The structure of a three-FET sensor cell is (from Wikipedia):

Now, how it works is very simple. The Mrst allows some current to Msf's gate, so there will be a fixed amount of charge there, then shuts off. Then the photodiode will draw some charge, the higher the light intensity, the more charge will be discharged. Msf will just amplify that final charge value and convert it to current, and Msel just select this cell when there is need to read this value.

Now, you will have to know how a photodiode works.

First of all, it should respond to incoming light intensity linearly:

(check out the left bottom quadrant).

However, photodiodes have wavelength-sensitivity:

so there is no clear one-to-one mapping between light intensity and pixel value.

Also, there are additional factors here, the most important is the Bayer filter. As you can see, R, G and B pixels have different intensity response areas:

This basically shows the diode sensitivity + the color filter effect.

So, to answer your question: is there a mathematical relation between these two? No, because

• sensitivities are frequency dependent,
• incoming light is not pure spectrally,
• sensors do not have a mathematical model (there are non-linearities, etc.)
• and let's not even go into the definition of "lumen", which involves frequency dependent intensity response of the human eye...

However, there are tools that can simulate photodiode physics and calculate the electrical response (latter is HSPICE or similar tool), and those can also simulate the entire circuit response to different incoming light intensity and spectra. (Randomly picking one: see this link).