Two analytical solutions for a model of pulsed arterial spin labeling with randomized blood arrival times

A fairly general theoretical model for pulsed arterial spin labeling perfusion methods has been available for some time but analytical solutions were derived for only a small number of arterial blood input functions. These mostly assumed a sudden and simultaneous arrival of the tagged blood into the imaged region. More general cases had to be handled numerically. We present analytical solutions for two more realistic arterial input functions. They both allow the arrival times of the molecules of tagged arterial blood to be statistically distributed. We consider cases of (1) a uniform distribution on a finite time interval and (2) a normal distribution characterized by its mean and standard deviation. These models are physiologically meaningful because the statistical nature of the arrival times reflects the distribution of velocities and path lengths that the blood water molecules undertake from the tagging region to the imaged region. The model parameters can be estimated from the measured dependency of the perfusion signal on the tag inversion time.