In Satipatthana we contemplate on earth, water, heat, pain and mental formations as the object of contemplation for meditation. However, Buddha chose breath as the anchor for his meditation. He might as well use any of others as his meditation object.
Is there a special reason for this?
For example in fourth Jhana breathing stop. Just imagine in fourth Jhana your blood circulation stops due to contemplation on water element.
“The investigators surmised that rather than regulating breathing, these neurons were spying on it instead and reporting their finding to another structure in the brainstem.
This structure, the locus coeruleus, sends projections to practically every part of the brain and drives arousal: waking us from sleep, maintaining our alertness and, if excessive, triggering anxiety and distress.
It’s known that neurons in the locus coeruleus exhibit rhythmic behavior whose timing is correlated with that of breathing. In a series of experiments, the Stanford researchers proved that the preBötC neurons that express Cadh9 and Dbx1 not only project to the locus coeruleus — a new finding — but activate its long-distance-projections, promoting brainwide arousal.”
(…)
“The preBötC now appears to play a key role in the effects of breathing on arousal and emotion, such as seen during meditation,” said Feldman. “We’re hopeful that understanding this center’s function will lead to therapies for stress, depression and other negative emotions.”
Buddha did not chose the breath. When mindfulness is established (i…e, keeping the mind free from craving) & the mind becomes silent, the breath automatically becomes the object of meditation because it is the coarsest/grossest sense object.
This is similar to a loud sound. If there is a loud sound, the loud sound automatically becomes the sense object of the mind because it is the coarsest/grossest sense object.
Please read the instructions with care:
And how, bhikkhus, is mindfulness with/when breathing developed and cultivated, so that it is of great fruit and great benefit?
Here a bhikkhu, gone to the forest or to the root of a tree or to an empty hut, sits down; having folded his legs crosswise, set his body erect and (1) establishes mindfulness in front of him; (2) ever mindful, he breathes in; (3) ever mindful, he breathes out.
MN 118
Note: “He” does not breath in & out. Instead, the body automatically breathes in & out.
With metta 
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I thought breath is the most subtle sense object.
So, in your opinion, what is the most subtle sense object?
The breath is the most subtle sense object when the mind is noisy with thoughts, hindrances & thoughts of “I”. But when the mind is silent & the eye, ear, nose, tongue & mind sense doors are not engaged, the breath (via the body sense door) is the grossest sense object.
It is the very thought intention to observe the breath that makes the breath too subtle for the mind to discern in a meaningful way. This intention to observe the breath is craving (tanha) & Mara.
I would suggest to read Ajahn Brahmavamso’s book about letting go & silence as the preliminary practicse to Anapanasati.
Every time the translation of ‘mindfulness of breathing’ is used, this is an obstacle to noble practise.
With metta
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It is important to highlight that Suttas indicate that anapanasati had a major role in the contemplative life of the Buddha. Or at least of the early Sangha. A piece of evidence towards that is the occurrence of the Pali expression:
… bhikkhu ānāpānassatiṃ bahulīkaronto nacirasseva akuppaṃ paṭivijjhatī
Such as found in AN5.96, AN5.97 and AN5.98:
(…) a monk pursuing mindfulness of breathing will in no long time penetrate the Unprovoked.
This is not mentioned in Ahahn Brahmavamso’s book.
Where you get this idea from?
From the 2nd noble truth (SN 56.11), which states craving-to-be is to be abandoned. From MN 118, which states the enlightenment factor of mindfulness has the quality of letting go (vossagga). From SN 48.10, which states jhana is reached from making letting go (vossagga) the meditation object. And from AN 11.2.
Kind regards
Buddha was breathing before Parinibbana. So what you are saying is Buddha did not eliminate craving?
No. Where did I say this? I answered your original question. Buddha did not chose the breathing. Instead, the breathing chose Buddha. When the mind is free from craving, silent & quiet, it will understand this. The breathing is a ‘sign’ (‘nimitta’) rather than a ‘choice’. Kind regards. 
So what you are saying is that Sutta instruction, “He mindfully breathe in and mindfully breathe out” is a craving and Mara?
[quote=“SarathW1, post:11, topic:5534, full:true”]
So what you are saying is that Sutta instruction, “He mindfully breathe in and mindfully breathe out” is a craving and Mara?[/quote]
I said intention to watch breathing is craving. So you are now saying mindfulness breathes? So you are now saying the ‘self’ or ‘he’ breathes? 
Thanks, I see your point.
