Overall significance of multiple downward trends? I am uncertain how to formally establish that I observe a negative relationship between X and Y in multiple sets of linked data points.
I have analyzed several cells in which values of interest (Y) can be estimated at a variety of distances (X) from its center. In any given cell, I expect Y to decrease at greater distances from the center. Despite the expected negative relationship between X and Y, it could be that Y(x=100) > Y(x=30) if these values are derived from different cells. 
Since the individual downward trends (one for each cell) appear to be shifted with respect to one another, I hardly see a negative relationship if I lump all my Y values together. It is important that I consider the linkage between different Y values which originate from the same cell, shown in a single color in the graph below:

I considered doing a repeated measures ANOVA, but fear this is impossible because the X values for which Y is obtained vary between different cells. 
What kind of data transformation or test could I use to establish that, for any given cell, Y tends to decrease as X increases?
 A: What you need is a mixed effects model.  If your response variable is continuous, you could probably use a linear mixed model.  You will need a cellID indicator variable and you will have random intercepts (and possibly slopes, etc.) for each cell.  If that is all of the data you have, you won't have enough to do anything very sophisticated, but you could fit an LMM with random intercepts, which might be good enough for your purposes.  (From the plot, you do have different slopes, and red seems to have a strongly curvilinear relationship between X and Y, so ideally, you would have fixed and random effects that could account for that.)  
A: To make inference about this relationship you would need multiple tests within a cell. Is that possible? If it is, I would suggest to measure the difference in Y and X relative to the baseline (the smallest X). This way, in each cell the smallest X represents 0 and bigger X's are computed relative to that point. This solution will be problematic if the relationship between X and Y is not a simple linear relationship.
