## 3 Results and Discussion
### Mathematical Modelling of Drug Release Kinetics  
The use of mathematical models enables the evaluation of the drug release process under different conditions and can aid in understanding the mechanism of release whether it be diffusion controlled, chemically controlled, osmotically controlled, or swelling and/or dissolution controlled; models can elucidate the main determinants to the rate of this release [55; 89]. Mechanistic models accurately and reliably describe drug release systems; however, they are complex models that incorporate multiple parameters and require large quantities of experimental data. The use of empirical models is more widespread as they incorporate only a few parameters and are simpler to interpret. In addition, they are more tolerant of experimental variation, and are largely applicable on the nanoscale. The use of empirical models is, however, less performant due to the associated assumptions [50; 90]. It is important to understand and to be able to predict the release behaviour of drug delivery systems in different conditions in vitro prior to further development. This enables the design and formulation of nanocarriers that are associated with optimal drug release and delivery [60]. This study uses four models to analyse the kinetic release data of DOX from the loaded nanocarriers. The zero-order model describes the drug release at a constant rate of time, in which the drug release rate is not dependent on the amount of drug remaining. Zero-order kinetics are best used to describe drug release from transdermal systems and low dosage matrix tablets; it is rare for complex nanocarriers to follow a zero-order release [50; 91]. The first-order model describes a system where the rate of release is dependent on the concentration of the remaining drug and has been used to describe absorption and elimination of some drug formulations, particularly water-soluble drugs [55; 92]. The Higuchi model is one of the most used models to describe the release rate from matrix systems and can be applied to planar systems as well as those with different geometries and porous structures [60; 93]. By using the Higuchi equation, the assumptions of its derivation must be considered which makes it rarely applicable for many controlled release systems, stimuli-responsive swellable matrices, and nanogels smaller in dimension than their payload [50]. Assumptions include \"The initial drug concentration in the system is much higher than the solubility of the drug, mathematical analysis is based on 1D diffusion and thus, edge effects must be negligible, the suspended drug is in a fine state such that the particles are much smaller in diameter than the thickness of the system, swelling or dissolution of the polymer carrier is negligible, the diffusivity of the drug is constant and perfect sink conditions are maintained\" [93]. Finally, the Weibull model is commonly used for comparing the release profiles of matrix-based systems and is useful due to its flexibility in fitting to experimental data [58; 92]. Due to this flexibility and the sensitivity of its parameters, the Weibull model has been applied to both rapid release systems and extended drug delivery systems [55]. The Weibull model has two parameters: \(t_{d}\) is the lag time before the onset of drug release,and the parameter \(\beta\) is a shape parameter that describes the shape of the release curve. When \(\beta=1\), it implies an exponential drug release profile; if \(\beta>1\), the release profile is sigmoidal. If \(\beta<1\), the graph is parabolic [58]. This model infers a descriptive account of a drug release profile but does not describe any dissolution kinetic properties. Further, it has limited use for establishing both in vitro and in vivo correlations [60; 91]. The calculated regression coefficient, \(R^{2}\) has been calculated for all experimental conditions for each of the analysed publications (Table 2).