Maxillary artery

The maxillary artery supplies deep structures of the face. It branches from the external carotid artery just deep to the neck of the mandible.
Structure.
The maxillary artery, the larger of the two terminal branches of the external carotid artery, arises behind the neck of the mandible, and is at first imbedded in the substance of the parotid gland; it passes forward between the ramus of the mandible and the sphenomandibular ligament, and then runs, either superficial or deep to the lateral pterygoid muscle, to the pterygopalatine fossa.
It supplies the deep structures of the face, and may be divided into mandibular, pterygoid, and pterygopalatine portions.
First portion.
The first or mandibular portion (or bony portion) passes horizontally forward, between the neck of the mandible and the sphenomandibular ligament, where it lies parallel to and a little below the auriculotemporal nerve; it crosses the inferior alveolar nerve, and runs along the lower border of the lateral pterygoid muscle.
Branches include:
Second portion.
The second or pterygoid portion (or muscular portion) runs obliquely forward and upward under cover of the ramus of the mandible and insertion of the temporalis, on the superficial (very frequently on the deep) surface of the lateral pterygoid muscle; it then passes between the two heads of origin of this muscle and enters the fossa.
Branches include:
Third portion.
The third or pterygomaxillary portion lies in the pterygopalatine fossa in relation with the pterygopalatine ganglion. This is considered the terminal branch of the maxillary artery.
Branches include: