All the branches of maxillary artery carry about 50% of the blood supply to the face and ear. Arising from the external carotid artery, the maxillary has a rather long and tortuous course. For this reason, it can be divided into three portions for a better comprehension.
First Portion (mandibular part)
As soon as it springs from the external carotid, the maxillary gives off five branches, with the first four arising from its superior side; 1) the deep auricular artery, which runs upwards to supply the auricular capsule of the mandibular joint, the inferior wall of the external auditory meatus, and the tympanic membrane; 2) the anterior tympanic artery, which travels up and enters the tympanic cavity through the squamotympanic fissure to supply its mucous membrane; 3) the middle meningeal artery, which is its largest branch, running upwards through the foramen spinosum into the skull to supply a large part of the dura mater that covers the frontal part of the brain; 4) the inferior alveolar artery, which is the only one that originates from the inferior side of first portion of maxillary as it travels downwards through the mandibular foramen to supply the chin and lower teeth.
Second Portion (pterygoid part)
The second part of the maxillary gives off several branches which originate both from its superior and inferior side. 1) The posterior deep temporal artery, which arises from its superior aspect and runs upwards to supply the temporal muscle, periosteum, and bone of the temporal fossa; 2) The anterior deep temporal artery, which also originates from the superior side of maxillary and irrigates the temporal muscle 3) the pterygoid branches, originating within the pterygoid bone, with one branch traveling upwards and the other downwards in the oppossite direction, to supply the lateral and medial pterygoid muscle respectively; 4) the messeteric artery, which descends through the mandibular notch to provided the messeter muscle with oxygenated blood; 5) the buccal artery, which also descends to supply the buccinator muscle and the upper portion of cheek.
Third Portion (pterygopalatine part)
The third and final section of the maxillary artery is located in the pterygopalatine fossa. There, several smaller branches arise from it to provide the upper and medial part of face with oxygenated blood. 1) The posterior superior alveolar artery, which originates from its inferior side and travels downwards and obliquely to supply the superior premolar and molar teeth, gingiva (gum), and maxillary sinus; 2) the descending palatine artery, which first gives off the artery of the pterygoid canal, then it divides into the greater and lesser palatine artery that supply the soft and hard palate as well as the gum and bone of the lower and posterior part of maxilla; 3) the infraorbital artery, which travels down to supply the roots of the superior incisors and canine teeth; 4) the sphenopalatine artery, being the terminal branch of the maxillary that runs through the sphenopalatine foramen to supply the posterior part of nasal cavity and the septum.
Below, schematic picture showing the branches of maxillary artery.