An apparatus for controlling the supply of an oxygen therapeutic gas, comprising: a nasal cannula adapted to be introduced into a nasal passage of a patient; a conduit to direct a pressurized oxygen therapeutic gas to the nasal cannula; a pressure sensor provided on the conduit, arranged to detect the pressure in the conduit; a valve, provided on the conduit, arranged to allow and block a fluid communication to the nasal cannula; a controller adapted to control the operation of the valve in synchronization with the respiration of a patient based on the changes in the pressure detected by the pressure sensor, to increase the volume of the oxygen therapeutic gas passing through the valve for each respiration, compared with a normal respiration condition, when the respiratory frequency increases, such that the volume of oxygen therapeutic gas supplied to the patient increases in steps according to the increase in the respiratory frequency, wherein the time period for opening the valve is given by T = F o / α R n f o wherein: f o is the flow rate of oxygen therapeutic gas through the valve when the valve is open under a predetermined pressure in the conduit upstream of the valve; R n is the respiratory frequency when the respiration of the patient is in normal condition; F o is a predetermined flow rate of oxygen therapeutic gas; and α is a non-dimensional constant that is an inverse of the ratio of the time period of inhalation relative to the time period of one respiration. a nasal cannula adapted to be introduced into a nasal passage of a patient; a conduit to direct a pressurized oxygen therapeutic gas to the nasal cannula; a pressure sensor provided on the conduit, arranged to detect the pressure in the conduit; a valve, provided on the conduit, arranged to allow and block a fluid communication to the nasal cannula; a controller adapted to control the operation of the valve in synchronization with the respiration of a patient based on the changes in the pressure detected by the pressure sensor, to increase the volume of the oxygen therapeutic gas passing through the valve for each respiration, compared with a normal respiration condition, when the respiratory frequency increases, such that the volume of oxygen therapeutic gas supplied to the patient increases in steps according to the increase in the respiratory frequency, wherein the time period for opening the valve is given by T = F o / α R n f o wherein: f o is the flow rate of oxygen therapeutic gas through the valve when the valve is open under a predetermined pressure in the conduit upstream of the valve; R n is the respiratory frequency when the respiration of the patient is in normal condition; F o is a predetermined flow rate of oxygen therapeutic gas; and α is a non-dimensional constant that is an inverse of the ratio of the time period of inhalation relative to the time period of one respiration. f o is the flow rate of oxygen therapeutic gas through the valve when the valve is open under a predetermined pressure in the conduit upstream of the valve; R n is the respiratory frequency when the respiration of the patient is in normal condition; F o is a predetermined flow rate of oxygen therapeutic gas; and α is a non-dimensional constant that is an inverse of the ratio of the time period of inhalation relative to the time period of one respiration.