Problem choosing an equation for a NLS model I need to present academically I want to be able to predict the optical density of a radiograph (x-ray image) based on the input factors of depth, SID, KVP, MAS. I'm a lot more familiar with linear models and have never had to give a model an equation before, and frankly, I have no idea what the equation should be.
I've used the equation below and the model is successful and does fit well (to an extent) with the experimental data, so it does what I need to do. However, I want to choose an equation the 'proper' way however that might be as I want to present my research academically. I can find little to no guidance on this however online.
sa1<-nls(OD~ ((KVP^a)*(MAS^b))/((SID^c)*(e*DEPTH^d)),
         data=bathdata,
         start=c(a=0, b=0, c=0, d=0.1, e=0.1))

For reference OD increases> exponentially with increasing MAS
exponentially with increasing KVP
AND
OD decreases> exponentially with increasing SID exponentially with increasing depth
[I'm an undergrad student trying to model OD as part of a project to optimise images and thus reduce radiation dose to bariatric patients]
Thank you for any responses


However, I want to choose an equation the 'proper' way however that
might be as I want to present my research academically.

That is commendable.
In my opinion, the "'proper' way", i.e., the most rigorous way works similar to a mathematical proof. You start with what you 'know' about the process that generated your data and derive a model of that process. This often results in a differential equation that can be solved for specific boundary and starting conditions. If you need examples for that approach, look into thermodynamics, e.g., see this Wikipedia entry.
Sometimes you can rely on analogies and don't need to do all that work. Fick's laws of diffusion were derived by positing an analogy with heat transfer. You can apply the solutions of differential equations describing heat transfer to diffusion problems.
If you don't know enough to derive a model for the data-generating process your model will always be at least partly empirical. Still, you might be able to convincingly derive at least some basic properties of the functional relationship from scientific knowledge or analogies (e.g., you might know that the relationship has to be convex, or maybe that there is saturation, or maybe that observed dependent values must be positive, or ...). Good examples of this apporach are some models for sorption isotherms.
If you can't do that, you could fit a non-parametric model and either use that or select non-linear functions that often occur as solutions of the rigorous approach (e.g. the exponential function, the log function or trigonometric functions) based on the shape of the non-parametric fit (or just based on plots of the data).
Finally, a linear model can be a surprisingly good approximation over a limited range of data. If you don't need to extrapolate, such a linear model might be sufficient even if the true relationship is non-linear.
