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Tube feeding is done by attaching a syringe to a catheter tube which is inserted in through the puppy's mouth down into the stomach. To figure out how far you'll need to poke the tube down the pup's throat, hold the tube alongside the pup and check the distance from the pup's mouth to the back of the rib cage. This is how far the tube needs to be inserted. With the tip of the tube held to the side of the pup, the tip should be at the end of the rib cage, and make a mark on the tube with a pen where the pup's mouth is. This will let you know when to stop when you are easing the tube into the pup's stomach.
The Difference Between Tube Feeding and Bottle Feeding Puppies
Although it is a lot quicker than using a bottle, tube feeding puppies should not completely replace bottle feeding. Bottle feeding satisfies the sucking reflex of the puppies. If you take away bottle feeding, they will suckle on the closest thing that resembles a nipple - and if there's no mother dog, the closest thing that resembles a nipple will be their brothers' and sisters' private parts. This is unsanitary and can lead to very sore little bottoms.
Feeding Newborn Puppies
Hold the puppy firmly around the body, making sure you have its head and mouth under control. The amount you feed depends on the puppy's weight, so consult your vet and the instructions of the milk replacement formula packet to figure out how much to feed.
You shouldn't have to push the tube into the pup's stomach - it will swallow the tube on its own. Coughing is a bad sign. If the puppy is coughing as you insert the tube or push the plunger down it may mean fluid is going into the trachea or lungs. This is bad news, because fluid in the lungs can lead to pneumonia, and pneumonia for a weak newborn puppy can spell death.
The formula you use should be warm to simulate milk. There should be more formula in the syringe than you need to feed the puppy - you don't want the syringe to be empty by the end of the feeding. This will make sure you don't inject and air bubble into the pup's stomach.
Make sure you talk to a veterinarian before you attempt to tube feed puppies. There is a bit of subtlety to this that can't be conveyed in words and you will need to make sure you have the right equipment. It's best to practice inserting the tube on its own before attaching it to a formula-filled syringe.
You may also want to read more about bottle feeding puppies.