In order to perform our analysis in a realistic case we have considered the problem of the separation of the circulating tumor prostanoid receptors (in particular MDA — MB 231 triple negative breast cells) from lymphocytes (LYM). The calculation of the f as a function of the frequency for two types of cells indicated that for the frequency of 60kHz: MDA are subjected to pDEP while LYM to nDEP. As a consequence at this frequency we should efficiently separate the two branches of cell components in a mixed colloidal suspension of them. Unlike the standard DEP, aDEP is effective also in the presence of uniform electric field. As a consequence, a simple configuration where these effects can be easily evidenced is the planar capacitor which generates a uniform electric field and we can observe the aDEP contribution only . However, our simulations were performed considering a realistic device geometry i.e. a channel with interdigitated electrodes as shown in Fig. 1 and an external non-uniform AC electric field IS imposed simulating the separation of particles at the indicated frequency of 60kHz.
As it has been explained in the introduction, the particles are suspended in a liquid medium, which is assumed to flow through the channel velocity field derived by the Navier–Stokes equation (see below). In order to determinate the dielectrophoretic velocity of the particle in the reference system of the fluid (with the standard dipole approximation or with the MST method) we use the Stokes formula and the steady state relationships as shown below :
where f is the friction factor that for the spherical particle became 6πμmediumR, Fdrag represents the flow resistance, and is the velocity field. F can be determined numerically solving the Poisson equation
and using the two methods indicated in Section 1.
For the effective medium velocity we need to solve the Navier–Stokes equation (with steady state and uncompressible condition (Eq. (10)))
with ρ indicating the density of the medium, η is the kinematic viscosity, represents the generic vector of our study, ε is the electric permittivity, and is the external electric field generated by the voltage V. The velocity field in the laboratory reference system is given by
This analysis is performed in Figs. 2 and 3 for MDA and LYM case respectively. In both figures we represent the velocity field (u) starting from a distance from the electrodes equal to the particles\’ radii.
In this section, we evaluate the anomalous DEP effects using a different approach to make evident the influence of the electrode proximity in the global cells\’ kinetics. We follow the motion of a given density of MDA and LYM entering from an inlet in an interdigitated channel device. We impose that particle density is governed by a drift–diffusion equation where the drift term is calculated by means of the Eqs. (1)–(12):
with Φ as the particles\’ density, the drift–diffusion current, is the total velocity field Eq. (12), and D(ϕ) represents the diffusion coefficient. In general, we can identify two terms: is the drift term while D(Φ)∇ is the diffusion term [5,6].
To better evaluate the influence of aDEP effects for MDA and LYM motion in the colloidal solution, we introduce the particles in the channel from two different positions for the inlets as indicated in Fig. 4:
As expected from the theory, in the aDEP zone we should have a different particle concentration space profile if the convection term is evaluated with the MST with respect to the standard method. In particular, we consider separately the two cases MDA and LYM and analyze in details the density field in the aDEP zones indicated in Figs. 2 and 3 (the working frequency 60kHz is the same for both configurations).
The snapshots in Fig. 5 represent the MDAs and the time dependence of the density in particular points out of the aDEP zone is shown in Fig. 6.