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Label-free optical biosensors based on integrated photonic devices have demonstrated sensitive and selective detection of biological analytes. Integrating these sensor platforms into microfluidic devices reduces the required sample volume and enables rapid delivery of sample to the sensor surface, thereby improving response times. Conventionally, these devices are embedded in or adjacent to the substrate; therefore, the effective sensing area lies within the slow-flow region at the floor of the channel, reducing the efficiency of sample delivery. Recently, a suspended waveguide sensor was developed in which the device is elevated off of the substrate and the sensing region does not rest on the substrate. This geometry places the sensing region in the middle of the parabolic velocity profile, reduces the distance that a particle must travel by diffusion to be detected, and allows binding to both surfaces of the sensor. We use a finite element model to simulate advection, diffusion, and specific binding of interleukin 6, a signaling protein, to this waveguide-based biosensor at a range of elevations within a microfluidic channel. We compare the transient performance of these suspended waveguide sensors with that of traditional planar devices, studying both the detection threshold response time and the time to reach equilibrium. We also develop a theoretical framework for predicting the behavior of these suspended sensors. These simulation and theoretical results provide a roadmap for improving sensor performance and minimizing the amount of sample required to make measurements.

Label-free optical biosensors based on integrated photonic devices are able to accurately detect chemical and biological molecules over a wide concentration range in real-time [

The performance of an integrated label-free biosensor is typically characterized according to its sensitivity or limit of detection for a given analyte of interest, with researchers continuously pushing the boundaries on sensitivity. In parallel, moderate consideration is usually given for specificity through the development of surface chemistries for targeting the analyte of interest. However, minimal effort is expended on optimizing the collection efficiency of the sensor. By increasing the collection efficiency, it is possible to significantly reduce the amount of sample required for a measurement and compensate for a lower sensitivity device. Minimizing sample consumption is critical for analytical applications, particularly those involving rare or valuable materials. As such, any improvements to the collection efficiency can reduce the cost and increase the ease of conducting experiments that allow for further optimization of sensitivity and specificity. Therefore, because of the balance between sensitivity and collection efficiency, the optimization of collection efficiency and the device sensitivity should occur in parallel.

In the present work, we explore how the vertical placement of a suspended optical device within a microfluidic channel can influence its collection efficiency through a series of finite element method simulations. This work is focused on integrated waveguide biosensors, which have demonstrated the ability to detect bacteria, cells and proteins in complex environments [

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 (

In order to compare sample delivery efficiencies of substrate-bound and suspended sensors, we varied the vertical position

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

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

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 [

The adsorption of analyte to the sensor surface is approximated as (first-order) Langmuir binding [_{s}_{s}]:

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 _{f}_{r}_{D}_{r}/k_{f}

Interleukin 6 and anti-IL6.8 were chosen as the representative analyte and receptor, respectively. For this system, we used the parameters listed 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 [

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 _{eq}_{s}]_{min}, and corresponding detection time _{d}_{s}]_{min} was set at 10 pg/mm^{2} (equivalently 3.85 × 10^{−10} mol/m^{2}, 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 [

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}^{−4} 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 _{1} 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

The problem of convection, diffusion and reaction to traditional surface-bound flat planar sensors has been studied extensively, both via simulation and experiment [

We begin by discussing flow around the suspended sensor. Consider the average velocity of fluid in the regions directly above and below the sensor (_{1} and _{2} respectively). In general, _{1} and _{2} depend on the size of the gaps _{1}_{2}

For a derivation of _{H}_{H}

Next we consider mass transport. The dimensionless flux (Sherwood number) is a useful generalized metric of the rate at which mass transport can deliver analyte molecules to the sensor surface [_{D}

This number compares the time needed for a particle to diffuse across a channel of height _{h}_{c}_{c}

In contrast to the embedded sensor, whose operation is characterized by a single Pe_{c} value, separate Pe_{c}_{c}_{s}^{−1/3}. Here:

Conversely, when Pe_{c}

Finally, we have the more complex scenario where Pe_{c,1}_{c,2}_{g}_{s}_{1} is the height of the upper channel (equivalently the height of the gap) and _{in}

For a given set of model parameters, we can easily calculate the equilibrium concentration, [C_{s}]_{eq}_{m}_{s}]:

Given enough time and an adequate supply of analyte, the concentration of analyte bound to the sensor surface will invariably approach this value. A quantitative analysis of convergence to [C_{s}]_{eq}_{D}^{−12} M arises as a natural concentration scale. When [A]_{0}/_{D}_{s}]_{eq}_{m}_{0}/_{D}_{s}]_{eq}_{m}_{0}/_{D}_{0} > 0.025 _{D}_{0} > ∼200 fM. However, since this estimate is highly conservative, it is possible that we will be able to detect solutions several orders of magnitude more dilute.

The approach to equilibrium is characterized by the Damköhler number, which is defined as the ratio of reactive flux to diffusive flux at the sensor surface [

Here _{0}/_{D}_{R}_{f}_{0})^{−1} and the reaction time-scale merely reflects how quickly the forward reaction proceeds. On the other hand, at low concentrations ([A]_{0}/_{D}_{R}_{r}^{−1}). When Da » 1, kinetics are mass transport-limited and the surface concentration increases linearly with time as analyte is delivered to the sensor surface. The binding time-scale in the transport-limited case is Daτ_{R}

To analyze the binding response of the present sensor, we performed simulations over a range of conditions with the sensor located at the mid-plane of the channel. Representative results are plotted in _{c}^{4}_{1}_{2}

The preceding analysis and the master curve of

_{R}

_{1}^{7} bound antigen molecules per millimeter length of sensor, irrespective of geometry. This particular value is equivalent to our earlier specification of [C_{s}]_{min}

The suspended sensor significantly outperforms the substrate-bound sensor in terms of detection time. At all but the slowest modeled flow velocities, the mid-channel placement offers a greater than twofold reduction in relative detection time over the conventional planar geometry. A long flow cell upstream of the sensor will diminish this performance difference by increasing all

We have developed a finite element model that simulates advection, diffusion and specific binding of IL-6 to the antibody-functionalized surface of a novel suspended waveguide biosensor in a microfluidic channel. We use this model to characterize sensor behavior for a range of average flow velocities, inlet antigen concentrations, and surface-immobilized antibody concentrations. Device performance is evaluated according to two common sensor metrics: the detection time and the equilibration time.

Our model predicts that the detection and equilibration times will have a weak dependence on flow velocity, while inlet analyte concentration can greatly impact the kinetic response of the device. Our results further show that reducing the surface antibody concentration can extend the range of conditions over which the sensor is reaction-limited. However, transport-limited conditions may be reached at sufficiently low flow rates, high analyte concentrations and high antibody concentrations on the surface. These conditions can lead to long equilibration times, require significant sample volumes in order to make measurements, and make it difficult to determine kinetic rate constants for the surface binding reaction, making them ill-suited for most biosensor measurements.

We also compare the specific binding to waveguide sensors suspended at varying heights in the channel with that for planar sensors on the channel floor. The sensor suspended at mid-height in the channel shows shorter detection times than the flat device due to its thinner boundary layers and increased active sensing area. Though equilibration times are similar for these two geometries, sensors elevated only slightly above the channel floor yield significantly slower equilibration times than either the planar substrate or the sensor at mid-height of the channel. The small gap between the suspended sensor and the channel floor at small elevations hinders the efficient mass transport via advection to the bottom side of the device. Our results indicate that suspended sensors can display a range of transient behaviors, with efficiencies either greater than or less than those of traditional planar sensors, depending upon their elevation. Given that lithographic methods are used to fabricate these and other semi-conductor based optical sensors, the elevation of the sensor within the channel can be optimized to locate the device within the region of highest sample delivery to ensure the largest possible performance improvements.

These findings, together with the general framework developed in this paper for analyzing and predicting suspended sensor behavior, provide a basis for optimizing the performance of novel suspended waveguide biosensors in microfluidic channels, leading to reduced sample consumption and improved response time of these devices.

The authors would like to thank Jason Gamba (University of Southern California and Intel) for helpful discussions. This work was supported by the National Science Foundation (1028440), the Office of Naval Research (ONR) Young Investigator Program (N00014-11-1-0910) and the Rose Hills Foundation Science and Engineering Fellowship.

Suspended waveguide splitter/coupler biosensor. (

(_{in}^{−4} m/s, [A]_{0} = 100 nM, [B]_{m}^{−9} mol/m^{2}, and others as in

(_{1}_{2}_{1} and _{2} are the average flow velocities through the regions above and below the sensor, and _{1}_{2}_{1} and _{2} respectively. Our approach is built on the intuition that the binding behavior of the suspended sensor might be reasonably approximated by the sum of the binding behaviors of these two flat planar sensors, which represent its upper and lower surfaces.

Average flow velocity in the region above or below the sensor as a function of the size of the gap on that side relative to the total channel height. The slow and fast limit expressions are compared with simulation data for a range of Reynolds numbers and geometry parameters.

Binding time _{R}_{R}^{−1} ≈ 63% of the equilibrium concentration. Results from a representative subset of the studies performed are presented, where the channel height _{R}

Binding curves for various sensor geometries (inlet bulk analyte concentration [A]_{0} = 100 nM, total surface concentration of binding sites [B]_{m}^{−8} mol/m^{2}, and average inlet flow velocity _{in}^{−5} m/s). All times are normalized with respect to τ_{R}

Response times as a function of sensor elevation/geometry and Sensor Péclet Number Pe_{l}_{in}l_{0} = 100 nM and total surface concentration of binding sites [B]_{m}^{−8} mol/m^{2}). Note that in these simulations,

Model parameters.

_{f} |
Association rate constant | 9 × 10^{6} L/mol·s |

_{r} |
Dissociation rate constant | 6 × 10^{−5} s^{−1} |

_{D} |
Dissociation equilibrium constant | 6.67 × 10^{−12} M |

[B]_{m} |
1.66 × 10^{−9} mol/m^{2} to 1.66 × 10^{−8} mol/m^{2} | |

Bulk diffusion coefficient of IL-6 in water | 1 × 10^{−10} m^{2}/s | |

[A]_{0} |
Inlet concentration of IL-6 in the bulk solution | 10^{−13} M to 10^{−7} M |

_{in} |
Inlet average fluid velocity | 5 × 10^{−5} m/s to 0.4 m/s |

Rate constant values are from Rispens

Upper limit of the range was estimated from the measured size of a single human IgG antibody [

Estimated from data on diffusion coefficients as a function of molecular weight [