Theoretical and experimental study of the role of cell-cell dipole interaction in dielectrophoretic devices: application to polynomial electrodes
© Camarda et al.; licensee BioMed Central Ltd. 2014
Received: 17 November 2013
Accepted: 23 May 2014
Published: 5 June 2014
We aimed to investigate the effect of cell-cell dipole interactions in the equilibrium distributions in dielectrophoretic devices.
We used a three dimensional coupled Monte Carlo-Poisson method to theoretically study the final distribution of a system of uncharged polarizable particles suspended in a static liquid medium under the action of an oscillating non-uniform electric field generated by polynomial electrodes. The simulated distributions have been compared with experimental ones observed in the case of MDA-MB-231 cells in the same operating conditions.
The real and simulated distributions are consistent. In both cases the cells distribution near the electrodes is dominated by cell-cell dipole interactions which generate long chains.
The agreement between real and simulated cells’ distributions demonstrate the method’s reliability. The distribution are dominated by cell-cell dipole interactions even at low density regimes (105 cell/ml). An improved estimate for the density threshold governing the interaction free regime is suggested.
As first defined by Pohl [1, 2], the term “dielectrophoresis” is used to describe the “ponderomotive” force exerted by a non-uniform electric field on polarizable neutral particles. Such force allows for the controlled manipulation of micro and nano-sized particles dispersed in colloidal solutions. Application fields include: cell partitioning and isolation [3, 4], bio-structure assembling , nanostructure (e.g. carbon nanotube) deposition , filtration systems for oils purification  etc. Among these, the separation of rare cells  is a specifically promising one as dielectrophoresis allows the capture/separations of cells without the use of biomarkers; relying, instead, in the strong selectivity of the dielectrophoretic (DEP) response  which depends on the particle mass, shape and composition. Indeed, recently, this selectivity has permitted to discriminate the tumor cell types of the NCI-60 panel from Peripheral Blood MonoNuclear cells (PBMNs) . However, although many intriguing micro-structures have been fabricated in research laboratories, DEP devices have hardly gone beyond the proof-of-concept stage . One of the problems that are hindering development and engineerization of the devices is the limited use of accurate numerical tools for their design which, in turn, is due to the computational complications arising by the particle-particle dipole interaction. Indeed, particle kinetics (i.e. the particle velocity field) in DEP devices can be easily calculated by mean of Poisson solvers and direct integration of the equation of motion only in the non-interacting particle approximation . This approximation is not valid in the accumulation regions of the DEP devices where, due to the increased particle concentration, dipole-dipole interactions become important and can promote the formation of clusters and significant rearrangements of the particle space distribution [13–16]. These many-particle effects can be accurately simulated solving directly the equations of motion in the few-particles limit  i.e. this approach is not applicable for the simulation and design of realistic systems. Another possible approach is the use of reaction–diffusion models [18–20] but this approach needs an “ad hoc” parameter calibration to effectively consider the dipole-dipole interactions in compact models.
Recently a coupled Monte Carlo-Poisson (MC-P) method  has been implemented which allows simulating a large number of particles in large active zones (within the experimental range), explicitly including particle-particle interactions. The MC-P method has pointed out the relevance of this inclusion in the modeling predictions for the simplified condition of Two Dimensional (2D) electric field distribution, where explicitly depends on two space coordinates as in the case of very long interdigitated electrodes. However, the possibility to apply this approach for the numerical design of devices exploiting more complex fully Three Dimensional (3D) electric field distributions has not been yet demonstrated. Moreover the MC-P predictions have never been compared with real cell distributions in dielectrophoretic devices, in order to confirm their reliability. Aiming to the two objectives of the model extension and validation, we have improved the application potentiality of the MC-P method to simulate the features of devices generating 3D electric field distributions. In addition we applied the simulation method to the case of polynomial electrodes which are known to produce well defined 3D non-uniform electric fields and are used for the study of negative dielectrophoresis  or for the determination of particle dielectrophoretic response through electrorotation analysis [23, 24]. We compare the simulated results with experimental distributions obtained in the same electrodes geometry to evaluate the role of p-p interactions and definitively demonstrate the predictive potential of this methodology.
A detailed description of the method can be found in ref. , here we summarize the key aspects of the simulations, specifically focusing on the 3D implementation.
Note that whereas the field in Eq. 1 is the field generated by the external electrodes only, in the Eq. 3 must be calculated considering all the particle presence. This direct calculation is not practically feasible in the kinetic simulation of large systems, since the particle distribution continuously changes in the space requiring an integration of the Poisson equation, , at each simulation step. A more efficient approach, that requires only the evaluation of the external field, can be implemented approximating the total distorted electric field with the sum of the field generated by the external electrodes plus the contributions of the dipoles induced in all the particles: .
The reliability of this approximation has been demonstrated in Ref.  with the aid of the full calculation based on Eq. 3 for the case of two spherical particles immersed in a uniform external electric field: the magnitude, the angular dependency and the scaling with the distance of the calculated force are similar to those derived in the interacting dipoles approximation .
where are the angles between the vectors , and and are the average polarizations for the i and j particles.
where is the effective diffusivity, N is the number of simulated particles, Δd is the elementary particle displacement, η is the medium viscosity, a is the particle radius, k B is the Boltzmann constant and T is the system temperature.
The polynomial electrode design described in the previous section, has been fabricated by deposition of 10 nm of Titanium followed by 200 nm of Nickel on a standard microscope glass. The electrodes were delineated by lithographic methods followed by wet etching. The device has been energized using a Protek 9205C signal generator which applied, consistently with the simulated systems, a sinusoidal voltage signal of 8Vpp value at 1 MHz for 180 sec (long time allow for cells equilibration). The final distribution was observed with a standard 10× phase contrast inverted microscope. The human breast cancer cell line MDA-MB-231 were cultured according to American Type Culture Collection (ATCC) instructions. The cells, just before DEP tests, were suspended in a low conductive buffer (used as the elute in all our experiments) composed of 9.5% ultrapure sucrose (S7903, Sigma-Aldrich), 0.3% dextrose (Fisher D-16), and 0.1% Pluronic F68 (P1300, Sigma-Aldrich) titrated to a conductivity of 30 mS/m (consistent with Monte Carlo simulations) by adding KCl with the aid of a conductivity meter. The buffer had an osmolarity of 320 mOs/L and a pH of 7 and the experiments were conducted at room temperature (22°C). The cells, suspended in DEP buffer at concentration of 5×105 cells/ml were pipetted into the chamber and occupied a total volume of about 100 μl when a cover slip was placed over the rubber o-ring.
From these results we can infer that particle-particle interactions compete with the dielectrophoretic force-field, which would otherwise massively trap (in p-DEP conditions) the particles in the regions where the gradient of the electric field is larger. Note also the cells in the region far away from the electrodes which are not trapped by the DEP field, this allows for the definition of a depletion volume as the region where cells are effectively attracted to the electrodes. The connection between depletion region, electrodes geometry and particle-particle interaction is currently under investigation.
Where 〈∇ z U eff (z)〉 is the average DEP force in the vertical direction at distance z from the bottom of the chamber and μ m is medium viscosity. In the case of the polynomial electrodes used and for a deposition time of 180 sec, r DEP ≅ 85 and z cap = 280μm ≅ (1/5)h so that the improved concentration threshold, to avoid dipole-dipole interaction, should be below 3 × 104 cells/ml. We performed Monte Carlo simulations in the 104 cells/ml range finding no significant evidence of cell chains formation (not shown), thus confirming that Eq. 14, together with Eq. 12, represent a better qualitative threshold to avoid cell-cell interaction requiring only a knowledge of the electric field in the DEP device.
In conclusion, we have demonstrated that the effects of particle-particle interactions play a crucial role in the kinetic evolution of colloidal systems in DEP devices even at low density regimes (105 cells/ml), lower than the ones currently used in DEP devices .
Monte Carlo methods allow for the simulation of sufficiently large systems in terms of size and number of particles (i.e. within the experimental scopes). The discrete approach (i.e. the particle resolution), as opposed to the fluid-flow methodologies, is the key ingredient of the method improvement. In the case of MDA-MB-231 tumor cells suspended in a static, low conductive, fluid under the action of a positive-DEP field generated by a polynomial schema we have elucidated the crucial role of particle-particle interactions on the trapping efficiency of the device, on the organization of cells in ordered chains and on the overall cell space distribution. We have also deduced a new qualitative concentration threshold to avoid cell-cell interaction which requires only a knowledge of the electric field in the DEP device. Clearly, to have an exact determination of the concentration threshold for the specific DEP device used, MC-P kinetic simulations varying the cells density, such as the one proposed in this paper, need to be performed.
Future works will be devoted to generalize the formalism here presented in order to include second-order effects such as cell-sedimentation and cell-stitching or including hydrodynamic forces to simulate cells distributions in dynamic separation systems.
We thank Salvo Di Franco of CNR-IMM for the lithographic fabrication steps and Caterina Grillo for the English Grammar and Spelling review.
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