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

If your understanding of the subject matter supports those "exponential" associations of your predictors with OD, then that seems to be an appropriate form of equation to start with.
Note that if you take the logs of both sides, you can convert this into a linear regression model using the logs of the current variable values:
log(OD) = a*log(KPV) + b*log(MAS) - c*log(SID) - e - d*log(DEPTH)

Thus you could model with lm() in R:
lm(log(OD) ~  log(KPV) + log(MAS) + log(SID) + log(DEPTH))

The coefficients (including the intercept) will all be related to those in your non-linear model. Technically, you are now modeling the mean of log(OD) values, not the mean of the OD values as you are in the non-linear model, but it might do what you need. This might be appropriate if errors around model predictions are relatively constant in magnitude on the log(OD) scale.
Alternatively, you could consider a Gaussian generalized linear model with a log link for the outcome OD and the log-transformed predictors, in R:
glm(OD ~ log(KPV) + log(MAS) + log(SID)+ log(DEPTH), family=gaussian(link="log"))

That models the log of the mean OD as a function of the log-transformed predictors. The glm could be more appropriate if error magnitudes are relatively constant in the OD scale, which is the assumption you are implicitly making in your non-linear model.
Either approach in the log scale will probably have less danger of numerical difficulties than will trying to solve the original non-linear equation with lots of exponents.
