## The formula for the Bühlmann algorithm, which has been valid for years, is as follows:

### pt tol i.g. = (p amb / b) + a

*or*

### p amb tol = (pt i.g. – a) * b

Legend | |
---|---|

pt tol i.g. | Highest still tolerated pressure of an inert gas in a given tissue/compartment (e.g. N2 or He) |

p amb | ambient pressure (air pressure + water pressure) |

a | coefficient a for a given tissue/compartment (in the pressure unit in which it is calculated) |

b | coefficient b for for a given tissue/compartment (linearity factor) |

pt i.g. | instantaneous pressure of the inert gas in the tissue |

p amb tol | Tolerated ambient pressure (absolute pressure, water + air pressure), pressure which must not fall below for the tissue) |

*The original Bühlmann model had first 12 and later 16 tissues/compartments with constant half-life (saturation/de-saturation velocity), thus also 12 resp. 16 pairs of a and b coefficients.*

*In the further developed decompression model for the dive computers 8 tissues/compartments with variable half-live times were used (the saturation/desaturation velocity depends on the blood flow through the tissues (cold/exertion). Thus, the pairs of coefficients also are variable and calculated upon the ambient situation.*

## Linear relationship between ambient pressure and symptomless tolerated inert gas overpressure.

For longer dive times and deep diving, the ascent to the surface must be delayed. At no time during decompression, whether performed continuously or in stages, the tolerated inert gas overpressure for the ambient pressure may be exceeded. When diving in mountain lakes, the question arises, if the tolerance limits must be adjusted as the surface ambient pressure is lower compared to the to sea level.

The assumption of a practically linear relationship between ambient pressure and tolerated inert gas overpressure in the tissue is obvious. The linear relationship can be formulated mathematically in simple terms:

#### pt tol i.g. = (pamb. / b) + a

#### p amb tol = (pt i.g. – a) • b

The numerical value of the coefficient a depends on the pressure unit. The coefficient b has no dimension, it determines the gradient of the relationship between ambient pressure (p amb) and tolerated inert gas pressure in the tissue (pt tol i.g).

The linearity can be tested for the “slowest” tissues with saturation dives. After saturation with a helium sat a pressure of 30 bar, the ambient pressure can be reduced to 27.4 bar in a few minutes without causing discomfort at this first decompression level. At an ambient pressure of 1.0 bar, a pt He of 1.59 – 1.60 bar is tolerated. With these 4 numerical values, the coefficient a is 0.512 and the coefficient b is 0.927.

For Nitrogen, the experience of aviation medicine can be used to get the a and b coefficients. The pN_{2} after an acclimatization of several days at normal pressure, is abour 0.74 – 0.75 bar. With this value, a person can ascend to an altitude of 5’500 m above sea level in a few minutes. In other words, the ambient pressure can be reduced to 0.50 bar. Even if the stay at this altitude lasts a few hours, joint pain rarely occurs. At an ambient pressure of 1.0 bar, the tolerated pt N_{2} is 1.27 bar. With these empirical 4 values, the coefficient a for the “slowest” tissues and N_{2} is 0.233 bar and the coefficient b is 0.965.

Regarding the experience of aviation medicine, it can be objected that pilots breathe oxygen at an altitude of 5’000 – 6’000m above sea level and often stay only a short time at this altitude. Therefore, the coefficients for the longest N_{2} half-lives were tested in particular. In Zurich, the ambient pressure at mid-year is 0.967 bar, and the pN_{2} in all tissues at saturation with air breathing is 0.714 – 0.720 bar.

16 subjects from the Zurich region were decompressed to 0.46 bar (6’200m altitude) within 15 min in the vacuum pressure chamber with air breathing. They remained at this altitude for 3 h and performed 125 W, some subjects 200 W, on the bicycle ergometer every hour for 10 min. Some subjects complained symptoms of hypoxia. However, pain did not occur in any case. With the mentioned coefficients a and b, the allowed ambient pressure would be 0.474 bar.

The “slowest” tissues have the lowest tolerance limits. The example from aviation medicine and the described experiment in Zurich explain the empirical procedure for the determination of the coefficients a and b. An analogy to extreme alpinism arises at this point. In the Himalayas, it is possible to transport mountaineers with insufficient adaptation to altitudes of 7’000 m above sea level and more for a short time and to drop them there. Even with O_{2} breathing, the occurrence of skin symptoms and especially pain have to be expected in these cases, as it has been shown with simulation tests.

If helium and nitrogen gas are breathed together (trimix) or in succession (inert gas exchange), pHe and pN_{2} must be added for each compartment:

#### pt_{He} +pt_{N2} = pt _{i.g.}

The coefficients a and b are determined according to the N_{2} / He shares in the pt _{ig.}. For compartment No. 16 with helium half-life time of 240 min and N_{2} half-life time of 635 min, a coefficient a of 0.366 and a coefficient b of 0.9435 are obtained for a nitrogen fraction of 0.50.