In order to accurately determine the collection efficiency of the suspended waveguide device, it is necessary to account for both the fluid flow around the sensor and the reaction kinetics at the surface of the sensor. This type of complex, interdependent modeling is ideally suited for COMSOL Multiphysics, a finite element simulation package, which can incorporate multiple physical phenomena interactively.
Specifically, finite element method simulations were performed using COMSOL Multiphysics 4.2 to solve the Navier-Stokes (i.e.
, momentum balance and continuity equations) and conservation of mass equations in a geometry representing a microfluidic flow cell containing a suspended optical waveguide sensor oriented transverse to the flow (). All dimensions were based on experimentally realistic conditions or previously determined values. For example, to accurately determine the dimensions of the optical waveguide, we used the results from previous work which first demonstrated this suspended waveguide device [12
]. Similarly, the total height H
of the channel was varied between 25 and 100 μm to reflect typical polydimethylsiloxane (PDMS) microfluidic device dimensions [14
Figure 2. (a) Schematic of a 2-D cross-section of the flow cell and suspended optical waveguide sensor (not to scale). The sensor is located in the middle of the flow cell lengthwise. The flow cell height H = 50 μm, length L = 300 μm; waveguide (more ...)
In order to compare sample delivery efficiencies of substrate-bound and suspended sensors, we varied the vertical position h
of the suspended sensor within the channel (). This height is determined by the combination of isotropic and anisotropic etchants which are used in the device fabrication process; therefore, there is extremely good control (sub-μm) over this parameter. The precise range of attainable values of h
has yet to be determined for this young technology, but values in excess of 50 μm are routinely achieved. The elevation values we consider here fall within the experimentally demonstrated range of values of h
, and also include the limiting value of zero elevation (). We also modeled a simple rectangular sensor that is embedded in the channel floor (). This geometry is similar to that of slab waveguide sensors and surface plasmon resonance (SPR) sensors. The flat SPR sensor, first commercialized by Biacore, is a benchmark for comparison, and also serves to verify our results with past studies [16
A finite element mesh was generated to focus computation power on regions of the flow cell where the dependent variables were most influenced by position. The model was tested over a range of mesh element sizes to check for convergence and to ensure that the model had sufficient spatial resolution to capture relevant phenomena. The accuracy of our model was determined based on its ability to reproduce analytical results for simple cases. Additional details of how the computational model was built and validated, along with specifics of convergence tests and details of the finite element mesh used are included in the online supplementary information
Several assumptions were made in order to simplify the process of solving for the fluid velocity and analyte concentration profiles in the system. First, the 3-D geometry was reduced to the 2-D cross section along the length of the channel shown in . This is an acceptable approximation when the effective sensing area of the waveguide is situated in the middle of the channel and away from the sidewalls, as it is for the devices of interest, or when the channel width is very large relative to the channel height [20
]. The simulated channel extends six sensor widths upstream and six widths downstream from the sensor in the direction of fluid flow, for a total length of 300 μm. Additionally, we consider only binding of the analyte to the functionalized sensor and ignore any non-specific adsorption to the sensor or channel walls. This simplification is supported by recent advances in surface functionalization chemistry for gold and silica surfaces that significantly reduce the amount of non-specific binding [21
]. We further assume incompressible, laminar flow that enters the channel with a fully developed parabolic velocity profile. Since this inlet flow profile is symmetric about the mid-height of the channel, it is only necessary to consider sensor elevations ranging from mid-channel to the flow cell floor.
It is important to note that the parabolic flow profile is characteristic of a pressure-driven flow, which is the conventional method used for PDMS microfluidic channels [24
]. An alternative method is electrokinetic-driven flow. The flow profile for electrokinetic flow is inverted, with the high flow velocity on the boundaries and slower flow in the middle of the channel [27
]. However, electrokinetic flow requires a fluid that contains charged molecules. As a result, when comparing the maximum achievable flow rates and utility of the two methods, it is widely acknowledged that pressure driven flow is able to achieve higher flow rates and is applicable to a broader range of biological fluids [28
]. Therefore, we have focused our efforts on pressure-driven flow.
The adsorption of analyte to the sensor surface is approximated as (first-order) Langmuir binding [30
] according to the following reaction between a freely diffusing protein A with concentration [A] and an unoccupied binding site B with surface concentration [B] forming a bound complex Cs
with surface concentration [Cs
Mass action kinetics allow us to express the rates of the forward and reverse reactions in terms of the forward and reverse kinetic rate constants kf
, respectively, as:
where the dissociation
equilibrium constant is KD
. Each bound analyte molecule is assumed to occupy a single binding site. Equations (2)
can be used to relate the concentration of freely diffusing analyte at the sensor surface to the surface concentration of the bound species, a boundary condition based on the conservation of mass (see Equation (S13) in the online Supplementary Information
Interleukin 6 and anti-IL6.8 were chosen as the representative analyte and receptor, respectively. For this system, we used the parameters listed in , varying certain parameters for different studies. Interleukin 6 was chosen because it is involved with a range of important biologic functions. For example, it plays a key role in the immune and neural systems, in hematopoiesis, and in acute phase response, where it is a sensitive physiological marker of systemic inflammation [31
]. Representative concentration profiles are shown in .
It is important to note that in optical devices, additional phenomena are present which can enhance the collection of particles by the sensor surface, including photophoresis and thermophoresis [33
]. The presence of these forces has been observed with micron-sized nanoparticles. However, the present work is studying IL-6, which is a nano-scale biological particle with minimal charge. Therefore, based on the fundamental governing equations for the forces, the effect would be negligible [36
We investigated two critical measures of sensor response time to characterize the collection efficiency: (1) equilibration time and (2) time of detection. For the present work, the equilibration time teq
was defined as the length of time required for the surface concentration of bound complex to reach 95% of its equilibrium value. This arbitrary but convenient percentage was chosen to minimize the error in identifying this threshold time that can arise as the surface concentration asymptotically approaches its equilibrium value, while still giving a realistic approximation of how long the device takes to equilibrate. The limit of detection [Cs
, and corresponding detection time td
, represent the lowest concentration of surface-bound analyte that can be detected, and the length of time taken to achieve this value, respectively. In our studies, [Cs
was set at 10 pg/mm2
(equivalently 3.85 × 10−10
, based on a representative molecular weight of 26 kDa for IL-6), which corresponds to approximately 2.5% coverage of the suspended sensor surface. This conservative value was selected based on the current experimental results using waveguide biosensors [7
]. Both of these metrics are commonly used in characterizing the performance of biosensors and other types of chemical detectors [2
In characterizing the sensor response, we also accounted for the time delay that may occur as analyte is carried from the flow cell inlet to the sensor via advection. For example, at in
m/s, it would take ~1 s for the antigen molecules introduced at the flow-cell entrance at time zero to travel 150 μm downstream to the waveguide and begin binding. We defined the start of binding as the time t1
when the average surface concentration is equivalent to a single molecule of bound analyte per micrometer of sensor length (for details, see Section 3.1 of the supplementary information
The problem of convection, diffusion and reaction to traditional surface-bound flat planar sensors has been studied extensively, both via simulation and experiment [44
], and powerful theoretical methods exist to characterize the regimes of operation of such devices based on the values of a few key dimensionless numbers, and thereby approximate their binding behavior over a vast range of operational parameters [20
]. Our general approach to analyzing suspended biosensor behavior in microfluidic channels builds upon this knowledge and intuition by treating the upper and lower surfaces of the suspended sensor as individual flat planar sensors located in an analogous two-channel system (). This analytical approach complements our simulation results in providing insights into the probable behavior of the new sensor geometry.
Figure 3. (a) Schematic of the suspended optical waveguide sensor in the flow cell (not to scale). h1 and h2 are defined as the distances from the mid-plane of the sensor to the top and bottom surfaces of the channel, 1 and (more ...)