Existing PIV and cPTV methods fail to track large, local deformation.
To test if existing PIV methods are able to track a large, local deformation, we created a ground-truth traction field consisting of a large traction, i.e., >10 kPa, and a few spots of small tractions (Fig. 1A). We built a synthetic bead image by randomly super-positioning 2D Gaussians (Fig. 1B, red) and applyed the ground-truth displacement field, calculated by using the Boussinesq solution assuming E= 8 kPa (Fig. 1C,D see Methods), to the individual beads to create another bead image, mimicking cell adhesion presence (Fig. 1B, green). We processed the image pair using the three different, widely-used, available PIV software, namely PIV Suite, Tseng’s PIV, and mpiv, to track the displacement field (see Methods, Fig. 1E-J). The three methods resulted in slightly different displacement fields overall, in terms of magnitude, direction and spacing between adjacent vectors. However, one common trend is that they failed to track the large displacement vectors around the center region (Fig. 1F,H,I). Specifically, PIVSuite produced systematic underestimated displacements in the center of large force regime (Fig. 1F, green vectors) while Tseng’s PIV resulted in missed several seed locations for displacement tracking (Fig. 1H, red circles). In a similar manner, mpiv missed more locations in large force region for displacement estimation (Fig. 1J). We also tested a performance of cPTV by applying it to the same pair of the synthetic bead images. cPTV identifies individual beads, from which an interrogation is performed by the cross-correlation. A significance criterion is used to determine whether the global maximum of the correlation score is significantly higher than the second largest local peak. When a strict criterion is used, many vectors in the large force regime were missing as in mpiv (data not shown). When a more generous significance criterion is used, many large vectors in the large force region were obtained (Fig. 1K), but they were incorrect (Fig. 1L). However, it was noticeable that cPTV was able to distinctly capture the displacement vectors in the region of low forces (Fig. 1K). Filtering out vector outliers removed most of large, wrong vectors in large force region but preserved well-tracked small vectors, e.g., on the small force area (Fig. 1M,N).
To assess the tracking accuracy, we used the mean-squared-deviation (MSD) over the entire field of view of the displacement field resulted from each tracking method against the ground-truth displacement field (Fig. 1O). mpiv showed the most amount of deviation followed by cPTV, filtered cPTV, PIVSuite and Tseng’s PIV. A large portion of deviation came from the missing or wrong vectors at the large force area (Fig. 1J,L,N). Quantifying MSD at only the large force area (Fig. 1C, red ellipse) indicated that cPTV created the largest deviation, which was reduced by the outlier filtering, followed by mpiv, PIVSuite and Tseng’s PIV (Fig. 1P). The low MSDs by Tseng’s PIV and PIVSuite at the large force area were likely attributed to the interpolation of vectors in their algorithms (Fig. 1F,H). At the small force area, however, cPTV, with or without filtering, has shown the exceptionally lowest MSD against all other PIV variants (Fig. 1Q). This result demonstrates the superiority of cPTV to PIV methods, at least for small-force-driven deformation, as the interrogation starts at the center locations of individual beads whereas PIVs do from the centers of the gridded windows. Together, these data demonstrate that all current PIV and cPTV methods fail to track a large, local displacement field, and if cPTV gains further accuracy in the large displacement tracking, it will be the method of choice for force-induced displacement quantification.
Correlation fails at the large, local displacement.
To investigate the reason behind the failure in large deformation tracking, we inspected the correlation score and its global vs. local maxima for small and large deformation cases (Fig. 2). For the small displacement, for example, we have picked a vector position whose true displacement is less than 30 pixels and plotted the cross-correlation score map (Fig. 2A), equivalent to a top view of the surface plot of the score. Then we depicted the shift position of the global maximum, i.e., the measured displacement (Fig. 2A, a red circle), as well as the shift position of the ground truth vector (Fig. 2A, a purple circle). The two circles being at the same location for the small displacement represents that the image interrogation finds the vector successfully. To check the image matching visually, we also directly mapped the center location of the bead of interest on the reference image (Fig. 2B) and its shifted position on the image in the deformed configuration (Fig. 2C). Cropping and enlarging the template windows in the reference (undeformed) image and at ground-truth and measured positions in the bead image of the deformed configuration confirmed that the template images in both ground-truth and measured positions are identical to that in the reference image, and so was the correlation score (Fig. 2D).
Applying the same visualization pipeline to a large displacement vector showed that ground-truth and measured vectors are at different positions on the correlation score map (Fig. 2E), indicating that the ground truth is not at the global maximum position for the large, local deformation. Mapping and zooming the template images in the reference and deformed configurations showed that the ground-truth displacement has much poorer correlation with the reference template than with the global maximum by more than ten folds (Fig. 2F-H). Similarly, for a vector that was determined as missing due to failure to pass the significance criteria, we identified that the correlation at the ground-truth displacement was not the global maximum (Fig. 2I). The correlation score of the ground-truth displacement was not as low as in the larger displacement case in Fig. 2E-H, as the cropped image also shows also similarity with the reference template image (Fig. 2J-L). However, the fact that the score is smaller than the global maximum but still large enough made the vector fail to pass the significance criterion. Together, this investigation shows that the large displacement caused by a large local traction is unable to be tracked via cross-correlation due to the poor correlation specific to TFM application.
cPTV-Retracking algorithm captures missing displacements using neighboring well-tracked vectors.
From this simulation experiment as well as our TFM force reconstruction experience from the experimental bead images, we noticed that failures in displacement tracking occurs exclusively in the large vectors due to concentrated forces. We also learned that the failed large displacement vector has at least the similar direction as the neighboring vectors, despite the poor cross-correlation score. We have thus implemented a new algorithm, referred to as cPTV-Retracking, that tracks the missed or filtered-out vector positions again using information from the well-tracked neighboring vectors (Fig. 3). The algorithm is based on the output displacement field from the initial cPTV for the entire field of view in the bead image pair, with a strict significance criterion to produce only well-tracked vectors, from which we identify bead locations that has missed tracking. Per missing location, the algorithm collects neighboring vectors with a certain search radius, from which the average and standard deviation of the magnitudes and the orientations are calculated. Then the cross-correlation score is calculated over the limited range in the magnitude and the angle based on the average neighboring vector. To accommodate the tendency that large vectors are usually missing and need to be retracked, a more generous upper limit is defined compared to a lower limit (Fig. 3B). From the new score map, local maxima are found and compared with the global maximum using the same significance criteria. We found that the reduced number of local maxima increases a chance of the global maximum passing the significance criteria. If it does not pass the criterion, all the local maxima are compared with the ‘model vector’, which is built from the median neighboring vector instead of the mean. The median was chosen because we observed the neighboring vector magnitudes followed non-Gaussian distribution where there is a consistent bias toward ‘larger’ magnitude, and we expected the missing vectors are at least equal or larger than the neighbors. To test whether a candidate displacement is close enough to the model vector, i.e., thus selectable as the displacement vector for the location, we used one standard deviation of both magnitude and angle within which we choose the vector. If the candidate vector does not pass this test, the algorithm reduces the template window size gradually, e.g., by 2 pixels in width and height, and iterate the candidate vector identification process (Fig. 3A), until the template window reduces it size into a size that can contain almost a single bead. Upon successful finding, the found vectors become new neighbor vectors for retracking of another missing bead position. Fig. 3C shows an example of gradual ‘filling-in’ of retracked vectors from the existing confident vectors at several selected iterations.
Enlargement factor enhances chances of finding larger displacements.
To make sure if the median-based model vector of the local neighboring vectors is truly representative as a model vector, we introduced a new variable, i.e., the enlargement factor, to adaptively increase the magnitude of the model vector above the median. Fig. 4A summarizes the related algorithm to use the enlargement factor for finding larger displacements. It begins with initializing the factor as one and obtain the optimal search radius to con contain at least 3 neighboring vectors. Then we loop through positions that have sufficiently close-enough neighbors for retracking as described in Fig. 3A. At the initial retracking, we use just median of the neighboring vectors as a model vector. When there are no more successfully tracked vectors, then the algorithm increases the minimum number of neighboring vectors for the search radius determination and performs the retracking. If it does not generate any retracked vectors, the algorithm decreases the template window length by 2 pixel and increases the enlargement factor by 10 % and retrack the missing locations. Applying the enlargement factor increased the chance of finding likely large vectors compared to retracking with only the median vector (Fig. 4B,C), as expected. To make sure if the newly-found large displacements help improve the tracking accuracy, we measured MSD of the displacement field and compared with the field from cPTVR with the median model vector and the field cPTVR with filtering (Fig. 4D). We found that the MSD value of cPTVR with the enlargement factor, or cPTVR-EF, is much smaller than the other two results in both entire field of view (Fig. 4D) or specifically in the large force area (Fig. 4E). Comparison of the accuracy (see Methods) between PIV methods and the cPTVR variants confirmed that cPTVR with both median and the enlargement factor exhibited much higher, e.g., >90%, accuracy than other PIV methods, where cPTVR with the enlargement factor showed the highest (~92%) accuracy (Fig. 4F). Together, these data suggest that a model vector larger than a median increases the rate of finding large displacements in the TFM setting.
cPTVR allows for detection of large traction in force reconstruction.
To seek whether the retracked displacement field actually leads to a more accurate traction field, we reconstructed the traction field out of the displacement field tracked by cPTVR and compared it with the ground-truth traction field along with tractions produced based on displacement fields by other existing PIV and earlier cPTV methods (Fig. 5, See Methods). Compared to the ground-truth (Fig. 5A,B), we found that the traction fields by all PIV methods, i.e., PIV Suite, Tseng’s PIV and mpiv, were largely underestimated or not colocalizing with designed force locations (Fig. 5C-H), most likely due to their failure in large displacement tracking. cPTV without retracking also resulted in the traction field mostly missing large forces whereas it detected small forces well (Fig. 5I,J). cPTV-Retracking, when performed with a median as a model vector, showed small improvement in rescuing large forces, but it did so insufficiently by missing many larger force vectors (Fig. 5K,L). However, cPTVR performed with the enlargement factor resulted in the traction field in which much more large force vectors were reconstructed as well as small force vectors (Fig. 5M,N). Quantifying the mean-squared deviation confirmed that cPTVR with the enlargement factor, or cPTVR-EF was able to produce the traction field with least deviation from the ground truth traction field (Fig. 5O). The lowest MSD exhibited by cPTVR-EF was attributed to the reduced deviation over the large force region (Fig. 5P) as well as over the region of small forces (Fig. 5Q). Quantification of the vector accuracy also confirmed that cPTVR-EF-based traction produced the most accurate field compared to other methods (Fig. 5R). Together, the traction reconstruction and comparison suggest that cPTVR-EF enables better detection of the large forces by being able to detect the large, local displacement vectors.
cPTVR enhances the resolving power of TFM for different-sized focal adhesions.
To test whether cPTVR-EF, which we refer to just cPTVR from now on, can track an experimental bead images for TFM, we conducted a TFM experiment with human bone osteosarcoma epithelial (U2OS) cells on a 4 kPa silicone gel coated with 40 nm-diameter fluorescent beads. We collected the bead images under total internal reflection fluorescence microscope when cells were adhered to the gel surface and after they are released (Fig. 6A). The pair of the bead images were processed with PIVSuite, cPTV and cPTVR for displacement measurement (Fig. 6B-D). We picked PIVSuite as a representative PIV method based on the average performance shown compared to other PIV methods (Fig. 1O-Q). The displacement tracked by PIV Suite showed mostly smooth field except for the failure at one corner (Fig. 6B). The displacement by cPTV showed patches of large (>40 pixel in magnitude) displacements (Fig. 6C). However, it showed even more places missing displacement tracking (Fig. 6C, magenta arrowheads), consistent with the failure observed in the simulation study (Fig. 1M,N). When cPTVR was used to quantify the displacement field, there were many more large displacements vectors detected, mostly near the cell edge (Fig. 6D).
To assess the reconstructed traction fields’ reasonability, we transfected the U2OS cells with paxillin-GFP, imaged its fluorescence under TIRF, and segmented and detected focal adhesions (FAs), focal complexes (FCs) and nascent adhesions (NAs) as done previously (Fig. 6E) (2,17). The traction maps reconstructed from the displacement fields showed generally similar characteristics as in the displacement fields (Fig. 6F-H). Specifically, the smooth displacement field by PIV Suite resulted in the smooth, likely underestimated, traction field with missing traction vectors on one location (Fig. 6F, a magenta arrowhead). The traction from the displacement tracked by cPTV showed a bit more detailed, and less underestimated field. However, the missing displacement vectors caused the tractions to be less organized by containing significant forces outside the cell boundary and positions of voids (Fig. 6G, magenta arrowheads). Contrary to this, the traction from the displacement field by cPTVR contained much larger, e.g., >1.5 kPa, tractions with more coherent-looking traction distribution (Fig. 6H). Comparison of the traction magnitude among the three traction fields revealed that whereas tractions at NAs were insignificantly different, the tractions at FCs and FAs were higher for cPTVR-based traction compared to both of the other two methods (Fig. 6I), demonstrating cPTVR’s outstanding performance in restoring large forces. Plotting the traction separately per tracking method revealed that whereas there is no or only a little difference in traction between FAs, FCs and NAs by PIV Suite or cPTV (Fig. 6J,K), there was a significant difference in traction at FAs compared to ones at FCs and NAs when cPTVR was used for TFM (Fig. 6L). Altogether, these data suggest that cPTVR method enables resolving large traction forces transmitted by mature focal adhesions on a soft substrate by being able to rescue locally large displacement vectors.