New titanium alloys with a low elastic modulus have been developed for biomedical applications to avoid the stress shielding effect of the artificial prosthesis. The newly developed alloys contained the transition elements like Nb, Ta, Zr which were non-cytotoxicity elements. These elements produced β, ω, and α'' phases with heat treatment conditions in titanium alloys and determined the elastic modulus of the alloys. However, the clear mechanism of the low elastic modulus alloys has not been known. In the present paper, the total energy and elastic modulus of β and α'' phases were calculated using a first principle calculation employing the generalized gradient approximation (GGA). The mechanism of the low elastic modulus was discussed with calculated values.