*2.5. Release Studies*

In vitro release studies were carried out using static Franz-type diffusion cells with an effective diffusion area of 1.76 cm<sup>2</sup> . A 500 µL of each formulation was added to the donor chamber, and the receptor chamber was filled with 12 mL of PBS (pH 7.4) (*n* = 6). Temperature was maintained at 32 ◦C throughout the experiment. A Spectra/Por® molecular porous membrane was used to separate donor and acceptor compartments. Both the donor compartment and the sampling port were covered with Parafilm® to avoid leakage and solvent evaporation. Samples of 400 µL were collected at 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 48, and 72 h. At every sampling time, the volume was replaced with pre-warmed PBS to guarantee sink conditions [56].

The released B12 was quantified using the same HPLC analytical method described above, and the cumulative amounts of B12 versus time were calculated. The total drug amount released was calculated according to the previous determination of the drug content and plotted versus time. The release profiles were fitted to different mathematical kinetic models: Higuchi, Korsmeyer–Peppas, Kim, Peppas–Sahlin, zero order and first order. For each case, the Akaike information criterion (AIC) was calculated to determine the optimal models to explain the experimental data. The correlation coefficient (*R* 2 ) of the most accurate model was reported as an indicator of the proportion of variation of the results that could be explained by the model [57,58].

The Higuchi model assumes that the release process is carried out only by passive diffusion and is governed by the following equation (Equation (2)) [59]:

$$\mathbf{M}\_{\rm t}/\mathbf{M}\_{\infty} = \mathbf{k} \cdot \sqrt{\mathbf{t}} \tag{2}$$

where M<sup>t</sup> is the amount released at time t, M<sup>∞</sup> is the maximum amount of drug released, Mt/M<sup>∞</sup> is the fraction of the amount of drug released at time t, k is the constant that governs the process.

The Korsmeyer–Peppas model or power-law is described with the following equation (Equation (3)) [60,61]:

$$\mathbf{M}\_{\mathbf{t}}/\mathbf{M}\_{\infty} = \mathbf{k} \cdot \mathbf{t}^{\mathbf{n}} \tag{3}$$

where M<sup>t</sup> is the amount released at time t, M<sup>∞</sup> is the maximum amount of drug released, Mt/M<sup>∞</sup> is the fraction of the amount of drug released at time t, k is the constant that governs the process and explains the characteristics of the system, and n is the diffusion release exponent. Values of n < 0.5 are indicative that the release process is carried out by passive diffusion; values of 0.85 < n < 1 show that the process is governed mainly by relaxation, and intermediate values 0.5 < n < 0.85 indicate the existence of both phenomena (anomalous transport).

As a variation of the Korsmeyer–Peppas equation, Kim et al. proposed a modification (Equation (4)) to assess the possible burst effect of the formulation [62]:

$$\mathbf{M}\_{\mathbf{l}}/\mathbf{M}\_{\infty} = \mathbf{k} \cdot \mathbf{t}^{\mathbf{n}} + \mathbf{n} \tag{4}$$

where b is the parameter corresponding to the burst effect.

The Peppas–Sahlin model considers that release may occur through the processes of passive diffusion and relaxation, each represented by a constant (Equation (5)) [63]:

$$\mathbf{M}\_{\mathbf{l}}/\mathbf{M}\_{\infty} = \mathbf{k}\_1 \cdot \mathbf{t}^{\mathbf{n}} + \mathbf{k}\_2 \cdot \mathbf{t}^{\mathbf{n}} \tag{5}$$

where k<sup>1</sup> and k<sup>2</sup> are, respectively, the constants associated with the processes of drug release by passive diffusion and relaxation, and n is the diffusion release exponent.

Representative theoretical models, zero order (Equation (6)) and first order(Equation (7)), are described by the following equations [64]:

$$\mathbf{M\_t/M\_\infty} = \mathbf{k\_d} \cdot \mathbf{t} \tag{6}$$

$$\mathbf{M}\_{\mathbf{t}}/\mathbf{M}\_{\infty} = 1 - \mathbf{e}^{-\mathbf{k}\mathbf{d} \cdot \mathbf{t}} \tag{7}$$

where k<sup>d</sup> is the constant of diffusion release that governs the process.
