Determining the Angle between cells

[Impartial D]

Hi,

I'm currently an undergraduate engineering student and am doing research for the summer writing programs to help analyze and study cell motility and cooperativity when introduced to certain growth factors. For specific growth factors there is a noticeable swarming movement amongst neighboring cells in our time lapse data while other growth factors induce distinct coordinating movement amongst groups of cells. As of right now I'm trying to write the portion of my program to calculate the angle between neighboring cells, the problem I'm having is maintaining a fixed origin and still returning the angle I desire. My data consists of x and y coordinates, thus my first instinct was to treat these points as 2D vectors and use the dot product to calculate the angle. However, in some of these paths there are very sharp turns where the angle I'd like to find is the obtuse angle of the turn. I've also thought of using the law of cosines but I'm not sure that it would render more accurate results than the dot product. As of now I'm not sure what direction to take my calculation and I was interested in hearing any other input on what type of calculation I possibly should think of using. Thanks again and please feel free to ask any questions as I will gladly provide more information or just answer any questions in general.

 

[pmsrw3]

Ha. This is actually very similar to something I'm working on myself. So, to clarify, you have, for each cell, a series of (t, x, y) data, with the t's equally spaced, and x, y presumably being center of mass? And when you say "the angle between cells", you mean changes in direction of movement? You're not actually trying to calculate some sort of statistic that reflects the interaction between two or more cells?

 

[Impartial D]

Actually, in an essence that's what we are trying to do. I personally am studying chemical and biological engineering and my graduate student is a biochemist so between the two of us we don't necessarily have the applied math resources to develop a full algorithm to describe these interactions. However, with the data we've collected and the strategies we've utilized in our analysis we believe we should be able to prove our hypothesis about motility and cooperativity nonetheless. Right now, I'd really like to find a way to measure the direction/ angle of the cell movement using our t,x, and y data points.

 

[pmsrw3]

Actually, in an essence that's what we are trying to do. I personally am studying chemical and biological engineering and my graduate student is a biochemist so between the two of us we don't necessarily have the applied math resources to develop a full algorithm to describe these interactions. However, with the data we've collected and the strategies we've utilized in our analysis we believe we should be able to prove our hypothesis about motility and cooperativity nonetheless. Right now, I'd really like to find a way to measure the direction/ angle of the cell movement using our t,x, and y data points.

But, for now at least, all you're trying to do is analyze the movement of single cells in isolation? (I don't mean the cells are isolated, but that the analysis is done one cell at a time.)

So, as I said, I've been working on something very similar. I'm attaching the handout for a lab meeting I did recently describing this. (This doesn't have formulas, but I can send those if you wish.) The straightforward way to get the change in angle is to find the rotation that sends (x1-x0,y1-y0) to the x-axis, then apply that same rotation to (x2-x1,y2-y1). This deals correctly with the obtuse/acute angle problem you mentioned, and even preserves information about the direction of the turns, if you care.

This method has a big problem. When the cells doesn't move much, the measurement of direction is very inaccurate. This is a problem if either (x1-x0,y1-y0) or (x2-x1,y2-y1) is close to your position noise. So if you do this in the straightforward way, you end up with lots of large measured angles that really just mean the cell wasn't moving much. The attachment describes a different data reduction that somewhat gets around this.

 

[Impartial D]

Yep, I am indeed looking at something very similar to your research. I'll take a look at your slides and get back to you after I've been able to read them. Thanks again for all your help.