The article talked about establishing a "breathing rhythm" as you run. Take two or three regular breaths, followed by a deep cleansing breath, then back to the regular breaths. You are trying to keep the lungs continuously filled with enough air to keep you going. But if you take an extra deep breath at regular intervals, you take in more air (oxygen), filling more alveoli than you normal breaths, thus giving you bloodstream a little extra boost of oxygen. In addition, the shallower breaths allow you lungs to more completely absorb the oxygen taken in during the deep breath.
It is often hard to breathe deeply when you are out of breath yet these deep breath are critical to keeping the muscles in your legs supplied with enough oxygen to keep going. More oxygen reaching the muscles translates to more endurance. Training yourself to take deep breath before you suddenly find yourself starved for oxygen is the key. Here is a quote by Everett Murphy, M.D., a runner and pulmonologist at Olathe Medical Center in Olathe, Kansas. "Exercise improves the conditioning of the diaphragm, the muscle that separates the chest from the abdomen, and the intercostal muscles, which lie between the ribs and enable you to inhale and exhale. When you take a breath, 80 percent of the work is done by the diaphragm. If you strengthen your diaphragm, you may improve your endurance and be less likely to become fatigued." Deep breathing strengthens the diaphragm.
Here is how I "manage" my breathing. This allows me to keep taking deep breaths even at the top of a long climb. I should actually do this all the time while running but I rarely think of it until I need more oxygen. When I do it properly I start this sequence before hitting a hill.
1. I establish a breathing rhythm based on my pace. I inhale and exhale normally, every four steps for twelve steps. That is (steps 1 &2 Inhale, steps 3 &4, exhale, steps 5 & 6 inhale, steps 7 & 8, exhale, steps 9 & 10 inhale, 11 &12 exhale.) On step 13, take a quick breath and on step 14 exhale quickly. Then on step 15 and 16, I take a very deep breath and exhale on 17 and 18. Then start the sequence all over again.
2. As the climb becomes steeper and I need more air, instead of three sets of normal breaths, I cut it down to two. That is, inhale and exhale on steps 1 to 4 and again on steps 5 through 8. Then breathe in quickly on step 9 and out on step 10, followed by the deep breath on 11 through 14.
3. If I still need more air I modify the breathing rhythm again as needed while still managing to keep that deep breath in there somewhere.
Here is a crude diagram of my breathing pattern tied to steps. Maybe it will make more sense.
Now, I don't count steps. You just get in a rhythm and breathe. The pattern basically establishes itself. Sometimes I find it necessary to drop the normal breaths all together. Then I just take the quick "inhale on step 1, exhale on step 2" and repeat that two or three times, then take the deep breath. The key is to manage to get that deep breath into your breathing patter. The breathing pattern will change with the terrain, too. As you start up hill, you will need more air and running downhill takes less oxygen so in reality, your breathing will constantly change.
Experiment with various breathing patters and find out what works. There is no right or wrong. When I remember to do it, I am always amazed at how much difference it makes in my hill repeats.