The maxillary artery is the larger terminal branch of the external carotid artery arising in the parotid gland behind the mandibular neck. It enters the infratemporal fossa by passing deep to the neck of the mandible and runs its course in three parts: before, on, and beyond the lateral pterygoid muscle. Each part gives 15 branches. The 10 branches of the first and third parts pass through foramina. The 5 branches of the second part are mainly muscular.
The first part
This runs anteriorly between the neck of the mandible and the sphenomandibular ligament giving the following branches:
Inferior alveolar artery: descends anteroinferiorly in front of the corresponding vein towards the inferior alveolar nerve which it meets at the mandibular foramen. The vessels and nerve then enter the foramen and run in the mandibular canal supplying the bone and lower teeth and gives a mental branch that emerges from the mental foramen to supply the lower lip and the chin.
Middle meningeal artery: ascends vertically embraced by the 2 roots of the auriculotemporal nerve to enter the middle cranial cavity via foramen spinosum.
Accessory meningeal artery: passes upwards through the foramen ovale to supply the dura of the floor of the middle cranial fossa and trigeminal ganglion. It may arise from the middle meningeal artery.
Deep auricular artery: passes posterosuperiorly to supply the external auditory meatus by passing between its bony and cartilaginous parts.
Anterior tympanic artery: passes through the petrotympanic fissure to the middle ear to participate in the circular anastomosis on the deep surface around the tympanic membrane.
The second part
This part runs anterosuperiorly deep to the temporalis insertion giving 5 muscular branches to temporalis (2), masseter (1), and the pterygoids (1). It also gives a buccal artery that accompanies the buccal nerve (1).
The third part
Runs deeply medially between the 2 heads of the lateral pterygoid to reach the pterygomaxillary fissure and enters the pterygopalatine fossa where it ends by giving its 5 branches:
Sphenopalatine artery: passes through the sphenopalatine foramen to enter and supply the nasal cavity.
Posterior superior alveolar artery: divides into several branches that accompany the corresponding nerve and pierce the posterior surface of the maxilla to supply the upper molar teeth and gum.
Greater (descending) palatine artery: gives rise to the lesser palatine arteries (to the soft palate) and then passes through the greater palatine foramen to supply the hard palate.
Pharyngeal artery: is very small and enter palatovaginal canal.
Artery of pterygoid canal: runs in its own canal.
After these 5 branches are given, the maxillary artery becomes the infraorbital artery which runs on the floor of the orbit and in the infraorbital canal to emerge with the corresponding nerve through the infraorbital foramen.