**3.4 Mathematical models of drug release**

Mathematical knowledge could assist in clarifying the drug release mechanism from polymer-based nanoparticles for controlled delivery. The exact mass transport and the relationship between drug release as a function of time can be described through various mathematical models. These models are able to predict analytics via data obtained from the drug release experiments to explain drug release kinetics. The coefficient of determination (R2 ) and adjusted coefficient of determination (adj.-R2 ) are investigated by statistical analysis to determine the relationship that the equation explains. The highest values of R<sup>2</sup> and adj.-R2 from curve fitting involving describing the optimal relations (maximum accuracy) as a function of release time can be predicted with these Equations [6, 21]. This chapter will discuss some of the most commonly used equations to predict the kinetics of release from polymeric nanoparticles, including the *zero*-*order*, *first*-*order*, *Higuchi*, *Hixson*-*Crowell*, and *Korsmeyer*-*Peppas* equations, as summarized in **Table 1**.

When predicting drug release rates, the *zero***-***order model* represents controlled drug release that occurs at a constant rate depending on only the time and is independent of the amount of polymer. In the *first***-***order model*, the amount of drug released tends to depend on the polymer concentration, which affects the swelling and porosity of the nanoparticles. The *Hixson***-***Crowell model* describes the rate of drug release according to the cube root of its volume, in which matrix dissolution occurs. As a result, the diameter and surface area of the nanoparticles decreases proportionally over time according to the cube root of the weight at that particular time [34]. The *Higuchi model* predicts the release rate via diffusion control based on Fick's first law, which is square root time-dependent (t1/2) [35].

Generally, the release mechanism can be predicted by the *Korsmeyer***-***Peppas model***.** This model determines the exponential relationship between the rate of drug release and time. It is based on polymeric matrices with different geometries, following the released exponential (n). The exponential n-value indicates three types


*Qt is the cumulative amount of drug release at time (t), Q0 is the initial amount of drug release at (t0), Mt is the cumulative amount of drug remaining at time (t), M0 is the initial amount of drug remaining at (t0), t is time, n is the released exponent of Korsmeyer-Peppas, and K0, K1, KHX, KH, and KKP are the zero-order, first-order, Hixson-Crowell, Higuchi, and Korsmeyer-Peppas constants, respectively.*

#### **Table 1.**

*Mathematical models of drug release [6].*

of release mechanisms (**Figure 6**). The first type is the mechanism of Fickian diffusion, where the rate of diffusion of drug release is considerably greater than the rate of polymeric chain relaxation. The second mechanism is anomalous transportation via diffusion and swelling/erosion via slow rearrangement of polymeric chains. The final type is only the swelling/erosion mechanism. The release mechanisms of swelling and erosion are affected by an expansion in porosity and polymeric chain cleavage, respectively. Thus, the anomalous transportation and swelling/erosion mechanisms are both non-Fickian diffusion [36].

Using equations to predict release mechanisms can be confusing. Therefore, we proposed the flowchart sequence of use steps for these equations to describe drug release from PLGA nanoparticles [6]. The initial step investigates whether the rate of drug release is independent or dependent on the various concentrations by comparing R<sup>2</sup> values between the *zero***-***order* and *first***-***order* models. Next, the *Hixson***-***Crowell equation* and *Higuchi equation* were compared to determine whether the release rate depends on the probability of matrix dissolution or is diffusion controlled. Finally, the mechanism of drug release was investigated via the *Korsmeyer***-***Peppas equation* to determine the exponential n-value, as proposed in **Figure 6.**
