A porous membrane is inserted into a ventricle of a heart. The porous membrane creates a relatively hemostatic volume in which a thrombus can grow. Blood can still pass through fenestrations of the membrane into and out of the hemostatic volume. The fenestrations reduce pressures that act on the membrane, and so reduce stresses within the membrane. The flow characteristics through the hemostatic volume promote growth of the thrombus from a base of the hemostatic volume. The thrombus grows to slightly larger than the original size of the hemostatic volume so as to provide support for the membrane. Any remaining stresses within the membrane are thereby substantially eliminated. The thrombus shrinks over an ensuing period of time, with the membrane merely acting as a barrier to which an outer wall of the myocardium retracts. The function of the membrane is then complete, and may be absorbed.