# Exact Analytical Relations for the Average Release Time in Diffusional Drug Release

## Abstract

**:**

## 1. Introduction

- To directly determine the release time scale during the design of a drug delivery device.
- To obtain the drug diffusion coefficient within the formulation, through an experimental estimate of the average release time ${t}_{av}$ by the measured release profile, given the size of the drug carrier (or the average squared size when there is a distribution of carrier sizes).

## 2. Methods

## 3. Results

#### 3.1. Release from a Sphere of Radius R

#### 3.2. Release from a Slab of Thickness L

#### 3.3. Release from a Cylinder of Height H and Radius R

#### 3.3.1. Very Long Cylinders ($H\gg R$)

#### 3.3.2. Very Short Cylinders ($H\ll R$)

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Fractional release curves ${M}_{t}/{M}_{\infty}$ (solid lines) and their complements $1-{M}_{t}/{M}_{\infty}$ (dashed lines) as functions of the corresponding dimensionless time $\tau $ in each case. Blue color curves represent release from spheres, red color release from slabs or flat discs, and green color release from long cylindrical rods. The respective average release time ${\tau}_{av}$ is shown by the vertical dotted line of the same color and it equals the area under the plot of the corresponding complement fractional release $1-{M}_{t}/{M}_{\infty}$ versus time.

**Figure 2.**Solid circles connected by the continuous black line show the dimensionless average time ${\tau}_{av}$ as a function of the aspect ratio A, obtained through the numerical calculation of the double sum in Equation (22). The horizontal green solid line and the inclined red solid line depict the limiting expressions of Equations (38) and (29), valid when $A\ll 1$ and $A\gg 1$, respectively. The blue dashed line represents the interpolating formula between these two limits, given by Equation (23). The magenta dotted curve corresponds to the difference between the approximation of Equation (23) and the exact result of Equation (22).

**Table 1.**Average release times for drug delivery formulations of different shapes. D denotes the drug diffusion coefficient.

Drug Carrier Shape | Characteristic Size | Average Release Time ${\mathit{t}}_{\mathit{av}}$ |
---|---|---|

Sphere | Radius R | $(1/15)\phantom{\rule{4pt}{0ex}}{R}^{2}/D$ |

Slab or thin film | Thickness L | $(1/12)\phantom{\rule{4pt}{0ex}}{L}^{2}/D$ |

Cylinder (general case) | Height H and Radius R | Equation (24) ^{3} |

Long cylindrical rod ^{1} | Radius R | $(1/8)\phantom{\rule{4pt}{0ex}}{R}^{2}/D$ |

Flat disc ^{2} | Height H | $(1/12)\phantom{\rule{4pt}{0ex}}{H}^{2}/D$ |

^{1}Very long cylinder ($H\gg R$), in practice when $H/R>10$.

^{2}Very short cylinder ($H\ll R$), in practice when $H/R<0.1$.

^{3}An approximate simple expression is given by Equation (25).

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**MDPI and ACS Style**

Kalosakas, G.
Exact Analytical Relations for the Average Release Time in Diffusional Drug Release. *Processes* **2023**, *11*, 3431.
https://doi.org/10.3390/pr11123431

**AMA Style**

Kalosakas G.
Exact Analytical Relations for the Average Release Time in Diffusional Drug Release. *Processes*. 2023; 11(12):3431.
https://doi.org/10.3390/pr11123431

**Chicago/Turabian Style**

Kalosakas, George.
2023. "Exact Analytical Relations for the Average Release Time in Diffusional Drug Release" *Processes* 11, no. 12: 3431.
https://doi.org/10.3390/pr11123431