Spreading in Distinct Directions and Comparison to Ellipse

As the holes in the structured substrates alternate with interspaces, the cell en-countered different conditions on these two regions of the substrate. Hence, the analysis has to describe the spreading behavior locally. One way to obtain a local description is to analyze the spreading along straight lines of different directions, which provides aspreading lengthdepending on the position along the cell outline.

One disadvantage of simply computing this spreading length is that, when com-paring these lengths, effects due to cell shape superimpose smaller differences that may show alterations in spreading behavior. To compensate for these influences, an ellipse was calculated as reference that had the same orientation, eccentricity and area as the cell. To this end, first, the orientation as well as major and minor axis of an ellipse that has the same normalized second central moments as the cell were calculated. Secondly, the length of minor and major axis were scaled so that the ellipse adopted the same area as the cell while keeping the eccentricity con-stant. By comparing the spreading length with the extension of the ellipse along the straight lines, effects of overall cell shape (e.g.eccentricity) can be neglected.

The nomenclature for this analysis is as follows (see also table 4.1): Points lying on the ellipse are denoted asq_ellipse(b,t), points lying on the cell outline asq_cell(c,t), points lying on the straight lines asq_line(d,α), points of intersection between the

vThis was only the case in the first images after attachment to the substrate.

Chapter 4 DATA ANALYSIS

straight lines and the cell outline as p_cell(β,t)and points of intersection between the straight lines and the ellipse as p_ellipse(β,t). b ∈ [1,N_ellipse] defines a specific point on the ellipse with N_ellipse being the total number of points on the ellipse.

c ∈ [1,N_cell] defines a specific point on the cell outline with N_cell being the to-tal number of points on the cell outline. Lastly, d ∈ [_1,Nline] defines a specific point on the straight lines with N_line being the total number of points on the line.

α ∈ [0^◦, 165^◦] with 15^◦ increments represents the angle the straight line encloses with thex-axis of the image andβ={α,α+₁₈₀^◦}^vi.

Table 4.1.: Abbreviations used in this section to describe variables needed to compute vectors from ellipse to cell outline.

q_line(d,α) points situated on the straight lines q_ellipse(b,t) points situated on the ellipse q_cell(c,t) points situated on cell outline

p_ellipse(β,t) points of interception between the straight lines and the ellipse p_cell(β,t) points of interception between the straight lines and the cell

out-line

~vcell-ellipse(β,t) vector between the points p_cell(β,t)andp_ellipse(β,t) lcell-ellipse(β,t) signed length of vector~v_{cell−ellipse}(β,t)

For easier comparison, the images of time lapse series on structured substrates were rotated by the imrotate function (method: bilinear, bounding box: loose) to align the rows of holes in the underlying substrate with the image borders. The angle needed to align the substrate with the image borders was found as described in section 4.4. Afterwards, the images of the platelets were binarized again^vii. Then, the cell area, the cell centroid as well as the length of major and minor axis and orientation of the above described ellipse that has the same normalized second central moments as the cell were computed using the functionsbwlabel (4-connected objects^viii) andregionprops. The major and minor axis of the ellipse were scaled so that the ellipse has the same area as the cell but the eccentricity was kept constant. Based on the centroid, the length of the major and minor axis and the orientation of the ellipse, points situated on the ellipse can be calculated following

viIf the straight lines go through any point that lies within the cell and not only touches the outline of the cell (see below), two points of interception were found and thus one was defined as belonging toαwhile the other one belonged toα+₁₈₀^◦. The same holds true for the ellipse.

viiDue to the interpolation when rotating the binarized image, especially pixels at the cell outline can have values differing from 255.

viiiThis means that a pixel belonged to the cell if at least one of its neighboring pixels had a value of non-zero. Neighboring pixels were those pixels directly above/below or left/right of the pixel and not the ones diagonal to it.

52

Analysis of Non-Fixed Platelets 4.3 the instruction given in [35]. The number of calculated points on the ellipseN_ellipse

was set to 120.

The straight lines, along which the spreading was evaluated, were computed by the following equation:

y(x) =y0+ (x−x0)·m (4.2) with m = tan(α). Additionally, the lines went through the mean center of mass (centroid) of the cell (x0,y0). The mean center of mass was calculated by aver-aging the center of mass positions of the cell in all frames containing a binarized and filled cell. An example for these lines is shown in figure 4.6, where also the definition of the angles is shown.

To ensure that for p_cell(β,t) only points were detected that were situated on the cell outline, this outline was extracted by the functionbwperim(8-connected neigh-borhood^ix) and used for further analysis. If all of the points q_cell(c,t)in an image had at least a distance of 2 pixels to the mean center of mass and the mean center of mass lay within the cell, the image was analyzed. Other images were omitted, since the large movement of the center of mass of the cell would have influenced the results. The further analysis detected points of interception between the cell outline and the straight lines as well as points of interception between the ellipse and the straight lines (p_cell(β,t) and p_ellipse(β,t), respectively). A sketch of how the pointsp_ellipse(β,t)were detected can be found in figure 4.7.

ixThis means that a pixel belonged to the outline if at least one of its neighboring pixels had a value of zero. Neighboring pixels were all pixels around the pixel.

Figure 4.6.: Plot of the computed straight lines for one cell.

The line of 0^◦ and the direction of following an-gles are marked. Fur-thermore, the mean cen-ter of mass is marked.

Chapter 4 DATA ANALYSIS

To maximize the number of points that were detected, the number of points on the lines N_line was increased by computing values for approximately every 0.2 pixel (inx-direction).

To detect p_cell(β,t) and p_ellipse(β,t), the x- and y-values of the points q_cell(c,t)as well as of the points q_ellipse(_b,t) were compared to those on the lines q_line(_d,α)_. To detect p_cell(β,t), those values on the cell outline were searched that had the same x- and y-values as the rounded values on the straight lines^x. The search was directly terminated after a point p_cell(_β,t) was detected and the same pro-cedure was performed again to find the second point of interception on the op-posite site of the cell outline. To detect p_ellipse(β,t), the x- and y-values of the points q_ellipse(b,t) were compared to those on the lines q_line(d,α). If the dis-tance between the points q_ellipse(c,t) and q_line(d,α) was smaller than 1.5 pixels,

xDue to the pixelated nature of the cell outline rounding to the nearest natural number was neces-sary.

Figure 4.7.: Illustration of detection of the points p_ellipse(β,t). The sketch shows the situation for one specic direction and one time point and illustrates how one point of interception was detected, the second point on the opposing site was detected accordingly.

a) The sketch shows a part of a cell outline (blue), a part of the corresponding ellipse (orange) and a part of the straight line (green) along which the points p_ellipse(β,t)were detected.

b) Detail of the sketch in a) showing how a point p_ellipse(β,t)was detected. 1. If a point on the line q_line(d,α) (green) lay less than1.5pixels (indicated by magenta circle) apart from a point on the ellipseq_ellipse(c,t), a pointp_ellipse(β,t)was detected. 2. The search was terminated after a point (nal point p_ellipse(β,t)marked with pink circle) was detected and the same procedure was performed again to nd the second point of interception on the opposing site of the ellipse.

For α=90^◦points on the ellipse were searched that did not deviate by more than1 pixel from thex-position of the mean center of mass (not sketched here). In order to prevent detection of the same point, the detected points on opposite sites of the ellipse had to be spaced more than 2.5pixels apart.

54

Analysis of Non-Fixed Platelets 4.3 a point p_ellipse(β,t)was detected. The search was directly terminated after a point

p_ellipse(β,t) was detected and the same procedure was performed again to find the second point of interception on the opposite site of the ellipse. In order to prevent detection of the same point, the detected points on opposite sites of the ellipse had to be spaced more than 2.5 pixels apart. As points p_ellipse(β,t) the points on the straight line were given. In the case of α = 90^◦xi, points on the ellipse p_ellipse(β,t) were detected that had an x-value that differed no more than 1 pixel from the x-value of the mean center of mass. Again, the detected points on opposite sites of the ellipse had to be spaced more than 2.5 pixels apart (see above). Asp_ellipse(β,t)the values for the points on the ellipse were given. For the cell outline and α = ₉₀^◦, points were searched that had the same x-value as the rounded x-value of the mean center of mass^xii. In order to determine whether the detected points p_cell(β,t)andp_ellipse(β,t) belonged to the angle β = α or to the angleβ=α+₁₈₀^◦ (see figure 4.6), the position of the points in relation to the po-sition of the mean center of mass was taken into account. Finally, the differences between p_cell(β,t)and p_ellipse(β,t)in x- andy-direction dx(β,t)/dy(β,t)as well as the lengths of the corresponding vectors~vcell-ellipse(β,t)between p_cell(β,t)and p_ellipse(β,t)were calculated. An example of the result of this analysis is shown in figure 4.8. In this figure the ellipse is shown in black, the cell in blue, the points of interception between straight lines and cell outlinep_cell(β,t)are shown in red, the points of interception between straight lines and ellipse p_ellipse(β,t) in cyan and the resulting vectors~vcell-ellipse(β,t)between the ellipse and cell outline are shown in magenta.

4.3.4 Further Data Analysis