On the Quantification of Cellular Velocity Fields

Abstract

The application of flow visualization in biological systems is becoming increasingly common in studies ranging from intracellular transport to the movements of whole organisms. In cell biology, the standard method for measuring cell-scale flows and/or displacements has been particle image velocimetry (PIV); however, alternative methods exist, such as optical flow constraint. Here we review PIV and optical flow, focusing on the accuracy and efficiency of these methods in the context of cellular biophysics. Although optical flow is not as common, a relatively simple implementation of this method can outperform PIV and is easily augmented to extract additional biophysical/chemical information such as local vorticity or net polymerization rates from speckle microscopy.

Figure 1

Techniques for flow extraction. (A) One-dimensional illustration of the intensity in an image at time t (blue) and t + Δt (yellow), corresponding to the net rightward motion of two bright features. (B) In PIV the user defines an interrogation area that encompasses identifiable features within the image. This region (dashed line) at time t is then rastered over the image at t + Δt and the cross-correlation function is used to determine the best match for the new location of that feature. (C) Optical flow constraint uses the change in intensity between two images along with the intensity gradient to determine the velocity. The image intensity is first blurred to spread out information in the image over a larger region (i.e., to produce greater overlap between the image features at t and t + Δt). The change in intensity is then described by Eq. 1, and one uses an LSM to determine the velocities within subregions of the image by computing the change in intensity and intensity gradients from processed image pairs. To see this figure in color, go online.

Figure 2

Validation of optical flow (cyan) and single-pass (yellow) and four-pass (magenta) PIV. (a–c) The accuracy of each methodology was determined by computing flow fluids from synthetic movies exhibiting (a) spatially uniform motion, (b) a dilute density of particles rotating in a single vortex, and (c) a dense population of particles rotating in multiple vortices. The overlays show discrepancies between optical flow and four-pass PIV. (d) L²-relative norm errors for all test cases. To see this figure in color, go online.

Figure 3

Recreating flows in collective cell migration using tracer particles. (a and b) The trajectories of simulated tracer particles, calculated from velocity fields extracted by optical flow using blurred (cyan) or unblurred (green) images, or extracted using single-pass (yellow) or four-pass (magenta) PIV, were used to determine the accuracy of the measured flow displacements in (a) confluent MDCK type I monolayers (scale bar, 50 μm) and (b) dense suspensions of E. coli (scale bar, 50 μm). These tracer particles are overlaid on the velocity field (blue) that was extracted using optical flow with blurring. To see this figure in color, go online.

Figure 4

Measuring F-actin flow and polymerization rates using optical flow. (a) The accuracy of measuring net assembly rates by adding a reaction term in optical flow was determined by comparing the known net polymerization rate of fluorescent synthetic particles (top panel) with one calculated using the augmented version of optical flow (bottom panel). The root mean square of the error between the computed and known polymerization rates normalized by the root mean square of the known polymerization rate is 0.5. (b) The flow of F-actin in newt lung epithelial cells was computed using optical flow augmented with a reaction term (top panel) and speckle tracking (bottom panel, from Ponti et al. (9)). Arrows show velocity vectors and colors represent the flow speed. (c) The net assembly rate of F-actin as a function of distance from the border that was calculated using optical flow agrees well with Ponti et al.’s (9) data that illustrate the rate profiles (s⁻¹) as a function of distance (microns) (inset, black line). The bottom panel of (b) and inset in (c) are reprinted with permission from Science and Gaudenz Danuser. To see this figure in color, go online.