I'm sorry! Um, it probably doesn't help that much of the relevant info hasn't been published yet; this patent is the best description that will be publicly available until the inventors get more funding. From the patent:
By replacing the volume of the vasculature (from 5 to 10 percent of the volume of tissues, organs, or whole organisms) with a gas, the vasculature itself becomes a “crush space” that allows stresses to be relieved by plastic deformation at a very small scale. This reduces the domain size of fracturing...
So, pumping the organ full of cool gas (not necessarily oxygen) is done for reasons of cooling the entire tissue at the same time, as well as to prevent fracturing, rather than for biological reasons.
ETA: To answer your last question, persufflation would be done on both cooling and rewarming.
Thanks!
This is great, by using small interlocking magnetic fields, you can keep the water in a higher vibrational state, allowing a "super-cooling" without getting crystallization and cell rupture
Subzero 12-hour Nonfreezing Cryopreservation of Porcine Heart in a Variable Magnetic Field
"invented a special refrigerator, termed as the Cells Alive System (CAS; ABI Co. Ltd., Chiba, Japan). Through the application of a combination of multiple weak energy sources, this refrigerator generates a special variable magnetic field that causes water molecules to oscillate, thus inhibiting crystallization during ice formation18 (Figure 1). Because the entire material is frozen without the movement of water molecules, cells can be maintained intact and free of membranous damage. This refrigerator has the ability to achieve a nonfreezing state even below the solidifying point."
http://mobile.journals.lww.com/transplantationdirect/_layouts/15/oaks.journals.mobile/articleviewer.aspx?year=2015&issue=10000&article=00005#ath