A novel model-based equation for size dependent mean recovery coefficients for spheres and other shapes

Background: In NM-imaging, theoretical curves for the recovery coefficient (RC) of the signal maximum and mean are known for spheres and cubes, if a 3D Gaussian PSF is assumed. The RC of the maximum is also known for cylinders. For these and other shapes empirical equations with one or two fit-parameters have been utilized. Methods: An equation for the RC for large objects of arbitrary shape is derived and generalized into an empirical equation for smaller objects, which is verified by numerical simulations. The proposed equation is compared to published results on SPECT kidney phantom measurements and to PET measurements on the NEMA IEC PET body phantom with six spheres. Results: The signal loss (1-RC) for large spheres is inversely proportional to the radius, where the slope is proportional to the FWHM of the spatial resolution. For non-spherical shapes the generalized instead of the volume equivalent radius should be utilized. For smaller objects, an equation with one added empirical fit-parameter is presented. It is demonstrated that the EANM-guidelines' two-parameter logistic function results in a poor fit if the theoretical slope and inverse proportionality are forced and it gives a suboptimal fit when both parameters are fitted. Conclusions: A novel model-based equation for the mean RC-curve is derived. It can be used for arbitrary shapes as long as the sphericity is taken into account and it is accurate down to RC = 10 %. One parameter is directly related to the spatial resolution, while the other is a shape depending fit-parameter.