Deep Love Ayu no Monogatari.

Ayu, a young prostitute has lived a meaningless life until she met a young boy named Yoshiyuki. They met when an old lady who taught her the meaning of life passed away, only briefly telling her of her grandson, Yoshiyuki. Knowing that the old lady wished only to see her grandson but is unable to grant the old lady her wish, Ayu then thought that she could take care of Yoshiyuki for her. Ayu and Yoshiyuki, despite his heart problems and her dark past love each other to the fullest. However, life wasn't easy for the both of them. Yoshiyuki's father forbids their relationship and kept them away from each other. But Ayu sacrificed herself by becoming a prostitute once again to earn money for Yoshiyuki's operation and to keep her promise to the old woman who changed her life.