Supplemental Information. On the Quantification of Cellular Velocity Fields. Dhruv K. Vig, Alex E. Hamby, and Charles W. Wolgemuth

Transcription

1 Biophysical Journal, Volume 110 Supplemental Information On the Quantification of Cellular Velocity Fields Dhruv K. Vig, Alex E. Hamby, and Charles W. Wolgemuth

2 Biophysical Journal Supporting Material Biophysical Perspective On the Quantification of Cellular Velocity Fields Dhruv Kumar Vig, 1 Alex E. Hamby, 1 and Charles W. Wolgemuth 1, * 1 Department of Physics, University of Arizona, Tucson, Arizona *Correspondence: wolg@ .arizona.edu

3 Supplemental Movie Captions Supplemental Movie 1 Synthetic movie of a population of fluorescent ellipsoidal-shaped particles moving at constant velocity. Supplemental Movie Synthetic movie of a dilute population of fluorescent ellipsoidal-shaped particles being transported by a prescribed single vortex. Supplemental Movie 3 Synthetic movie of a population of fluorescent ellipsoidal-shaped particles being transported by a prescribed array of four vortices. Supplemental Movie 4 Following epithelial cell movements using tracer particles. The velocities in a confluent monolayer of MDCK cells were extracted using optical flow with blurring (cyan) or without blurring (green), single-pass PIV (yellow), and four-pass PIV (magenta). We then simulated the motion of tracer particles and overlaid the positions onto the original movie. The total movie duration is 5 hr, with image acquisition every 5 minutes. Images were acquired in DIC using a 0x air obective. Total width of the domain is 69 µm. Supplemental Movie 5 Following the swimming dynamics of a dense suspension of E. coli. The velocities were extracted in a similar manner as in Supplemental Movie 4. The total movie duration is 10 s, with an acquisition rate of 3 fps. Images were acquired in DIC using a 40x water-immersion obective. Total width of the domain is 30 µm.

4 The L relative error norm was calculated using the equation below. L ( vxm, vxp, ) + ( vym, vy, p) ( vxp, ) + ( vyp, ) where v x,m (v y,m ) is the x(y)-component of the measured velocity (Optical Flow determined velocity) and v x,p (v y,p ) is the x(y)-component of the prescribed velocity (velocity defined when generating the synthetic movie). I. Instruction manual for optical flow General Information The optical flow algorithm is free to use and is distributed in a hope that it will be useful to the broad research community. Copyright 015 Arizona Board of Regents on behalf of The University of Arizona. The use of this algorithm requires MATLAB, but it is not necessary to have MATLAB s Image Processing Toolbox. To get started place the OpticalFlow.m file in the directory that contains the movies to be processed or set the directory path for the file in MATLAB. II. Description of the optical flow algorithm The main inputs to the optical flow algorithm are a time-lapse sequence of images (in our MATLAB code we allow for inputs in either AVI or TIF format), a binary mask that defines the ROI where the velocity is to be computed, and parameters that govern the blurring, box size, velocity smoothing, and output velocity vector density. The ROI mask can be input as either a single image file or a stack of images (with a different mask for each frame). The time-lapse images are handled in pairs, e.g., the first frame with the second frame, the second with the third, and so on. To account for uneven lighting in the images, we begin by

5 subtracting the background from the images. The background is determined by fitting the intensity profile in the image to a cubic function using a Least Squares Algorithm. Each image in the pair is blurred using a radially symmetric Gaussian blur with user-defined kernel size with standard deviation BlurSTD. Blurring spreads the intensity profile over a distance comparable to the characteristic displacements in the movie. In the provided algorithm we defined BlurSize as ceil (3.5*BlurSTD) + mod (ceil(3.5*blurstd,) + 1. Over finite time intervals, t, the advection equation is approximately equal to ( ) ( ) ( v ) I I t+ t I t t I. The left-hand side of this equation is computed by subtracting the first image of the image pair from the second. The gradient in the intensity on the right-hand side of the equation is computed using a central difference approximation for the spatial derivatives of the average intensity profile for the image pair: I x I y t+ t t t+ t t ( Ii+ 1, + Ii+ 1, ) ( Ii 1, + Ii 1, ) 4 x t+ t t t+ t t ( I I + 1 ) ( I 1 + I 1) 4 x where the subscript denotes the x,y pixel location of the node and x is the linear pixel size. Over small subregions of the image (of size BoxSize BoxSize), the velocity is approximately constant. The BoxSize should be set in such a way that there is at least one image feature per subregion. Within these subregions, the advection equation should be approximately correct; i.e., ( v ) I + t I R, where R is a small residual and the velocity v is a constant vector. To determine the velocity, we then minimize the squared residuals in the subregion with respect to ( ) the velocity. That is, we define an error function χ I + t( I ) 1 v subregion Minimizing this error with respect to the velocity leads to a linear system of two equations and two unknowns whose solutions are

6 v v x y I I C I B I subregion x + subregion y t AC ( B ) I I A I + B I subregion y subregion x t AC ( B ) The sums in the above equation are calculated using an image filtering operation, and, if a binary ROI has been used in the input, the velocities are only calculated within the masked ROI. This algorithm is conceptually similar to a method previously developed 1 ; however, our use of blurring and a least squares minimization over a small window allows for the calculation of instantaneous velocity fields where movements occur over tens of pixels between frames. The velocity field is then output on a coarser domain with spacing defined by the user (the arrow spacing is defined as ArrowSize). Figure 1 summarizes the optical flow algorithm.

7 Figure 1. Flow-Chart describing the optical flow algorithm. III. Using optical flow to generate flow fields. OpticalFlow.m requires the user to define the following variables: Parameters Definition MovieName Name of the image sequence file to be analyzed, can be either AVI or TIF format. BinaryMask Name of the region-of-interest image sequence files, can be either AVI or TIFF format. If an ROI mask is not necessary then the input is [ ]. scale Converts pixels to microns, scale is defined in microns per pixel. dt Time interval between frames BoxSize Sets the linear size of the subregions (in pixels) where the velocity is computed. Should be set to be large enough that each subregion contains at least one identifiable image feature. BlurSTD Sets the size of the standard deviation for the Gaussian blur. Should be set to half maximum velocity between two images in pixels. ArrowSize Used to define a coarser output grid for the velocity vectors. Defines the spacing (in pixels) between output velocity vectors. Optical To access an augmented mode of Optical Flow enter Rotation to determine the Flow local voriticity (ω ο ) or React to measure the effects of an added source term (γ). Method To run the standard default version use none or omit this input parameter. 1. Input the desired values for these parameters in MATLAB s command prompt window by replacing the name of each parameter with their desired value, an example is shown below. >>[X,Y,Vx,Vy,Mov] OpticalFlow ('Movie1.tif',[],13,0,0.3145,5, none ). The outputs of the code are vectors X and Y which are the x and y position of the velocity vectors (V x, V y ) defined on the coarse grid. The velocities are output as M N matrices where the columns correspond to time and the rows correspond with the spatial positions stored in (X, Y). A MATLAB movie matrix, Mov, is also output that contains the overlaid

8 microscope image sequence with optical flow extracted velocities. This movie can be played back using the command movie (Mov). It can also be stored to the current directory by using >> movieavi (Mov, FileName.avi, compression, none ) This AVI video file is playable in ImageJ. 3. When optical flow with Rotation is used, there are two additional outputs. First is a matrix containing the Vorticity, which has the same format as V x and V y. The second is a movie file that shows the velocity field overlaid on top of a colormap depicting the vorticity. 4. When optical flow with React is used, there is one additional output, a matrix containing the reaction rate, which has the same format as V x and V y. Analyzing several movies within a directory. The file DriveOF.m (also provided in the supplemental software) can be used to automate optical flow and analyze several time-lapse image sequences in series. The file needs to be stored in the same directory as the movie files and OpticalFlow.m (or set the directory path for the file in MATLAB.) In the MATLAB command prompt launch DriveOF.m by typing: >> open DriveOF.m 5. In this driver the user will see the same list of parameters shown in the above Table. These variables are now set within the driver and not through MATLAB s command prompt. 6. It is now necessary to define a new variable TotalMovies, which corresponds to the total number of movies that the user wishes to analyze in series. 7. Automation is performed by assuming that all the movies within the directory have the same prefix and format, for example Movie1.av Movie.av etc. or Movie1.tif, Movie.tif, etc. The ROI Mask and Image sequences to be analyzed do not have to have the same format, but should have the same prefix. 8. An example using the driver in MATLAB s command prompt window is shown below. >> DriveOF ( Movie1.avi, [])

9 9. When DriveOF is used, workspace files and overlaid movie files for each movie will be created and stored in the current directory. These files will have the same name as the input movies and will be labeled as FileNameXX.mat (workspace file) and MovieXXTracked.avi (overlay movie file). Using extensions of optical flow Using DriveOF.m, it is possible to implement extensions of optical flow. One application where this can be useful is in fluorescent speckle microscopy, where speckles appear or disappear due to polymerization or depolymerization 18. In optical flow, it is straightforward to add a net source/sink reaction rate γ to Eq. 1 i.e. di dt v I +γ. Optical flow can also be augmented to determine the local vorticity within a flow field by assuming that the velocity consists of translational and rotational components, v v + ( r) o ω, where r xxˆ+ yyˆ and ω ω z o ˆ, where ω 0 is the vorticity. This form for the velocity is then used in Eq. 1 and the squared residuals are minimized with respect to v x, v y and ω o. The resulting system of equations can be solved analytically to determine the velocity and vorticity. To use these different versions of optical flow, the user needs to type Rotation or React into the DriveOF command in the MATLAB command window, as shown below. >> DriveOF ( Movie1.avi, Movie1Binary.tif, Rotation ) or >> DriveOF ( Movie1.avi, Movie1Binary.tif, React ) To use the standard version optical flow simply omit the third entry above or input none.