Calibrated functional magnetic resonance imaging (fMRI) is a method to independently measure the metabolic and hemodynamic contributions to the blood oxygenation level dependent (BOLD) signal. This technique typically requires the use of a respiratory challenge, such as hypercapnia or hyperoxia, to estimate the calibration constant, M. There has been a recent push to eliminate the gas challenge from the calibration procedure using asymmetric spin echo (ASE) based techniques. This study uses simulations to better understand spin echo (SE) and ASE signals, analytical modelling to characterize the signal evolution, and in vivo imaging to validate the modelling. Using simulations, it is shown how ASE imaging generally underestimates M and how this depends on several parameters of the acquisition, including echo time and ASE offset, as well as the vessel size. This underestimation is the result of imperfect SE refocusing due to diffusion of water through the extravascular environment surrounding the microvasculature. By empirically characterizing this SE attenuation as an exponential decay that increases with echo time, we have proposed a quadratic ASE biophysical signal model. This model allows for the characterization and compensation of the SE attenuation if SE and ASE signals are acquired at multiple echo times. This was tested in healthy subjects and was found to significantly increase the estimates of M across grey matter. These findings show promise for improved gas-free calibration and can be extended to other relaxation-based imaging studies of brain physiology.