The deep branch is susceptible to damage during thyroidectomy or cricothyrotomy , as it is immediately superior to the external artery of thyroid gland . The deep branch lies susceptible to damage during thyroidectomy or cricothyrotomy , as it is immediately superior to the external thyroid artery . The external branch is susceptible to damage during thyroidectomy or cricothyrotomy , as it lies immediately deep to the superior thyroid artery .