Hmm, this could potentially be solved using a mix of pixel classification and cell (nucleus) detection.
First, create the appropriate stain vectors for the PAS staining via Preprocessing > Estimate stain vectors. This will be useful in improving the classification accuracy of the subsequent pixel classifier.
Next, we need to identify cell borders. Either train a pixel classifier based on manually annotating regions of cytoplasm, nucleus, and cell membrane areas, and include the deconvolved stains as some of the channels to train on. Alternatively, if your deconvolution looks crisp enough to delineate the structures using the individual dyes making up PAS, create a pixel thresholder based on the stain corresponding to cell membrane. Whichever method, the end result should be a single annotation corresponding to membrane staining.
As previously mentioned, not all membranes are clearly visible in the image, rather most appear to be fragmented. Find the distance of the largest gap, and dilate the annotation by that amount via Annotations > Expand Annotations. This should bridge any gaps. Perform an erosion of the annnotation by the same amount, by repeating the previous step but supplying the negative value of the aforementioned distance. You should now have an annotation corresponding to all cell membranes, with most gaps patched.
Perform nucleus detection via Cell Detection > Cell Detection. Set the cell size growth to 0, so that we only have nuclei detected. Now, to identify the cell area, we’ll need to calculate the distance of each detected nuclei to the nearest cell membrane annotated area. I’m not sure if this needs to be done via scripting, but if that’s the case, you may have to adapt this script originally meant for calculating annotation-to-annotation distances, into nucleus-to-annotation distances Making Measurements in QuPath · GitHub. Could potentially achieve this by transforming nucleus detections into annotations, but it’s up to you.
Once we have the distances of each nucleus to the nearest cell membrane area, that’ll serve as our radius for calculating cell area. We’ll have to make a very bold assumption that the cells can be loosely modeled as a circle, and square and multiply by pi (i.e. pi*r^2 formula for determining area of a circle given a radius). It’s not ideal, but the closest I can think of given the high degree of variability in membrane staining.
Or, have a gaggle of summer students painstakingly annotate the thousands of cardiomyocytes in this image, outlining every cell membrane and nucleus in the image. And hey, maybe you could train a cell segmenter with those annotations in Ilastik/Cell Profiler, who knows!