When measuring avidity should you use the ligand concentration or the nanoparticle concentration?
In the case where there is not cooperativity between binding events, the Scatchard Equation is the generalized solution to multiple ligand binding to a macromolecule. The Scatchard plot uses the free ligand concentration. 

There are multiple models used to address cooperative binding. The Hill Equation is the most frequently encountered in biochemistry literature:
$$ \theta = \frac{[L]^n}{K_d+[L]^n} $$
Where $[L]$ is the concentration of free ligand, $\theta$ is the fraction of bound ligand-receptor complex, $K_d$ is the apparent dissociation constant, and $n$ is the Hill coefficient, describing cooperativity.