The tissues are during their physiological function, e.g., in the course of growth, adolescence, and aging, subjected to a cyclic mechanical loading and to large displacements and rotations as well. A tissue free of all external tractions is in a state that minimizes its internal power. In the course of aging of the tissues, for instance in the wall of the aorta, the vein, and also in the myocardium or heart valves, the decrease of the water content and increase of the collagen content occurs; while in compact and trabecular bone the contents of both mineral substances and collagen, undergo reduction. In accordance with it, the strain energy function and the constitutive equations of living tissue based on the hyperelasticity theory using rotationless strain were studied. On the base of the proposed eigenvalue decomposition of the rotationless strain tensor and hyperelasticity the strain energy function was formulated as depending on biological time of tissue. The quantity of strain energy function per unit of the biological time, which essentially characterizes the velocity of change of mechanical response of tissue in the course of its aging, was also defined. The coefficient of tissue aging is the further diagnostic parameter, which is independent of the rotationless strain tensor and expresses the relative change of mechanical response of tissue during the biological time. The corresponding constitutive equation of tissue depending on the biological time is also determined. On the base of the regression analysis the theoretical stress-strain curves for myocardium and blood vessels were determined. The numerical results reveal that the coefficient of aging progressively increases in hardening tissues (coronary artery, vena cava inferior) whereas at the softening tissues it has a relatively slow increase at the dependence on tissue aging.