# Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems

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## Abstract

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## 1. Introduction

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- diffusion phenomena, when the support is represented by molecules;
- -
- convective transport by currents in fluids, described by fluid mechanics;
- -
- radiative transport of elementary particles.

## 2. Mathematical Methods for Solving Transfer Equations in Initial and Boundary Conditions Imposed by Particular Systems

#### 2.1. Diffusion Equation

#### 2.2. Initial and Boundary Conditions

- Different combinations of phenomenological conditions can lead to the same initial and boundary conditions and, consequently, to the same mathematical solutions. It frequently happens that experimentally determined release kinetics to fit a theoretical law are deduced in completely different phenomenological conditions.
- Derivation of solutions essentially implies the initial and boundary conditions, such that the in-depth analysis of phenomena and prediction possibilities are best achieved in connection with understanding of the mathematical aspects.

#### 2.3. Release in an Infinite Medium from an Interface where Concentration Is Kept Constant: Laplace Transform Method

_{s}for the constant concentration at the interface. This suggests the case of “saturation concentration” (Figure 1), which helps us concretize mathematical phenomenon; however, the mechanisms for keeping a relative constant concentration at an interface are surely diverse and multiple.

#### 2.4. Transfer at Liquid/Liquid Interfaces: Release from Microemulsions

#### 2.4.1. Stationary State Models

_{pd}is the drug partition coefficient between membrane and microemulsion aqueous phase, and k

_{pr}is the membrane release medium partition coefficient.

#### 2.4.2. Compartmental Models

#### 2.5. Diffusion in Membranes: Method of Separation of Variables

#### 2.5.1. Diffusion in a Domain Bordered by Two Interfaces where Concentration Is Kept Constant

#### 2.5.2. Diffusion in a Domain Bordered by Two Interfaces of Constant but Different Concentrations

#### 2.6. Diffusion Equation in Spherical and Cylindrical Coordinates

#### 2.6.1. Solutions of Diffusion Equation in Spherical Coordinates

#### 2.6.2. Release from a Non-Degrading Polymer

#### 2.6.3. Release from Lipid Dosage Forms

- The beads do not significantly swell or erode during drug release.
- The beads are spherical in shape.
- The drug is initially homogeneously distributed within the spheres.
- Perfect sink conditions are provided throughout the experiments.
- Mass transfer resistance due to liquid unstirred boundary layers at the surface of the spheres is negligible compared to mass transfer resistance due to diffusion within the systems.
- Drug dissolution is rapid and complete upon exposure to the release medium.
- Diffusion with time- and position-independent diffusion coefficients is the release rate-limiting mass transfer step.

_{0}< C

_{s}and maintained in a solution of constant concentration ${C}_{1}$.

#### 2.6.4. Release from Lipid Implants with Cylindrical Geometry

- The implants do not significantly swell or erode during drug release.
- The implants are cylindrical in shape.
- Diffusional mass transport occurs in radial and axial direction, with the same diffusivities.
- The drug is initially homogeneously distributed within the implants.
- Perfect sink conditions are provided throughout the experiments.
- Mass transfer resistance due to liquid unstirred boundary layers at the surface of the implants is negligible compared to mass transfer resistance due to diffusion within the systems.
- Drug dissolution is rapid and complete upon exposure to the release medium.
- Diffusion with time- and position-independent diffusion coefficients is the release rate-limiting mass transfer step.

#### 2.7. Release Controlled by Transfer across Membranes, Considered as Coupled Interfaces: Release from Liposomes

## 3. Mechanistic and Empirical Models in Systems with Moving Boundaries

#### 3.1. Matrix Systems

#### 3.1.1. Stefan’s Problem

#### 3.1.2. Steady-State Higuchi’s Moving Boundary Model

_{s}at the interface with the core which was not attained by solvent, to concentration zero (sink conditions) at the matrix–dissolution medium interface (x = h) (Figure 7).

#### 3.1.3. Release from a Spherical Matrix

#### 3.1.4. Boundary Layer Effect

_{s}, obtained the following equation:

_{a}is the diffusion coefficient in water.

^{2}, and the approximate solution results in

#### 3.2. Swellable Polymers

#### 3.2.1. Intrusion of Water into Matrix

_{t}and M

_{∞}represent the absolute cumulative amounts of drug released at time t and infinite time, respectively; ${q}_{n}$ are the roots of the Bessel function of the first kind of zero order, and R and H denote the radius and height of the cylinder. The release strongly depended on the wettability of the material [50].

#### 3.2.2. Swelling Component of Release from Polymers

#### 3.3. Erodible Polymers

#### 3.3.1. Kinetics of Release from a Sheet of Thickness 2l

#### 3.3.2. Kinetics of Release from a Sphere of Initial Radius ${r}_{0}$

#### 3.3.3. Kinetics of Release from a Cylinder of Radius ${r}_{0}$ and Height 2${h}_{0}$

#### 3.3.4. Empirical Surface Erosion Models

_{0}is the initial drug concentration in the system, and k

_{0}is the equilibrium rate constant.

^{3}and area is proportional to a

^{2}. El-Arini and Leuenberger [97] modified the Hopfenberg model by accounting for the lag time (${t}_{lag}$) before the release process to start.

#### 3.3.5. Mechanistic Surface Erosion Models

_{e}are the drug concentration and effective diffusivity in liquid-filled pores, respectively; k is the drug dissolution rate constant, ε is the porosity of the polymer matrix, and $\epsilon {C}_{s}$ is the saturation concentration in the solution filling the pores.

#### 3.4. Complex, Multiparameter Release Models

#### 3.4.1. Concomitant Depolymerization, Erosion, and Diffusion

#### 3.4.2. Monte Carlo Simulation Models

#### 3.4.3. Artificial Neural Network Models

## 4. Release Models Based on Fick’s First Law

#### 4.1. Noyes–Whitney Model

#### 4.2. “Empirical” Extensions

#### 4.3. Applications of “Empirical” Models in Describing Release from Micro- and Nanostructured Carriers

#### 4.3.1. Micro-Sized Polymeric Carriers

#### 4.3.2. Nano-Sized Polymeric Carriers

- -
- the release models developed for transfer across plane surfaces are no longer applicable;
- -
- their curvature implies specific properties, primarily high free energy and aggregation tendency;
- -
- continuum models lack the ability to describe the kinetics of drug release as the concentration of the drug in the nanosystems fluctuates and the notion of concentration profile becomes meaningless.

#### 4.3.3. Liquid Crystals

#### 4.3.4. Liposomes

^{®}90H, and citric acid in phosphate-buffered saline (pH 7.4, PBS) using a dialysis technique was best described by the Weibull model [147].

#### 4.3.5. Solid Lipid Nanoparticles and Lipid Dosage Forms

^{0.5}” versus the square root of time for in vitro release of interferon a (IFNa) from lipid cylindrical matrices based on tetraglycerol tripalmitate (squares), tetraglycerol monopalmitate (filled triangles), tetraglycerol dipalmitate, tetraglyerol distearate, or tetraglyerol monostearate led to a linear dependence [201].

#### 4.4. Selection of the Mathematical Release Model

^{2}(0.9924 and 0.9972).

^{2}≥ 0.9724).

## 5. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

#### Diffusion in a Domain Bordered by Two Interfaces where Concentration is Kept Constant

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**Figure 1.**Spatial distribution of the active substance at different (t

_{n1}, …, t

_{nk}) time points.

**Figure 2.**Release kinetics of a drug from microemulsions in an experiment using a dialysis membrane.

**Figure 3.**Spatial distribution of the concentrations of active substance at different time intervals: (

**a**) case ${c}_{1}<{c}_{0}$, transfer into the membrane; (

**b**) case ${c}_{1}>{c}_{0}$, transfer out of the membrane.

**Figure 4.**Distribution of concentration in a membrane separating two domains with constant concentrations.

**Figure 5.**Radial transfer across a hollow sphere in a release medium where the concentration of drug has a constant value ${c}_{1}$.

**Figure 8.**Higuchi’s model for release from a spherical tablet of radius R, in the condition of a moving solvent front.

**Figure 9.**Swelling of a spherical polymer particle following the intrusion of solvent across the outer surface.

**Figure 11.**Black-box model of transfer (weighting) function, defined in the space of image functions obtained after the application of an integral transformation.

**Table 1.**Examples of the application of empirical models in describing release kinetics from micro-sized polymeric carriers.

Drug | Supramolecular System | Main Excipients | Release Experiment | Empirical Model | Reference |
---|---|---|---|---|---|

Cefpodoxime proxetil | Micro-balloons (hollow microspheres) | Hydroxypropylmethyl cellulose (HPMC) ethyl cellulose (EC) | Method (M): United States Pharmacopoeia (USP) paddle apparatus Dissolution medium (DM): 0.1 N HCl (pH 1.2) | Higher values of correlation coefficients were obtained in the case of Higuchi’s square root of time kinetic treatment; diffusion was the predominant mechanism of drug release. | [156] |

Nimodipine Coumarin | Microparticles | PLGA | DM: 50/50 (w/w) mixture of phosphate-buffered saline (PBS), pH 7.4 and ethanol | Higuchi model | [157] |

Ethinyl estradiol (EE) Drospirenone (DRSP) | Microparticles | PLGA | M: dialysis sac method DM: USP phosphate buffer pH 7.4 + 8% 2-Hydroxypropyl--β-cyclodextrin | EE release from PLGA microparticles was faster than DRSP release; EE release is assumed to be primarily controlled by drug diffusion. | [158] |

Sodium fluorescein (hydrophilic compound) | Spray-dried microparticle | Poly(glycerol adipate-co-ω-pentadecalactone), l-arginine, l-leucine | DM: PBS, pH 7.4 (n = 3) | Higuchi model | [159] |

Levonorgestrel | Microparticles | PLGA; Methocel Polyvinyl alcohol | DM: 0.9% sodium chloride + 0.5% sodium dodecyl sulfate | Release kinetics followed predominantly a zero-order release profile. | [160] |

Anastrozole | Microparticles | PLGA | M: modified dialysis method DM: 0.1 N HCl (pH 1.2) and phosphate buffer (pH 7.4). | An initial burst release phase was followed by a gradual release phase with good correlation coefficients for the Higuchi model. | [161] |

Centchroman | Microparticles | Glutaraldehyde Glyoxal | NA | A burst release of 29% centchroman within an initial period of 40 h was seen, and the remaining 70% was released in the next 60 h following zero-order release kinetics. | [162] |

5-fluorouracil (5-FU) | Microspheres | Bovine serum albumin Galactosylated chitosan (coating) | M: dynamic dialysis DM: phosphate buffered saline (pH 7.4, PBS) | Attenuated burst release in comparison with uncoated microspheres. Release followed Higuchi’s square root model. | [163] |

Methotrexate (MTX) 5-fluorouracil (5-FU) | Microspheres | Chitosan | DM: PBS, pH 7.4 | Biphasic release (more prominent for MTX microspheres). 5-FU release followed Higuchi’s model, whereas MTX was released more slowly with a combination of first-order kinetics and Higuchi’s square-root model | [164] |

Vitamin B_{12} | Microparticles | Bovine serum albumin (BSA) | M: dialysis technique DM: pH 2, pH 6 and pH 10 buffers | First stage: power law and Weibull equations. The second stage: super case II transport mechanism, as a result of diffusion, relaxation, and erosion. Application of Hixson–Crowell model confirmed the erosion mechanism. | [165] |

Aspirin | Microcapsules | Ethyl cellulose, Cellulose Acetate Phthalate | M: USP apparatus 2 DM: pH-1.2 for 2 h followed by acetate buffer at pH 6.0 for 7 h | The best fit was the Higuchi model, indicating diffusion-controlled release. The n in Korsemeyer–Peppas model varied between 0.5 and 0.7, suggesting a diffusion-controlled release. | [166] |

**Table 2.**Examples of application of empirical models in describing release kinetics from nano-sized polymeric carriers.

Drug | Supramolecular System | Main Excipients | Release Experiment | Empirical Model | Reference |
---|---|---|---|---|---|

Docetaxel | Nanoparticles | Chitosan | Method (M): dialysis sac method Dissolution medium (DM): PBS pH 7.4 | Higuchi’s square-root and Korsmeyer–Peppas; 0.45 ≤ n ≤ 0.89 indicates a combination of both diffusion of drug through the polymer and dissolution of the polymer. | [167] |

Ofloxacin | Nanoparticles | Carboxymethyl gum kondagogu; Chitosan | M: dialysis sac method DM: phosphate buffer solution pH 7.4 | Higuchi model; ‘n’ exponent of Peppas equation (n < 0.43) suggested diffusion-controlled mechanism. | [168] |

Aceclofenac | Nanoparticles | Eudragit RL 100- | M: dialysis sac method DM: Sorenson’s phosphate buffer | Higuchi model (0.43 < n < 0.85) | [169] |

Ellagic Acid | Biodegradable nanoparticles | PLGA polycaprolactone (PCL) | M: dialysis technique DM: phosphate buffer pH 7.4 | An initial burst release was followed by Higuchi’s square-root pattern in the case of PLGA and PCL nanoparticles. | [170] |

Estradiol | Nanoparticles | PLGA | M: dialysis technique DM: phosphate buffer pH 7.4 | Zero order for low-molecular-weight nanoparticles; it was considered that degradation plays a dominant role and controls the release rate. High-molecular-weight nanoparticles showed the best fit into the Higuchi’s model. | [171] |

Doxorubicin | Nanoparticles | Gelatin cross-linked with genipin Fe _{3}O_{4} | DM: PBS pH 7.4 | A correlation between the quantity of released drug and swelling of the nanoparticles was established using a power-law model. | [172] |

Chloroquine phosphate | Nanoparticles | Gelatin | DM: PBS pH 7.4 and distilled water | Fick’s power law allowed establishing a correlation between the quantity of released drug and swelling of the nanoparticles. | [173] |

Indomethacin | Nanocapsules | Pluronic F127 Polylactide (PLA) Labrafac CC | M: dialysis technique DM: PBS pH 7.4 | The release pattern was found to follow a power-law model, with n values ranging between 0.35 and 1.03 (depending on the preparation method). | [174] |

Tigecycline | Nanoparticles | Calcium phosphate (CP) PLGA | DM: physiological solution at 37 °C under static conditions | The tigecycline content was released within a 35-day period. The in vitro data were best fitted with the Weibull model, and the release was defined as non-Fickian transport. | [141] |

Moxifloxacin | Nanosuspensions | PLGA | M: USP apparatus 1 DM: simulated tear fluid (pH 7.4) | All formulations followed Korsemeyer–Peppas release kinetics with n values between 0.45 and 0.89 (anomalous behavior). | [175] |

**Table 3.**Examples of experiments concerning release from liquid crystals, described by empirical models.

Drug | Supramolecular System | Main Excipients | Release Experiment | Empirical Model | Reference |
---|---|---|---|---|---|

Alpha lipoic acid (ALA) | Cubosomes loaded gel | Glycerol monooleate (GMO) Poloxamer P407 | M: USP Apparatus 5, paddle over disk assembly DM: hydro-alcoholic solution (1:1), 700 mL | Higuchi model ALA release from cubosomes in gels was shown to be primarily controlled by diffusion through the matrix. | [192] |

Doxorubicin | Bicontinuous lipidic cubic phases (LCPs) | GMO Phytantriol (PT) | DM: pH 7.4 and pH 5.8 buffer | Higuchi model was n > 0.5 in all cases, indicating non-Fickian anomalous transport in which both diffusion and matrix effects. | [193] |

Capsaicin | Cubic phase gels | GMO: propylene glycol (1,2-propanediol, PG): water | DM: isotonic phosphate buffered solution (PBS) | Release kinetics were determined to fit Higuchi’s square-root equation indicating that the release was under diffusion control. The calculated diffusion exponent showed the release from cubic phase gels was anomalous transport (n = 0.57–0.60) | [194] |

Salicylic acid | Cubic phase gels | GMO Myverol 18–99 ^{®} distilled monoglycerides | M: USP app I DM: Isotonic phosphate buffer | Release mechanism could be fitted to both Higuchi and first-order models. | [195] |

2-pyrrolidone (model) | In situ cubic phase forming monoglyceride drug delivery systems | Monoglyceride (GMO or glycerol monolinoleate) Cosolvents (ethanol, PEG 300, 2-pyrrolidone, DMSO) | DM: 0.1 M phosphate buffer, pH 7.4, with 0.1% sodium azide as preservative | The release of oligonucleotide from the fully swollen cubic phase matrix followed a diffusion-controlled release mechanism square-root Higuchi model in 24-h intervals for all formulations. | [196] |

Carbamazepine | Nanoemulsion | Castor oil; Lipophilic emulsifier (lecithin or polyoxyl 35 castor oil); Tween 80 | M: dialysis technique DM: phosphate buffer pH 7.4 | Higuchi model best characterized the release profiles for the nanoemulsions and for the free drug, and drug release was described as a diffusion process based on Fick’s law. | [197] |

l-glutathione | Microemulsions Liquid crystal systems | - | NA | Higuchi model | [198] |

**Table 4.**Examples of experiments concerning release from solid lipid nanoparticles and lipid dosage forms, described by empirical models.

Drug | Supramolecular System | Main Excipients | Release Experiment | Empirical Model | Reference |
---|---|---|---|---|---|

Etofenamate | Solid Lipid Nanoparticles (SLN) | Compritol 888 ATO Precirol ATO 5 | NA | Higuchi model for Compritol 888 ATO SLNs; Zero-order release for Precirol ATO 5 SLNs | [202] |

Curcuminoids | SLN | Poloxamer 188 Dioctyl sodium sulfosuccinate Stearic acid Glyceryl monostearate | M; vertical Franz diffusion cells DM: 50% (v/v) ethanol | 25% burst release of the curcuminoids within 10 min followed by controlled release pattern following Higuchi’s square-root model for 12 h | [203] |

Bixin | SLN | Trimyristin Glycerol monostearate | M: diffusion using Franz diffusion cells Receptor medium: Sorensen buffer pH 7.7 | The release was first-order diffusion-controlled. The n-values obtained from the Korsmeyer–Peppas model (n = 0.697) indicated the release mechanism was non-Fickian type. | [204] |

Gatifloxacin | SLN | Stearic acid (SA)/ Compritol/Gelucire Poloxamer-188 Sodium taurocholate | M: Automated transdermal diffusion cells Receptor medium: phosphate buffer (pH 7.4) | The release pattern was found to follow Korsmeyer–Peppas model (n = 0.15). | [205] |

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**MDPI and ACS Style**

Mircioiu, C.; Voicu, V.; Anuta, V.; Tudose, A.; Celia, C.; Paolino, D.; Fresta, M.; Sandulovici, R.; Mircioiu, I.
Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems. *Pharmaceutics* **2019**, *11*, 140.
https://doi.org/10.3390/pharmaceutics11030140

**AMA Style**

Mircioiu C, Voicu V, Anuta V, Tudose A, Celia C, Paolino D, Fresta M, Sandulovici R, Mircioiu I.
Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems. *Pharmaceutics*. 2019; 11(3):140.
https://doi.org/10.3390/pharmaceutics11030140

**Chicago/Turabian Style**

Mircioiu, Constantin, Victor Voicu, Valentina Anuta, Andra Tudose, Christian Celia, Donatella Paolino, Massimo Fresta, Roxana Sandulovici, and Ion Mircioiu.
2019. "Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems" *Pharmaceutics* 11, no. 3: 140.
https://doi.org/10.3390/pharmaceutics11030140