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Valerio Vitali^{a},
Tie Yang^{b} and
Paolo Minzioni*^{a}
^{a}University of Pavia, Dept. of Electrical, Computer and Biomedical Engineering, Via Ferrata 5A, 27100 Pavia, Italy. E-mail: paolo.minzioni@unipv.it; Fax: +39 0382 422 583; Tel: +39 0381 985 221
^{b}School of Physical Science and Technology, Southwest University, Chongqing 400715, China

Received
25th October 2018
, Accepted 13th November 2018

First published on 20th November 2018

The development of lab-on-chip microfluidic systems based on acoustic actuation, and in particular on the acoustophoretic force, has recently attracted significant attention from the scientific community thanks, in part, to the possibility of sample sorting on the basis of both geometrical and mechanical properties. It is commonly recognized that sample prefocusing and launch-position optimization have a substantial effect on the performance of these systems but a clear explanation of how these two parameters influence the system efficiency is still missing. In this manuscript we discuss the impact of both the sample launch position and the sample distribution at the input by the theoretical analysis of a simplified system and by numerical simulations of realistic configurations. The results show that the system performance can be greatly improved by selecting the proper microchannel dimensions and sample-launch position, offering relevant guidelines for the design of micro-acoustofluidic lab-on-chip devices.

Acoustofluidics has been successfully applied to many different research studies, ranging from micro-droplet production and manipulation to micro-particle (or cell) sorting, focusing, separation, mixing and arraying.^{16–27} Among these possible applications, cell separation has attracted a lot of attention including substantial effort devoted towards enabling the isolation of circulating tumor cells from human blood samples.^{28–36}

Several groups have demonstrated the possibility of isolating target cells from a given sample containing a mix of cells with different characteristics using acoustofluidics.^{37,38} A theoretical analysis of particle separation efficiency in acoustophoretic devices was recently reported and it showed that both intrinsic factors, related to the sample itself, and extrinsic factors, related to the microfluidic system, can strongly affect the separation result. Among all the factors, the sample launch position into the active region (i.e. where the acoustophoretic force is present) plays a critical role.^{39}

The aim of this paper is to investigate how the launch position of the sample inside the channel changes the separation efficiency, and to demonstrate how the best launch position depends on different parameters: the aspect ratio of the channel, the cross-section occupied by the sample distribution at the channel input and the radius of the target bead/cell sample. In the following we focus our attention on investigating the challenging situation where the micro-objects to be selected and separated show a small deviation of their properties from the other beads/cells flowing along the channel.

It is worth underlining that in our analysis we neglect the effect of gravity, which can be helpfully used to produce a sample separation in the vertical direction,^{41} and the acoustic streaming-induced drag force (generally relevant for particles much smaller than cells).

One of the main parameters affecting the movement of particles exposed to acoustic waves is the so called acoustic contrast factor (φ), which is given by the following equation where ρ_{p} and ρ_{f} are the densities of the suspended microparticles and fluid, respectively, and β_{p} and β_{f} are the corresponding compressibilities.^{40}

(1) |

The acoustic contrast factor is one of the main quantities appearing in the expression of the acoustophoretic force applied to a microsphere in a plane standing wave, see eqn (2), where R is the beads/particles radius, y is its position in the transverse direction, E_{ac} is the acoustic energy density in the microchannel, and k_{y} represents the acoustic wave number.

(2) |

By imposing the Stokes drag force to be equal to the acoustic force it is possible to calculate the transverse coordinate of the particle as a function of time, as shown in eqn (3).^{40} In that equation we identify as y_{0} the position, along the y-axis, occupied by the particle when it enters the area of the microchannel where the acoustic wave is present.

(3) |

Regarding the particle movement in the x-direction, we assume that it moves together with the fluid, whose velocity v_{x}(y,z) in the microchannel is given by eqn (5).^{39} To derive that equation we assume a rectangular microchannel of height h, width w, we identify with ∇P the pressure gradient and with v_{c,∞} a constant factor, defined by eqn (4).

(4) |

(5) |

Regarding the z-direction, conversely, we assume that no significant movement occurs during the time that the microbeads flow along the microchannel. We also initially set Δz_{0} = 0, even if we will remove this assumption in Section 4.2.

The above equations allow us to calculate the movement in 3D of a particle flowing in the considered microfluidic system, and thus they also allow us to define the bandwidth (BW), displacement (D) and separation-efficiency (SE) parameters, as already reported in ref. 39. To briefly recall those definitions, we identify as D_{i}(x) the distance traveled by beads belonging to the i-th population along the y-direction with respect to the starting position, y_{0}, and we name BW_{i}(x) the spreading, in the y-direction, of the i-th population.

Starting from these two quantities, the SE parameter is calculated, at each position in the x direction, by the ratio of two distances as shown by eqn (6). As a numerator we use the difference between the distances traveled by two beads, with different characteristics, and injected in the microchannel at the same position y_{0}. Conversely at the denominator we sum the two single-side bandwidths with the bead radius. As a result the SE parameter calculated by eqn (6) is greater than 1 when the two populations are completely separated by a distance larger than one bead radius (see Fig. 2).

(6) |

Fig. 2 Left scale: transverse position along the width direction of two bead populations with different acoustic contrast factors (80% and 100% of that associated with polystyrene beads in water) while they flow along the x-direction. The inset shows a zoomed-in view of the injection region (“C” stands for the center of the injection area, “W” stands for the border of the injection area closer to the microchannel wall and “N” stands for the border closer to the node of the acoustic wave). Right scale: the SE calculated according to eqn (6), and as a function of the x coordinate, shows a peak at about 800 μm where the two populations are well separated. |

An analysis of the intrinsic (i.e. sample-related) and extrinsic (system-related) parameters affecting the SE can be found in ref. 39.

(7) |

This allows us to rewrite eqn (3) as:

y(t) = k^{−1}arctan{tan[ky_{0}]exp[At]}
| (8) |

We use this compact form to develop an analytical study of the simplified 1D-system which, although an over-simplification, provides useful insights into the impact of different parameters on the separation efficiency (SE), as defined in eqn (6). Using this notation, a change in the particle properties is reflected by a change of the A parameter, and it is thus possible to define an average A and a deviation ΔA. Similarly, as anticipated in Section 2, Δy_{0} represents the uncertainty of the input position, i.e. the maximum variation from the desired value y_{0}.

(9) |

Using this definition, it is possible to rewrite eqn (6) as:

(10) |

If we then approximate the differences appearing in the eqn (10) with the corresponding first-order differential terms we obtain:

(11) |

The above equation can be rewritten in a more useful form by calculating the two derivatives so as to obtain an explicit expression for the SE evolution as a function of time, or of flown-distance, as we consider uniform flow speed in the channel. If we then set ΔA = ΔA_{r} × A, where ΔA_{r} is the relative variation of the A parameter we obtain:

(12) |

In the following we identify with an asterisk (*) the values yielding the best performance, thus and t* are the launch-position and time-instant corresponding to the maximum separation efficiency (SE*) achievable between two populations. In particular, thanks to eqn (12), it is possible to derive some non-trivial considerations about , t* and SE*:

• C.(1) The role of ΔA_{r}:

With the given hypothesis the SE(t) (and hence SE*) is directly proportional to ΔA_{r}, but ΔA_{r} has no effect on the y_{0} value maximizing the SE. It thus means that the optimal launch position of the sample in the channel doesn't depend on how large the sample property variations are.

• C.(2) The maximum-separation instant (t*):

The time instant giving the maximum SE cannot be explicitly calculated as the resulting equation is transcendental, but we observe that in eqn (12) the time always appears in the product At. This implies that the maximum-separation instant t* is inversely proportional to A, i.e. the product At* does not depend on A. It is thus useful to rewrite eqn (12) in a more compact way by defining the following terms:

(13) |

The SE achieved at t* can thus be calculated as shown in eqn (14), which has two important characteristics: E is a constant while both F and G only depend on the value of Δy_{0}/R.

(14) |

• C.(3) The ratio Δy_{0}/R:

Following the above considerations we observe that if both Δy_{0} and R are multiplied by the same quantity, SE (t*) is left unmodified and thus also the optimal launch position doesn't change. Nevertheless, it is important to highlight that even if we keep Δy_{0}/R constant, a variation of R affects both the value of t* and that of SE(t*) as they are proportional to R^{−2} and R^{−1}.

In this simplified situation depends only on the ratio Δy_{0}/R. By analyzing Fig.(3) it is interesting to notice that the ideal injection position is close to the border when the ratio Δy_{0}/R is almost zero, it rapidly increases when Δy_{0}/R grows from 0 to 1 (going from 0% to 12% of w) and then it almost saturates around 20% of w when Δy_{0}/R gets much larger than 1.

After this study of the 1D system it becomes relevant to analyze, by means of numerical simulations, how the system properties are affected by moving to a 2D and 3D system, so as to verify if the above considerations are still valid.

Symbol | Value | |
---|---|---|

Water | ||

Density | ρ_{f} |
998 kg m^{−3} |

Compressibility | β_{f} |
4.48 10^{−10} Pa^{−1} |

Viscosity | η_{f} |
8.94 10^{−4} Pa s |

Sound speed | ν_{f} |
1483 m s^{−1} |

Beads | ||

Density | ρ_{p} |
1050 kg m^{−3} |

Compressibility | β_{p} |
2.49 10^{−10} Pa^{−1} |

Beads radius | R_{p} |
3.75 10^{−6} m |

Microchannel | ||

Pressure gradient | ∇P | 20 Pa m^{−1} |

Acoustic energy density | E_{ac} |
1.0 J m^{−3} |

All the simulations were carried out using a custom MATLAB script based on the equations reported in Section 2.2. The beads trajectory at the x–y (2D systems) and x–y–z (3D systems) coordinates as a function of time were calculated combining eqn (3) and (5), thanks to the use of “ODE45” function of MATLAB. The trajectories were calculated over a time vector, composed by 2000 uniformly spaced values, whose time-step depends on microbeads properties and on the channel geometry, as they impact on the time required by beads to reach the microchannel center. As a general indication, the calculation of each point appearing in the figures from 4 to 6 required about half an hour of computation time on a 4-cores processor at 3.50 GHz and with 16 GB of RAM.

It is possible to investigate different microchannel aspect-ratios in two different ways: keeping one dimension fixed and varying the other one (e.g. fixed height and variable width), or by simultaneously modifying both dimensions so as to keep the cross-sectional area constant. In the following, unless otherwise specified (in the final part of Section 4.2), we keep the microchannel area fixed at 9 × 10^{−2} mm^{2}, corresponding to a square microchannel with a side of 300 μm. The dependencies and trends observed for this specific area are of general validity.

The numerical analysis was initially carried out to verify whether or not the considerations expressed in Section 3 are still valid in the 2D case. In order to verify the previously reported consideration C.(1), we considered in our numerical simulations distinct beads populations having a φ (see eqn (1)) different from that of standard PS. In Fig. 4 we show the values as a function of the microchannel aspect-ratio (w/h) and considering four different bead-population pairs.

Fig. 4 (Best injection position expressed as a percentage of the channel width) as a function of channel's aspect ratio. Parameters used for this simulation: R = 3.75 μm; φ = 0.5; Δy_{0} = 10 μm. Values in the legend correspond to ΔA_{r} value in eqn (12). |

The first pair is composed by beads having an acoustic contrast factor (φ) equal to 98% and 102% of the nominal PS value, and it is thus indicated as ±2% in the legend (corresponding to the ΔA_{r} value in Section 3). In an analogous way, we considered population-pairs with an increased difference of φ, up to the ±20% case, which corresponds to bead-populations having an acoustic contrast factor equal to 80% and 120% respectively of that of PS. As reported in Section 3, it is possible to notice that the position doesn't depend on how large the sample properties variations are, provided that the difference is not too big. As it is evident considering the ±20% case, if the φ variation becomes too large the position may start to vary, as the first-order approximation used to derive eqn (11) is not sufficient anymore.

Nevertheless, as the most critical separation situation is when small differences are present between the sample populations, this limitation is not particularly relevant for our study. It is also interesting to notice that the value depends on the aspect ratio of the microchannel cross-section (indicated as w/h in the figures), but it becomes almost constant when w > 10h, as it approaches the 1D-situation theoretically analyzed in Section 3. As a comparison it is interesting to notice that data used to create Fig. 4 yield a ratio Δy_{0}/R ≈ 2.7 which corresponds in Fig. 3 to , exactly matching the numerically calculated position for large aspect-ratios.

Subsequently we also verified that the value does not depend on the absolute value of φ. To analyze this aspect, we considered three different bead-population pairs, with largely different values of the nominal φ factor (0.05, 0.5 and 5), while keeping Δφ = ±5%. The results show that no change of the value is induced by modifying the nominal φ of the populations, independently of the channel aspect ratio (see Fig. 5).

Fig. 5 vs. channel's aspect ratio (w/h) when three different values of φ are considered: 0.5, 5 and 0.05. Other simulation parameters: R = 3.75 μm; Δφ = ±5%; Δy_{0} = 10 μm. |

We then moved to verify the dependence of on the Δy_{0} and R parameters (i.e. consideration C.(3) of Section 3). According to what previously reported we expect the value to depend on Δy_{0}/R, but not on Δy_{0} and R separately. To assess this dependence, we compared the ideal for five different configurations of Δy_{0} and R, while keeping constant the acoustic contrast factor of the two populations (φ = 0.5; Δφ = ±5%). The five parameter-set considered in the numerical simulations are schematically reported in Table 2. We included three different combinations of Δy_{0} and R yielding the same Δy_{0}/R ratio (8/3), and two different combinations yielding a four-times increase and decrease of the Δy_{0}/R value (32/3 and 2/3 respectively).

Color | Symbol | Δy_{0} [10^{−6} m] |
R [10^{−6} m] |
Δy_{0}/R |
---|---|---|---|---|

Blue | Triangle | 10 | 3.75 | 2.67 |

Red | Diamond | 20 | 7.5 | 2.67 |

Green | Square | 5 | 1.875 | 2.67 |

Black | Circle | 20 | 1.875 | 10.67 |

Cyan | Triangle | 5 | 7.5 | 0.67 |

The results clearly highlight that even in this case, as in the 1D situation previously considered, the optimal launch position depends on Δy_{0}/R, and is thus unmodified if both values are multiplied by the same factor (see Fig. 6).

Fig. 6 as a function of w/h considering different Δy_{0}/R combinations. The values of Δy_{0} and R are as reported in Table 2. Other simulation parameters: φ = 0.5; Δφ = ±5%. |

To complete the 2D-approximation analysis of the launch position we created a figure to show the overall dependence of the parameter on Δy_{0}/R and w/h, which are the only two parameters affecting the value. The result of the numerical simulations is reported in Fig. 7 (left panel) as a color-map. Calculations were carried out considering a nominal φ = 0.5, Δφ = ±5% and R = 5 μm, but the reported results have a much more general validity as derived by the above reported analysis.

Fig. 7 Optimal injection position (, left) and corresponding separation efficiency (SE*, right) as a function of Δy_{0}/R and of channel aspect ratio in the 2D case. |

In the same conditions we also calculated the SE* value, defined according to eqn (6), achievable by proper selection of the injection point, as a function of Δy_{0}/R and w/h. The obtained results, reported in right panel of Fig. 7, show the benefit of using large aspect ratios and the advantages given by a reduction of Δy_{0}, which can be achieved, as an example, by using a prefocusing section.^{42}

The results of this analysis demonstrate two important aspects: (i) the importance of optimizing the launch position to improve the SE and (ii) that the value, expressed as a percentage of the microchannel width, depends only on microchannel aspect ratio (w/h) and on the Δy_{0}/R ratio, but not on other sample factors. Anyway, it is important to highlight that the whole analysis reported up to this section completely neglects the vertical dimension of the microchannel.

In particular we define the BW parameter of each population as the maximum distance (in the y-direction) between two beads, i.e. considering at each section along the x-axis the bead closer to the microchannel wall (at half of the channel height) and the one closer to the microchannel center (and closer to microchannel floor). The distance |D_{1} − D_{2}|, which represents the distance between the “centers” of the two beads distributions, is calculated as the distance between the centers of the two population-bands.

It is important to notice that, as it is evident by Fig. 8, choosing an injection height different from the center of the channel can only worsen the system performance. The flow-speed gradient in fact becomes larger as we move away from the middle-height position and thus the beads distribution becomes wider. For this reason, we considered in our analysis the impact of a vertical spreading Δy_{0} while keeping the center of the injection channel fixed at half-height of the microchannel. Regarding the Δy_{0} parameter it should be noticed that it is possible to define it as a given percentage of the microchannel height, or by its own value (in μm).

To keep consistency with the previous analysis we initially consider the case of Δy_{0} defined as a fixed percentage of the microchannel height. We decided to start our analysis considering a Δz_{0} equal to 5% of the channel height, while keeping all the other parameters set as for the final 2D simulations: nominal φ = 0.5, Δφ = ±5% and R = 5 μm. As in the previous case we calculated the optimal launch position , and the corresponding separation efficiency (SE*) as a function of the microchannel aspect ratio and of the Δy_{0}/R value.

The data reported in Fig. 9 (left panel and right panel) show two partially surprising results: the obtained in the 3D case exactly matches that obtained in the 2D approximation and also the SE* figure matches that obtained in the 2D case, once rescaled by a constant factor. The reason for these results is that the presence of a vertical spreading implies the presence of beads flowing at a different height, where the flow speed is simply scaled (by a factor smaller than 1) with respect to the flow speed at half-height of the channel. As no distortion of the speed profile is introduced, the value for beads flowing at half height and for those flowing at any distance from the channel bottom is the same, provided that beads' interaction with the bottom surface can be neglected. A direct consequence of this is that even analysis carried out using a larger vertical spreading (e.g. Δz_{0} equal to 10% or 15%) would yield the same results and thus do not bring any additional information.

The obtained values show that even a minor vertical spreading can have a significant impact on the achievable SE in case of microchannels with high aspect ratio: as an example, a 5% vertical spreading in the microchannel with aspect ratio 25 corresponds to a Δz_{0} as small as ±3 μm, and yields a SE* reduction almost by a factor of 2. On the other side, the use of microchannels with a smaller aspect ratio, although yielding a lower SE* value in the ideal case of Δz_{0} = 0, is expected to be significantly more tolerant to the vertical spreading. We thus investigated the performance of microchannels with different aspect-ratios while fixing Δz_{0} equal to ±5 μm and ±10 μm.

The color-maps obtained in these conditions do not add any relevant information with respect to Fig. 9 and they are thus not reported in the manuscript. Conversely, it is interesting to analyze the data reported in Fig. 10, showing the achievable SE in the above described conditions and with a fixed value of Δz_{0}. The reported maps highlight that, once Δy_{0}/R and Δz_{0} are given, it is possible to identify the ideal channel cross-section and then the achievable SE.

It is interesting to notice that while in the 2D case an aspect ratio as large as possible was desirable (see right panel of Fig. 7), in the 3D case the presence of a non-negligible Δz_{0} suggests the use of higher channels, so as to mitigate the effect of the vertical spreading of the sample. As a consequence of the necessity to find a trade-off between the mitigation of horizontal and vertical spreading, the ideal aspect ratio has a non-obvious dependence on both Δy_{0}/R and Δz_{0}.

Additionally, it is worth mentioning that in the above reported discussion we exclusively focused our attention on the acoustic radiation force, applied on the flowing particles because of the sound-waves scattering, while we neglected the acoustic streaming effect and the related drag-force. Following the analysis reported by Muller et al.^{43} it is possible to show that, in order to neglect the acoustic streaming effect, the particle diameter must exceed by a few times the boundary-layer thickness δ.

(15) |

(16) |

As a reference value, considering cells with φ = 0.15 and R = 3 μm the maximum value of w allowing to neglect the acoustic streaming effect is ≈1000 μm, which may thus impose a limitation on the achievable microchannel aspect-ratio (w/h).

It is interesting to notice that thanks to a careful optimization of the injection position of the sample, high SE values can be obtained even in case of no prefocusing techniques. This allows largely simplifying the design and the operation of the microfluidic systems.

- N. de Souza, Nat. Methods, 2011, 9, 35 CrossRef.
- C. Liberale, G. Cojoc, F. Bragheri, P. Minzioni, G. Perozziello, R. La Rocca, L. Ferrara, V. Rajamanickam, E. Di Fabrizio and I. Cristiani, Sci. Rep., 2013, 3, 1258 CrossRef CAS PubMed.
- K. Ahn, C. Kerbage, T. P. Hunt, R. Westervelt, D. R. Link and D. A. Weitz, Appl. Phys. Lett., 2006, 88, 024104 CrossRef.
- K. Khoshmanesh, S. Nahavandi, S. Baratchi, A. Mitchell and K. Kalantar-zadeh, Biosens. Bioelectron., 2011, 26, 1800–1814 CrossRef CAS PubMed.
- P. Minzioni, R. Osellame, C. Sada, S. Zhao, F. Omenetto, K. B. Gylfason, T. Haraldsson, Y. Zhang, A. Ozcan, A. Wax, F. Mugele, H. Schmidt, G. Testa, R. Bernini, J. Guck, C. Liberale, K. Berg-SÃÿrensen, J. Chen, M. Pollnau, S. Xiong, A.-Q. Liu, C.-C. Shiue, S.-K. Fan, D. Erickson and D. Sinton, J. Opt., 2017, 19, 093003 CrossRef.
- T. Yang, F. Bragheri and P. Minzioni, Micromachines, 2016, 7, 90 CrossRef PubMed.
- N. Bellini, K. Vishnubhatla, F. Bragheri, L. Ferrara, P. Minzioni, R. Ramponi, I. Cristiani and R. Osellame, Opt. Express, 2010, 18, 4679–4688 CrossRef CAS PubMed.
- S. Z. Hua, F. Sachs, D. X. Yang and H. D. Chopra, Anal. Chem., 2002, 74, 6392–6396 CrossRef CAS PubMed.
- T. Yang, Y. Chen and P. Minzioni, J. Micromech. Microeng., 2017, 27, 123001 CrossRef.
- N. Pamme and A. Manz, Anal. Chem., 2004, 76, 7250–7256 CrossRef CAS PubMed.
- N. Xia, T. P. Hunt, B. T. Mayers, E. Alsberg, G. M. Whitesides, R. M. Westervelt and D. E. Ingber, Biomed. Microdevices, 2006, 8, 299 CrossRef CAS PubMed.
- J. Friend and L. Y. Yeo, Rev. Mod. Phys., 2011, 83, 647–704 CrossRef.
- G. Destgeer and H. J. Sung, Lab Chip, 2015, 15, 2722–2738 RSC.
- H. Bruus, J. Dual, J. Hawkes, M. Hill, T. Laurell, J. Nilsson, S. Radel, S. Sadhal and M. Wiklund, Lab Chip, 2011, 11, 3579–3580 RSC.
- H. Bruus, Theoretical Microfluidics, Oxford University Press, Oxford, 2008, p. 364 Search PubMed.
- J. Brenker, D. Collins, H. V. Phan, T. Alan and A. Neild, Lab Chip, 2016, 16, 1675–1683 RSC.
- Y. N. Cheung, N. T. Nguyen and T. N. Wong, Soft Matter, 2014, 10, 8122–8132 RSC.
- I. Leibacher, P. Reichert and J. Dual, Lab Chip, 2015, 15, 2896–2905 RSC.
- D. J. Collins, A. Neild and Y. Ai, Lab Chip, 2016, 16, 471–479 RSC.
- J. Shi, H. Huang, Z. Stratton, Y. Huang and T. J. Huang, Lab Chip, 2009, 9, 3354–3359 RSC.
- C. Devendran, I. Gralinski and A. Neild, Microfluid. Nanofluidics, 2014, 17, 879–890 CrossRef CAS.
- X. Ding, J. Shi, S.-C. S. Lin, S. Yazdi, B. Kiraly and T. J. Huang, Lab Chip, 2012, 12, 2491–2497 RSC.
- C. Devendran, N. R. Gunasekara, D. J. Collins and A. Neild, RSC Adv., 2016, 6, 5856–5864 RSC.
- R. W. Rambach, V. Skowronek and T. Franke, RSC Adv., 2014, 4, 60534–60542 RSC.
- M. K. Tan, L. Y. Yeo and J. R. Friend, EPL, 2009, 87, 47003 CrossRef.
- C. E. Owens, C. W. Shields, D. F. Cruz, P. Charbonneau and G. P. López, Soft Matter, 2016, 12, 717–728 RSC.
- S. H. Kim, M. Antfolk, M. Kobayashi, S. Kaneda, T. Laurell and T. Fujii, Lab Chip, 2015, 15, 4356–4363 RSC.
- F. Petersson, L. Aberg, A.-M. Swärd-Nilsson and T. Laurell, Anal. Chem., 2007, 79, 5117–5123 CrossRef CAS PubMed.
- A. H. J. Yang and H. T. Soh, Anal. Chem., 2012, 84, 10756–10762 CrossRef CAS PubMed.
- X. Ding, Z. Peng, S.-C. S. Lin, M. Geri, S. Li, P. Li, Y. Chen, M. Dao, S. Suresh and T. J. Huang, Proc. Natl. Acad. Sci. U. S. A., 2014, 111, 12992–12997 CrossRef CAS PubMed.
- O. Jakobsson, S. S. Oh, M. Antfolk, M. Eisenstein, T. Laurell and H. T. Soh, Anal. Chem., 2015, 87, 8497–8502 CrossRef CAS PubMed.
- P. Patil, M. Madhuprasad, T. Kumeria, D. Losic and M. Kurkuri, RSC Adv., 2015, 5, 89745–89762 RSC.
- M. Antfolk, C. Magnusson, P. Augustsson, H. Lilja and T. Laurell, Anal. Chem., 2015, 87, 9322–9328 CrossRef CAS PubMed.
- P. Augustsson, C. Magnusson, M. Nordin, H. Lilja and T. Laurell, Anal. Chem., 2012, 84, 7954–7962 CrossRef CAS PubMed.
- C. Grenvall, C. Magnusson, H. Lilja and T. Laurell, Anal. Chem., 2015, 87, 5596–5604 CrossRef CAS PubMed.
- M. Antfolk, C. Antfolk, H. Lilja, T. Laurell and P. Augustsson, Lab Chip, 2015, 15, 2102–2109 RSC.
- A. Urbansky, P. Ohlsson, A. Lenshof, F. Garofalo, S. Scheding and T. Laurell, Sci. Rep., 2017, 7, 17161 CrossRef PubMed.
- P. Augustsson, J. T. Karlsen, H.-W. Su, H. Bruus and J. Voldman, Nat. Commun., 2016, 7, 11556 CrossRef CAS PubMed.
- T. Yang, V. Vitali and P. Minzioni, Microfluid. Nanofluid., 2018, 22, 44 CrossRef.
- R. Barnkob, P. Augustsson, T. Laurell and H. Bruus, Lab Chip, 2010, 10, 563–570 RSC.
- T. Kanazaki and T. Okada, Anal. Chem., 2012, 84, 10750–10755 CrossRef CAS PubMed.
- G. Nava, F. Bragheri, T. Yang, P. Minzioni, R. Osellame, I. Cristiani and K. Berg-Sørensen, Microfluid. Nanofluid., 2015, 19, 837–844 CrossRef CAS.
- P. B. Muller, R. Barnkob, M. J. H. Jensen and H. Bruus, Lab Chip, 2012, 12, 4617–4627 RSC.

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