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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Finite element method analysis was applied to the characterization of the biomolecular interactions taking place in a microfluidic assisted microarray. Numerical simulations have been used for the optimization of geometrical and physical parameters of the sensing device. Different configurations have been analyzed and general considerations have been derived. We have shown that a parallel disposition of the sensing area allows the homogeneous formation of the target molecular complex in all the active zones of the microarray. Stationary and time dependent results have also been obtained.

In the past two decades microfluidics has emerged as a powerful tool for biosensing [

Many works concerning the modeling of a microfluidic biosensor have appeared recently. The main aim of these studies usually was to improve some aspect of the sensing performance, such as sensitivity, time response, and dependence on external factors. How the assay parameters determine the amount of captured analytes [

Among biosensors, the microarray technology has demonstrated a great potential in drug discovery, proteomics research, and medical diagnostics. The reason of this success is the very high throughput of these devices due to the large number of samples that can be analyzed simultaneously in a single parallel experiment. The microarray technology is based on the immobilization of a huge amount of bioprobes on a solid platform, which can be obtained by

The convergence between microfluidics and microarrays has been relatively straightforward due to their multiple shared features, but the implementation of a microfluidic circuit on an array device is not trivial nor simple: a specific design is often required to meet biological constraints and fabrication technique demands. In this context, numerical simulations by finite element methods (FEM) allow a space and time characterization of the biomolecule distribution and interaction in the circuit. Hu

In this work, we present a numerical study by FEM analysis of the binding interaction between active sites on the array surface elements with biochemical species in microfluidic networks. While the literature works generally consider interactions between biochemical species under flow conditions, in our simulation we have also considered the binding kinetics under static conditions, with an initial step involving flow of a liquid solution to fill the channel, followed by a flow velocity decreasing to a zero value, and we have compared the results with respect to the dynamic approach. Many experiments, especially those requiring consumption of a very low volume of reagent for economic or technical reasons, are driven in static, or quasi-static, steady flow conditions, so this is a useful design tool for both situations. On the basis of the results obtained, we also propose a new microfluidic layout for parallel flow to provide efficient and uniform analyte distribution on the sensing part of microfluidic assisted microarrays.

The modeling of what happens before transduction of a biomolecular interaction in a biosensor requires considering at least three physical processes: (1) the surface reactions, ^{2}), bound to the sensing area, and a second chemical species B (mol/m^{3}), present in a buffer solution, producing a complex C created by the two molecular species, can be described by the first order time-dependent Langmuir Equation [^{2}, k_{a} is the association rate constant (M^{−1}s^{−1}), and k_{d} is the dissociation rate constant (s^{−1}). This equation can be used for antigen-antibody [_{eq} can be expressed as:
_{d}/k_{a}_{eq} in the microfluidic configuration assigned in order to maximize the sensor response as a function of the fabrication parameters.

The fluid flow can be modelled using the Navier-Stokes equations with the incompressibility condition:
^{3} kg/m^{3} and μ = 10^{−3} Pa·s. The flow is considered laminar with a parabolic profile at the inlet and an average velocity u_{0}, since the flow in the microchannel is in the low Reynolds number region. Boundary conditions for the equations are p = 0 at the outlet and no-slip walls (u = 0) elsewhere.

Moreover, the transport of the chemical species B in bulk liquid phase is described by the convection and diffusion equation:
^{2}/s) is the diffusion coefficient of the chemical species B in bulk phase. Complete boundary conditions are the following:

In order to simulate a static process, we have multiplied inlet conditions B_{0} and u_{0} for the function 1-H(t-t_{fill}), where the H(t) is the Heaviside step function and t_{fill} is the necessary time to fill the channel given by L/u_{0} where L is the total device length. In this way we can simulate the injection of the solution of B for t_{fill} sec in the microchannel and then the subsequent static incubation. Under dynamic conditions a constant flow velocity in the microchannels is assumed. The numerical calculations have been performed using the FEMLAB™ (Comsol Inc.) finite element software package combining the three differential equations into a single model.

The microarray that we have considered is composed by sixteen elements arranged in 4 × 4 matrix: each circular element has a radius of 100 μm and they are spaced 600 μm apart. In this work, we propose two different microfluidic configurations; an example of the first one is sketched in _{a} and k_{d} (k_{a} = 5 × 10^{5} M^{−1} s^{−1}, k_{d} = 10^{−4}s^{−1}) [^{−11} m^{2}/s and an active site surface concentration A = 1 × 10^{−8} mol/m^{2}.

We have also assumed B_{0} = 10 nM and u_{0} ranging from 0.1 to 10 mm/s: we have studied the formation of compound C in the sensing region by changing the inlet velocity. We have chosen to change this parameter because it doesn’t directly influence the equilibrium complex concentration C_{eq} [

The simulation has shown that there is a clear decrease in the formation of compound C which is proportional to the inlet distance from the first to the fourth element, respectively. The effect is due to a concentration decrease along the channel of the chemical species B: by increasing the inlet velocity up to 10 mm/s more homogeneous values among the four active surfaces can be obtained. A change of the inlet position will cause only a time shift in the graphs of

The second layout that we propose is viewed as an improvement of the device using the same element configuration, but changing the microfluidic network. In this design, we propose a parallel approach [see

Since under static flow conditions, a homogeneous distribution of C along the channel depends on the average inlet velocity, if we want the same density of C in the parallel active sites, we must have the same local velocity. The pressure driven, steady-state flow of an incompressible fluid through a straight channel can be described by the Hagen-Poiseuille law:
_{hyd}_{i}, L_{i}, h_{i}, w_{i}_{1} = Q_{2} = Q_{3} = Q_{4}_{c} = 3R_{a}_{b}_{a}_{c}_{c}_{a}_{b}

From these calculations, it results that the four current flows are equal within a confidence range of less than 5%. The electrical network analogy can be thus used for the fabrication of a compact microfluidic circuit which feeds the chemical substance B in parallel. The comparison between the binding kinetics of formation of C in the four active areas in the case of the two microfluidic layouts for u_{0} = 1 mm/s is presented in

In a dynamic regime, the four active areas reach the same amount of C are different time points, and the saturation condition is obtained with a time difference of 48% between the last element (2,040 s) with the respect to the first element (1,380 s). By parallel microfluidics it is possible to almost cancel this delay: all the elements saturate in the same interval (the time delay is less than 1%). We have also investigated how the binding kinetics under dynamic flow conditions depend on the inlet velocity in the case of a parallel microfluidic system; the results are shown in

The simulations have confirmed that there are no substantial differences among the four elements in this case, and also a substantial decrease of saturation time can be noted upon increasing the inlet velocity: the saturation value is reached in 2,880 s for u_{0} = 0.1 mm/s, in 1,380 s for u_{0} = 1 mm/s, and in 1,020 s for u_{0} = 10 mm/s. We can thus conclude that the inlet velocity plays a fundamental role in the optimization of the microfluidic microarray both for static and dynamic regimes.

We have analysed the binding kinetics of the formation of a complex C in the case of a generic molecular interaction which could happen in the channel of a pressure driven microfluidic circuit used to assist and enhance the performances of a microarray. We have found the conditions required to optimize the uniformity of the chemical species distribution on the sensing area. Different microfluidic layouts have been proposed to improve the sensing performance. The dynamic flow condition approach seems to be the best in terms of homogeneity and time parameters for the microfluidic biosensor, but the static approach can be useful in case where very low sample consumption is necessary.

Comparison of the formation of complex C simulating a static incubation from the first to the last element in a linear microchannel for different inlet velocity values.

Comparison of C formation binding kinetics obtained by simulating a dynamic incubation from the first to the last element in linear and parallel microfluidic system, respectively.

Comparison of the formation of complex compound C simulating a dynamic incubation from the first to the last element in a parallel microfluidic system for different inlet velocity values.