## 4 Materials and Methods
### Model for the Movement of Particles  
The most commonly available forms of nanocarriers for clinical use are liposomal formulations [13]; in fact, today, the main types of transporter-mediated anticancer agents are liposomes [19]. Liposomes are vesicles that form in water, with a diameter of 25 to 500 nm and a thickness of about 5 nm. In the said nanocarrier, the hydrophobic drug is enclosed in the aqueous process of the liposomes, while the hydrophilic drug is found in the bilayer of lipid functionality [17]. Liposomes have been used in different investigations for the treatment of colorectal cancer with results indicating that coating their surface with polymers and ligands increases their mucopenetration capacity, especially when liposomes modified with polyethylene glycol (PEG) are used [20; 21; 22]. These polymeric coatings also allow oral liposomal formulations to resist degradation in the harsh environment of the GI tract, including bile salts and enzymes that would normally dissolve the lipid bilayer [23]. For this investigation, liposomes were modeled as spherical particles with a diameter of 200 nm and a density of 1000 \(\frac{kg}{m^{3}}\)[50]. Likewise, a mean maximum inhibitory concentration of liposomes was used, which was \(\mathrm{C_{P}}=13.5\frac{kg}{m\mathrm{L}}\)[32; 33]. A Brownian force model represented by Equation (7) was considered as the initial behavior of the particles:  
\[\mathrm{F_{b_{i}}}=\zeta_{i}\sqrt{\frac{\pi\mathrm{S_{0}}}{\Delta t}} \tag{7}\]  
where \(\zeta_{i}\) is a Gaussian random number with a mean zero (independent of unit variance); \(\Delta t\) is the timestep of the particle; and \(\mathrm{S_{0}}\) is a function of the Stokes-Cunnigham slip correction coefficient, temperature, kinematic viscosity of the pore fluid, particle diameter and density. The forces that act on the particle affect its acceleration; this acceleration is due to differences in the velocities between the particle and the fluid, and are represented in Equation (8):  
\[\mathrm{m_{p}}\frac{\mathrm{d}\vec{\mathrm{v}_{\mathrm{p}}}}{\mathrm{d}t}=\vec{ \mathrm{F}_{\mathrm{D}}}+\vec{\mathrm{F}_{\mathrm{B}}}+\vec{\mathrm{F}_{ \mathrm{R}}}+\vec{\mathrm{F}_{\mathrm{R}}}+\vec{\mathrm{F}_{\mathrm{P}}}+\vec{ \mathrm{F}_{\mathrm{E}}} \tag{8}\]  
where:  
* \(\vec{\mathrm{F}_{\mathrm{D}}}\): drag forces acting on the particle;
* \(\vec{\mathrm{F}_{\mathrm{B}}}\): buoyancy forces;
* \(\vec{\mathrm{F}_{\mathrm{R}}}\): forces due to rotation;
* \(\vec{\mathrm{F}_{\mathrm{P}}}\): Force associated with the pressure gradient. This is the force applied to the particle due to the pressure gradient in the fluid surrounding the particle caused by the acceleration of the fluid. This is only significant when the density of the fluid is comparable to or greater than the density of the particle;
* \(\vec{\mathrm{F}_{\mathrm{E}}}\): force associated with the electrostatic interaction of the particle immersed in the electric field produced by the negative charge on the surface of the malignant cell. The transfer of momentum from the continuous phase to the discrete phase is calculated by examining the change in the particle's momentum as it passes through the control volume in the model. This change in momentum is calculated as follows:  
\[\mathrm{F}=\Sigma\Bigg{(}\frac{18\mu C_{\mathrm{D}}\mathrm{Re}}{\rho_{\mathrm{ p}}\mathrm{d}_{\mathrm{p}}^{2}24}\big{(}\mathrm{u_{p}}-\mathrm{u_{f}}\big{)}+ \mathrm{F}_{\mathrm{additional}}\Bigg{)}\dot{\mathrm{m}}\Delta\mathrm{t} \tag{9}\]  
where:  
* \(\mu=\) fluid viscosity;
* \(\rho_{\mathrm{p}}=\) particle density;
* \(\mathrm{d_{p}}=\) particle diameter;
* \(\mathrm{Re}=\) Reynolds number;
* \(\mathrm{u_{p}}=\) particle velocity;
* \(\mathrm{u_{f}}=\) fluid velocity;
* \(\mathrm{C_{D}}=\) drag coefficient;
* \(\dot{\mathrm{m}}=\) particle mass flow;
* \(\Delta\mathrm{t}=\) timestep or step of time. * \(\mathrm{F_{\mathrm{additional}}}=\) additional forces of interaction, such as the electrostatic force that induces the selective coupling [51].