Deformation-induced actuation of cells in asymmetric periodic flow fields

Analyzing and sorting particles and/or biological cells in microfluidic devices is a topical problem in soft-matter and biomedical physics. An easy and rapid screening of the deformation of individual cells in constricted microfluidic channels allows, for example, the identification of sick or aberrant cells with altered mechanical properties, even in vast cell ensembles. The subsequently desired softness-specific segregation of cells is, however, still a major challenge. Moreover, aiming at an intrinsic and unsupervised approach raises a very general question: How can one achieve a softness-dependent net migration of particles in a microfluidic channel? Here, we show in a proof-of-principle experiment that this is possible by exploiting a deformation-induced actuation of soft cells in asymmetric periodic flow fields in which rigid beads show a vanishing net drift.

Figure 1

Representative bright-field images of RBCs in a microfluidic channel (width $10\mu \mathrm{m}$, height $100\mu \mathrm{m}$) with flow rates (a) $Q_1=800\mu \mathrm{l}/\mathrm{h}$ and (b) $Q_2=200\mu \mathrm{l}/\mathrm{h}$ pointing to the right and left, respectively. Stronger deformations at higher flow rates are clearly visible; scale bar: $2\mu \mathrm{m}$. (c) Quantifying the cell deformation $\delta$ [Eq. (1)] confirms this visual impression: Deviations of individual RBCs from a circular shape (for which $\delta =0$) were significantly larger for $Q_1$ (red squares) than for $Q_2$ (black circles). Solid lines and shaded regions indicate the respective mean and standard deviation.

Figure 2

(a) Applying a periodic and asymmetrically switching dichotomous flow rate $Q\left(t\right)$ in the microfluidic channel (blue line; forward: $Q_1=1600\mu \mathrm{l}/\mathrm{h}$ for ${\tau }_1=10$ s; backward: $Q_2=400\mu \mathrm{l}/\mathrm{h}$ for ${\tau }_2=40$ s) induces average maximum bead velocities that oscillate between plateau values $\langle v\rangle \approx +1.4$ mm/s and $\langle v\rangle \approx -0.4$ mm/s (black line). Please note that the periods ${\tau }_1$ and ${\tau }_2$ are sufficiently long to allow for a fair relaxation of the average velocity $\langle v\left(t\right)\rangle$ to the stationary value after reverting the flow. (b) Integrating $\langle v\left(t\right)\rangle$ over time yielded an average oscillatory excursion path $x\left(t\right)$ for this asymmetric periodic flow rate (black line). For comparison, the path obtained for a symmetrically switching flow rate ($Q_1=Q_2=400\mu \mathrm{l}/\mathrm{h}$ and ${\tau }_1={\tau }_2=10$ s) is also shown (red line, multiplied by five for better visibility). In both cases, the minima in $x\left(t\right)$ do not show signatures of a net drift. (c) In line with this observation, the deviation $\mathrm{\Delta }x=x\left(nT\right)-x\left(0\right)$ from the initial position after $n$ cycles is nonzero in both cases due to experimental imperfections, but no progressive increase (as expected for an actuation) is observed. This is in qualitative and quantitative contrast to soft RBCs [cf. Fig. 3].

Figure 3

(a) Comparing the ensemble-averaged position $x\left(t\right)$ of beads (black) and RBCs (red) in the same experiment reveals a marked progression of the minimum position for RBCs already after a few periods $T$ (highlighted by the magnification in the blue rectangular region; dashed-dotted lines indicate the minima positions). (b) During the strong forward flow phase, RBCs attain a higher velocity than beads (red vs black line), whereas the softer but longer backward flow during the second phase is very similar. Gray dashed lines indicate the times at which flow rates were reverted. (c) As a result, RBCs show a significant net actuation from the initial position for progressing periods, as quantified via the minima positions $x_{\text{min}}\left(nT\right)$ while beads remain basically at the initial position [cf. Fig. 2]. The dashed line is a linear guide to the eye.

Figure 4

Tracking individual RBCs (see main text and Supplemental Material [16] for details) reveals a net actuation $\xi$ that grows linearly with the number of periods, in agreement with the average drift of an ensemble of RBCs displayed in Fig. 3. In addition, fluctuations around the linearly growing average reflect the individual cell's response to the deforming flow as RBCs vary in size, stiffness, etc.