Question

why is there a weak relationship between breathing rate and time breath is held?

Answer 1

In the normal breathing zone of 15.3-16.1% oxygen in the lungs, the blood reaches approximately 99% saturation. If, for some reason, the oxygen concentration in your lungs were to fall to 10%, your blood would still be 95% saturated, a level at which you would experience some discomfort but not be at any immediate health risk. If the oxygen in your lungs falls to 7% and the corresponding blood saturation falls to 85%, you are likely to pass out and fall to the floor.

If you take a breath and then hold it, you start with 3.5 liters of gas in your lungs with an oxygen concentration of 16.1%. As your heart keeps pumping, your blood continues to extract oxygen from your lungs. After about 200 seconds, the concentration of oxygen in your lungs will fall to 7%, causing the oxygen saturation level in your blood to drop to 85%. At this point, you pass out, fall over, and start to breathe, so all is well.

So, holding your breath for a bit doesn't make much difference to blood oxygen saturation and thus the heart doesn't get a feedback to increase pumping of blood. But holding your breath for more than 2 - 3 minutes will surely increase your heart rate. But also it depends on indevidual and the practice of breath holding he/she has. The more you practice the your body adapts to conserve oxygen.