Training without eating, what effects on liver ( Type 2)

Hi, I have type 2 for 6 years (age 56 now), Originally on metformin, 3 tablets a day, but now off them completely. 12 months Test 41. Does anyone know whether training without eating may be harmful. I did it as an experiment , 12 hours after evening meal, get up and go for a tough 3 mile run up and down hills. When I returned home by sugars was 7 so had to wait a few hours before eating at 5.6. The liver obviously kicked in to stop me going too low, is this good practice or could i be harming my liver?

Thanks for that Dave .I am on holiday in Spain at the moment so an early run on the hills of the Costas is the only way( 37 C is a bit ho)t. Also I find when I am in a hot country my blood sugars tend to be lower and easier to control.. The effects of heat on blood sugars, thats another interesting one!!