An extended model for extracting measures of brain perfusion from pulsed arterial spin labeling (ASL) data while considering transit effects and restricted permeability of capillaries to blood water is proposed. We divided the time course of the signal difference between control and labeled images into four phases with respect to the arrival time of labeled blood water at the voxel of interest (t A ), transit time through the arteries in the voxel (t ex ), and duration of the bolus of labeled spins (). Dividing the labeled slab of blood water into many discrete segments, and adapting numerical integration methods allowed us to conveniently model restricted capillary-tissue exchange based on a modified distributed parameter model. We compared this four-phase single-capillary stepwise (FPSCS) model with models that treat water as a freely diffusible tracer, using both simulations and experimental ASL brain imaging data at 1.5T from eight healthy subjects ( Arterial spin labeling (ASL) MRI, which uses endogenous arterial water as a tracer for blood flow, is entirely noninvasive and is increasingly being used to study brain perfusion and function (1,2). Quantification of tissue perfusion with ASL MRI is generally accomplished by modeling the signal evolution of ASL based on pharmacokinetic information, such as the exchange of blood water between capillaries and brain tissue, and the Bloch-Terry equations, which describe spin magnetization in the presence of relaxation and flow (3-15). However, early applications of ASL (3-6) relied on single-compartment kinetics for quantification, which limited accuracy for several reasons. First, it was assumed that labeled water delivered to the brain would instantaneously enter the capillary bed for exchange. Second, permeability of the capillaries was assumed to be infinitely high for water. Together, these assumptions require an instantaneous equilibrium for water in intra-and extracapillary spaces.More recently, two conceptually different models have been proposed to account for restricted water exchange between intra-and extracapillary compartments. One model (14,15) assumes well-mixed concentrations of ASL water in each of the two compartments. The other model, also known as a distributed model, accounts for variations of ASL water concentration along the capillary path (or along both intra-and extracapillary spaces) (12,13). In practice, however, this approach may not be applicable, since some parameters (e.g., capillary length) are not accurately known (12). Although both models focus on the effect of restricted water exchange, they do not systematically include transit effects for labeled spins, such as the time to arrival (t A ) and the bolus duration (). Both parameters are generally associated with heterogeneous values for labeled spins because they travel along different trajectories of the feeding vessels.After the labeled spins arrive at the image voxel, there may also be an additional delay (t ex ) before water exchange in the capillary bed can begin, because labe...