ELISA is a multifaceted bio-physical phenomenon combining liquid-solid adsorption, biochemical binding (specific and nonspecific), and mass transfer (oh yeah, the world is not perfect!). Each of these is affected by thermodynamic (temperature for instance) and kinetic (molecular crowding and heterogeneity to name a few) factors. To make this beautiful soup quantitative, we use a clean/pure antigen with a known concentration as the standard. However, things can be less than ideal. 

In the ideal world, one might expect to observe an adsorption isotherm (such as Langmuir) or a pseudo-first-order reaction for the ELISA standard curve. However, there are a few factors that deviate reality from expectations, such as heterogeneity in a mobile analyte or in ligand population on the surface, and mass transfer limitations. It is hard to quantify these because we use an optical biosensor (such as a plate reader) to measure the output signal. I found this paper very informative on this topic https://www.cell.com/biophysj/pdf/S0006-3495(03)75132-7.pdf  Briefly they “… explored whether it is possible to retrieve information on the combined distribution of affinity and kinetic parameters of heterogeneous populations of analytes or immobilized sites”, and concluded that “… the obtained two-dimensional kinetic and affinity distributions have a higher resolution than corresponding affinity distributions based on the isotherm analysis alone.”

Each ELISA is different. For some assays, a linear curve might work well, but for most cases, we use a four-parameter logistic (4PL) when analyzing ELISA data. I personally found a cubical polynomial regression model might be good enough in some cases. Here is a paper that compared the quadratic, cubic, and 4-parameter logistic models for fitting sandwich-ELISA data. https://pubmed.ncbi.nlm.nih.gov/18822292/

In my opinion, any curve that satisfies the following conditions can be used as a standard curve with an acceptable accuracy:

• Cover the measurement range. Any curve might fail to predict when extrapolated outside of its’ measured range.
• CV<20% between repeats of each dilution. In most cases, wet-lab variabilities are more to blame than the fitted model. Ensure that the standard curve is replicable.
• Have good recovery for standard values. The model needs to have a recovery in the 80% to 120% range when back-calculated for standard values. This also can help to define the range of the calibration curve.
• For best predictability, try to have at least two dilutions for samples in the linear range of the calibration curve.
• The predictability of the calibration curve (which is obtained from a pure/clean analyte) might be compromised due to the matrix effect in real sample solutions. Try to find the OD range that generates the best linearity for two consecutive sample dilutions. 

The Four-Parameter Logistic (4PL) model, also known as the Hill equation, is defined by the following equation.