Carotid artery disease, narrowing of the arteries in the neck that supply the brain with blood, is a gradual process that slowly blocks the artery, and therefore often does not present any warning signs until the artery is almost totally closed.When blood flow is blocked through the artery, a stroke results.A stroke occurs when brain tissue begins to die because of a lack of oxygenated blood. A stroke or a mini-stroke (transient ischemic attack) might be your first indicator that your carotid arteries are not healthy.hen blood flow is blocked through the artery, a stroke results. A stroke occurs when brain tissue begins to die because of a lack of oxygenated blood. A stroke or a mini-stroke (transient ischemic attack) might be your first indicator that your carotid arteries are not healthy.