## 2 Materials and Methods
### Mathematical Analysis of pDNA Release  
Release analysis of active components from biodegradable systems is a combination of a number of factors. These factors determine the contribution of mechanisms of release involved, such as initial burst, NPs degradation-relaxation and diffusion. [PERSON] et al. reported a comprehensive review that considers several physical factors, such as temperature and pH of the system, size and shape of NPs, drug solubility, and others [38]. The pDNA release analysis from biodegradable NPs usually considers those factors, resulting in more than one mechanism of release to describe the entire release process. One of the most relevant mechanisms of release is initial burst, which is attributed to a process of interfacial diffusion between solid sphere surface and liquid media. By considering that initially all pVAX1-NH36 is incorporated in NPs, kinetics of initial burst follow an exponential relationship as reported in literature [53; 54].  
\[\frac{M_{t}}{M_{\infty}}=1-\exp\left(-k_{b}\,t\right) \tag{1}\]  
where \(M_{t}\) is the cumulative amount of pVAX1-NH36 released at time \(t\), \(M_{\infty}\) is the cumulative amount of pDNA released at infinite time and \(k_{b}\) is the initial burst constant, incorporating factors such as concentration of pVAX1-NH36 on NPs surface, surface area, interphase properties, solubility of pVAX1-NH36 and electrostatic interactions between pDNA and carrier [40].  
Other mechanisms associate the amount of pDNA released with polymeric matrix degradation, which depends on factors such as pH of the media, incubation temperature, and larger particles degrading at faster rates than smaller ones [55]. Degradation of PLGA in aqueous solutions was carried out by hydrolysis, resulting in relaxation of NPs as evident in literature [56]. PLGA NPs degradation-relaxation effects have been explained by using the Prout-Tompkins equation [40; 41; 54; 55]. This equation can also be used to describe pDNA release and could be rewritten in the following form:  
\[\frac{M_{t}}{M_{\infty}}=\frac{\exp\left[k_{r}\left(t-t_{max}\right)\right]}{ 1+\exp\left[k_{r}\left(t-t_{max}\right)\right]} \tag{2}\]  
where \(k_{r}\) is the rate of degradation-relaxation constant and \(t_{max}\) is the time to achieve a maximum rate of pDNA release or the time to achieve 50% of release.  
Hence, a linear combination of Equations (1) and (2), that simultaneously consider the stages of initial burst and NPs degradation-relaxation, was used to describe the entire pDNA release process from biodegradable NPs according to Equation (3):  
\[\begin{array}{ll}\frac{M_{t}}{M_{\infty}}=&\theta_{b}\left[1-\exp\left(-k_{ b}\,t\right)\right]\\ &+\left(1-\theta_{b}\right)\left\{\frac{\exp\left[k_{r}\left(t-t_{max}\right) \right]}{1+\exp\left[k_{r}\left(t-t_{max}\right)\right]}\right\}\end{array} \tag{3}\]  
where the initial burst contribution fraction over the entire pDNA release process is \(\theta_{b}\). Equation (3) contains four unknown parameters that were determined by adjusting the equation to experimental data of pVAX1-NH36 release by using nonlinear least-squares algorithm in MATLAB(r) (R2010b, MathWorks, Natick, MA, USA).