intracranial circulation in patients with ischemic stroke. Phase-contrast MRA Phase-contrast (PC-MRA) can be used to encode the velocity of moving blood in the magnetic resonance signal's phase. The most common method used to encode velocity is the application of a bipolar gradient between the excitation pulse and the readout. A bipolar gradient is formed by two symmetric lobes of equal area. It is created by turning on the magnetic field gradient for some time, and then switching the magnetic field gradient to the opposite direction for the same amount of time. By definition, the total area (0th moment) of a bipolar gradient, , is null: (1) The bipolar gradient can be applied along any axis or combination of axes depending on the direction along which flow is to be measured (e.g. x). , the phase accrued during the application of the gradient, is 0 for stationary spins: their phase is unaffected by the application of the bipolar gradient. For spins moving with a constant velocity, , along the direction of the applied bipolar gradient: (2) The accrued phase is proportional to both and the 1st moment of the bipolar gradient, , thus providing a means to estimate . is the Larmor frequency of the imaged spins. To measure , of the MRI signal is manipulated by bipolar gradients (varying magnetic fields) that are preset to a maximum expected flow velocity. An image acquisition that is reverse of the bipolar gradient is then acquired and the difference of the two images is calculated. Static tissues such as muscle or bone will subtract out, however moving tissues such as blood will acquire a different phase since it moves constantly through the gradient, thus also giving its speed of the flow. Since phase-contrast can only acquire flow in one direction at a time, 3 separate image acquisitions in all three directions must be computed to give the complete image of flow. Despite the slowness of this method, the strength of the technique is that in addition to imaging flowing blood, quantitative measurements of blood flow can be obtained. Flow-independent angiography Whereas most of techniques in MRA rely on contrast agents or flow into blood to generate contrast (Contrast Enhanced techniques), there are also non-contrast enhanced flow-independent methods. These methods, as the name suggests, do not rely on flow, but are instead based on the differences of T1, T2 and chemical shift of the