6.2 Network morphologies in time-lapse studies
6.2.3 The convex hull of networks
We carried out time-lapse studies with the device T using the confocal micro-scope. The resulting images provide details on the networks of vimentin fila-ments (Fig. 6.8). The challenge is to quantify the different morphologies. In this analysis many factors play a role. Most importantly, bleaching effects occur due to repeated imaging of the same filaments. Therefore, analysis methods that sen-sitively depend on absolute intensity are not suitable.
For instance, the absolute value of the standard deviation depends on the overall intensity in the drop. For the measurements in the device C, this is not an issue, as the networks are imaged once and bleaching does not play a role (Fig. 7.1). The area of the binarized images is also in principle a measure for the expansion of the network in the drops. However, this method is highly influenced by bleaching.
For example, as the visible diameter of a bleached filament shrinks over time. Ad-ditionally, this parameter depends also on the imaging details, like the step size of the pixels in a scan. There are other methods that account for the mesh sizes or measures for the inter-filament distances in networks. For example, ‘bubbles’
are fitted into meshes giving an estimate of the mesh size of the networks. [86,89]
Statistics on the filament to filament distance are considered. [9, 62, 63] However, our networks are in most cases so dense that most of the details of the meshes are simply not resolved, and cannot be analyzed with these methods.
One distinct property of the networks is their pronounced compaction over time (Fig. 6.8, from a to d). Within this compaction the area of an envelope to the whole networks gets smaller over time. For a clear quantification of the area of such an envelope, the convex hull1 is considered. This quantification has two ad-vantages: First, the convex hull has a clear mathematical definition and is there-fore unique for each binary image. Second, the way the raw data is converted to binary images (Sec. 6.2.2) and therefore prepared for the convex hull analysis, is much more robust against bleaching effects compared to the above mentioned techniques. The convex hull for a binary image is found by the usage of the MAT-LAB function ‘regionprops’, which is fast and straightforward to use.
In Fig. 6.9, the examples of convex hulls are shown. In the first row, it becomes obvious that, if the convex hull is taken for the total binary image, it is
dispro-1The convex hull of a binary image is the smallest set of pixels that contains all non-zero pixels as well as all pixels that are in between a straight line between two arbitrarily chosen non-zero pixel of the binary image.
6 DATA ANALYSIS
100%
50%
90%
95%
time
network in hull area +
convex hull area network outside hull area
+ + + drop area color
code
outlier IC
A= 0.90
A= 0.92 A= 0.85 A= 0.62 A= 0.05
A= 0.90 A= 0.83 A= 0.29 A= 0.04
A= 0.04
A= 0.04 A= 0.83 A= 0.22
A= 0.90 A= 0.83 A= 0.22
Figure 6.9: Convex hull of binary images. For the binary images from Fig. 6.8, the convex hull is given. In the dierent rows, the convex hull is chosen in a way that it covers at least IC= 100%,95%,90% or 50%of the total network intensity (IC: intensity conserved). For cases of, IC = 90% and lower, the calculated hulls are robust against outliers that distort the hulls disproportionally. Additionally, the relative drop hull areaA is given. (Partly adapted from reference [35] with permission from The Royal Society of Chemistry)
portionately influenced by outliers. To make the convex hull a robust quantity, the intensity of the network is taken into account. The binary images consist of one or more connected regions (‘islands’). For the ith island, the intensity in the corresponding region of the raw data image is summed up to a value Si. The total intensity S of the network is then the sum over allS. Starting with the
is-Network morphologies in time-lapse studies 6.2
lands with the smallestSi, the islands are excluded from the convex hull until a certain fraction IC of total intensity in the drop is still conserved by the remaining islands. For example, for IC = 90%, small islands are removed untilat least90%
of the total intensity is still there. This concept leads to different outcomes of the convex hulls (Fig. 6.9). It is found that IC= 90% is a value, at which most of the low intensity outliers are excluded from analysis, while the characteristic part of the network is enclosed in the convex hull. For lower values than IC = 90%, the hull does typically not change, as the largest part of the intensity is in one big island.
The area of the convex hull (IC = 90%) characterizes the expansion of the char-acteristic part of the network in the drop. To compare between drops of different sizes, it is reasonable to normalize this area to the area of the drop, leading to the relative hull areaA(Fig. 6.9). For instance, when two drops of different sizes have a network that is spread everywhere in the drop, the networks should be characterized by the same quantitative value. The relative hull areaAis then the same for both networks, although the absolute hull area is not.