that contribute to the studied processes, for example, changes in granule particle size during compaction, and percolation effects or wicking between the disintegrant fibers. When comparing the in vitro drug release of formulations A3 and A4 and uncompacted granules as shown in Figure 3, the formulations compacted at higher compressive stresses (i.e., 99 MPa, formulation A3), showed a faster drug release than formulation A4 compacted at lower compressive stress (i.e., 45 MPa). Similar behavior is reported for the uncompressed granules. For this reason, it can be thought that the granules were damaged during the compaction process, and the produced fine fractions resulted in faster drug release profiles. Compressive stress seems to play an important role in maintaining granulometric composition within a tablet, which is well supported in the literature [46,47,48]. However, the damage to granular structures after compressive stress application cannot be seen on the microtomographic acquisitions from these tablets. By contrast, the unchanged granular patterns can be seen in the consecutive cross-sectional images (see horizontal and vertical cross-sectional images shown in Figure 2). Therefore, further investigations to consider the influence of the compression dwelling time, the material mechanical properties, and the behavior of granular breakage may be necessary for a better understanding of the effect of granular partitioning on the disintegration and release rates.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The molecular dynamics set-up is shown for simulating the diffusion process of a single molecule of the mefenamic acid in aqueous media; (

**b**) the root-mean-squared deviations of the target molecule within 7 ns of simulation time. The slope is the first derivative by time and was used to estimate the diffusion coefficient.

**Figure 2.**(

**a**) The horizontal cross-sectional image of formulation A2 analyzed by microtomography. (

**b**) The vertical cross-sectional image of formulation A2 analyzed by microtomography. (

**c**) The results of the volume rendering from microtomographic reconstruction for formulation A2, a diameter of 2 mm (red voxel color corresponds to mefenamic acid), and (

**d**) algorithmically created component arrangement, a diameter of 2 mm (blue voxels correspond to virtual mefenamic acid particles). (

**e**) The skeletonized drawing of the particle distribution (only drug component) is shown after 10 s of simulated dissolution.

**Figure 5.**Comparison between in silico and in vitro release profiles obtained from simulations with algorithmically created tablet component arrangements and the reconstructed matrices with the help of microtomography for formulations A3 (

**left**) and A2 (

**right**).

**Figure 6.**Comparison between simulated release curves obtained from algorithmically created components’ arrangements and corresponding experimental data for formulations A1–A4 (

**a**–

**d**).

Formulation Composition | True Density (g/cm^{3}) | Formulation | |||
---|---|---|---|---|---|

mg | %, w/w | ||||

Granular composition | |||||

Mefenamic acid | 1.2554 | 250.0 | 50.0 | ||

d-mannitol | 1.4888 | 165.0 | 33.0 | ||

Microcrystalline cellulose | 1.5701 | 50.0 | 10.0 | ||

Croscarmellose sodium | 1.5757 | 10.0 | 2.0 | ||

Hydroxypropyl cellulose | 1.2334 | 15.0 | 3.0 | ||

Granulate | - | 490.0 | 98.0 | ||

External phase composition | |||||

Croscarmellose sodium | 1.5757 | 5.0 | 1.0 | ||

Magnesium stearate | 1.0539 | 5.0 | 1.0 | ||

Tablet weight | - | 500.0 | - | ||

Tablet Parameters (N = 9) | - | A1 * | A2 | A3 | A4 |

Tablet porosity (%, v/v) | - | 6 | 9 | 14 | 23 |

Compressive stress (MPa) | - | 210 | 150 | 99 | 45 |

Component | True Density (g/cm ^{3}) | Type Identifier | Component Code | C_{1} Constant * | C_{2} Constant |
---|---|---|---|---|---|

Mefenamic acid | 1.2554 | 1 | API | 22,082 | Not used in simulation algorithm for types 1–9 |

d-Mannitol | 1.4888 | 10 | Non swelling, soluble filler | 200 | Not used in simulation algorithm for types 10–19 |

Microcrystalline cellulose | 1.5701 | 31 | Non-swelling or negligible swelling, insoluble fillers | insoluble | Not used in simulation algorithm for types 30–39 |

Croscarmellose sodium | 1.5757 | 61 | Fibrous disintegrant | insoluble | 2 ** |

Hydroxypropyl cellulose | 1.2334 | 41 | Hydrophilic swelling matrix | 1 × 10^{8} | Swelling of hydrophilic matrix components (types 40–49) was not included into this simulation algorithm |

Magnesium stearate | 1.0539 | 71 | Hydrophobic ingredient | insoluble | Not used in simulation algorithm for types 70–79 |

_{1}reflects the reciprocal dissolution rate of the solid in contact with the simulated dissolution medium (refer to Equations (4)–(6)); ** C

_{2}indicates the range of disintegration (refer to stage 2 of the disintegration simulation algorithm).

Formulation | Resultant Compressive Stress (MPa) | Tensile Strength (MPa) (n = 3) | Disintegration Time (s) (n = 6) | Porosity (%, v/v) |
---|---|---|---|---|

A1 | 210 | 3.31 ± 0.13 | 543 ± 37 | 5.6 |

A2 | 150 | 2.53 ± 0.06 | 311 ± 16 | 9.5 |

A3 | 99 | 1.48 ± 0.04 | 160 ± 4 | 13.7 |

A4 | 45 | 4.72 ± 0.01 | 53 ± 2 | 23.1 |

**Table 4.**Results of statistical analysis of dissolution rates at 10 min, 15 min, and 30 min for the formulations.

Source of Variance | F-Value | p-Value | Tabulated F-Value |
---|---|---|---|

Dissolution rates at 10 min | 31.19322 | 2.18 × 10^{−09} * | 2.75871 |

Dissolution rates at 15 min | 7.89681 | 2.93 × 10^{−04} * | 2.75871 |

Dissolution rates at 30 min | 26.35112 | 1.20 × 10^{−08} * | 2.75871 |

Formulation | A1 | A2 | A3 | A4 | Uncompacted Granules |
---|---|---|---|---|---|

A1 | - | 61 | 45 | 48 | 48 |

A2 | 61 | - | 57 | 60 | 56 |

A3 | 45 | 57 | - | 67 | 55 |

A4 | 48 | 60 | 67 | - | 73 |

Uncompacted granules | 48 | 56 | 55 | 73 | - |

**Table 6.**Summary of similarity factors (f

_{2}) between in vitro and in silico drug release profiles.

Tablet | A1 | A2 | A3 | A4 | |
---|---|---|---|---|---|

Similarity factor (f_{2}) | X-ray reconstructed matrices | NA * | 54 | 72 | NA * |

Algorithmically created matrices | 67 | 68 | 73 | 71 |

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**MDPI and ACS Style**

Yokoyama, R.; Kimura, G.; Schlepütz, C.M.; Huwyler, J.; Puchkov, M.
Modeling of Disintegration and Dissolution Behavior of Mefenamic Acid Formulation Using Numeric Solution of Noyes-Whitney Equation with Cellular Automata on Microtomographic and Algorithmically Generated Surfaces. *Pharmaceutics* **2018**, *10*, 259.
https://doi.org/10.3390/pharmaceutics10040259

**AMA Style**

Yokoyama R, Kimura G, Schlepütz CM, Huwyler J, Puchkov M.
Modeling of Disintegration and Dissolution Behavior of Mefenamic Acid Formulation Using Numeric Solution of Noyes-Whitney Equation with Cellular Automata on Microtomographic and Algorithmically Generated Surfaces. *Pharmaceutics*. 2018; 10(4):259.
https://doi.org/10.3390/pharmaceutics10040259

**Chicago/Turabian Style**

Yokoyama, Reiji, Go Kimura, Christian M. Schlepütz, Jörg Huwyler, and Maxim Puchkov.
2018. "Modeling of Disintegration and Dissolution Behavior of Mefenamic Acid Formulation Using Numeric Solution of Noyes-Whitney Equation with Cellular Automata on Microtomographic and Algorithmically Generated Surfaces" *Pharmaceutics* 10, no. 4: 259.
https://doi.org/10.3390/pharmaceutics10040259