Bühlmann decompression algorithm: Difference between revisions

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Building on the previous work of [[John Scott Haldane]]<ref name="haldane" /> (The Haldane model, Royal Navy, 1908) and Robert Workman<ref name="Workman65" /> (M-Values, US-Navy, 1965) and working off funding from [[Shell Oil Company]],<ref name=Pressure94/> Bühlmann designed studies to establish the longest [[Half time (physics)|half-times]] of nitrogen and helium in human tissues.<ref name="AAB1984" /> These studies were confirmed by the ''Capshell'' experiments in the [[Mediterranean Sea]] in 1966.<ref name="Pressure94" /><ref name="pmid6053671" />

===Tissue inert gas exchange===

The basic idea (Haldane, 1908)<ref name="haldane" /> is to represent the human body by multiple tissues (compartments) of different saturation half-times and to calculate the partial pressure <math>P</math> of the inert gases in each of the <math>n</math> compartments (Haldane's equation):

Inert gas exchange in haldanian models is assumed to be perfusion limited and is governed by the ordinary differential equation

<big><math>P = P_0 + (P_{gas} - P_0) \cdot (1 - 2^dfrac{-\frac{t_mathrm{expd}P_t}{t_\mathrm{1/2d}t} = k(P_{alv} - P_t)</math></big>

This equation can be solved for constant <math>P_{alv}</math> to give the so-called Haldane equation

with the initial partial pressure <math>P_0</math>, the partial pressure in the breathing gas <math>P_{gas}</math> (minus the vapour pressure of water in the lung of about 60&nbsp;mbar), the time of exposure <math>t_{exp}</math> and the compartment-specific saturation half-time <math>t_{1/2}</math>.

<math>P_t(t) = P_{t0} + (P_{t0} - P_{alv}) \cdot e^{-kt}</math>

When the gas pressure drops, the compartments start to off-gas.

which is frequently expressed in decompression theory literature as

===Nitrogen (air, nitrox) set of parameters===

To calculate the maximum tolerable pressure <math>P_{t.tol}i.g.</math>, the constants <math>a</math> and <math>b</math>, which are derived from the saturation half-time as follows (ZH-L 16 A):

<math>aP_t(t) = \fracP_{2\,\textt0} + (P_{atm}alv} - P_{t0})\sqrt[3]{t_{cdot (1/2}} - e^{-kt})</math>

===Alveolar inert gas pressure===

<math>b = 1.005 - \frac{1}{\sqrt[2]{t_{1/2}}}</math>

The Bühlmann model uses a simplified version of the respiratory equation to calculate alveolar inert gas pressure

are used to calculate M-Value (<math>P_{t.tol}i.g.</math>)<ref>{{Citation |last=Bühlmann |first=Albert A. |title=Dekompressionstabellen |date=1990 |work=Tauchmedizin |pages=109–117 |url=http://dx.doi.org/10.1007/978-3-642-97256-0_9 |access-date=2024-05-20 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |isbn=978-3-540-52533-2}}</ref>:

<math>P_{t.tolalv}i.g. = \frac{[P_{amb.}} - P_{H_{b2}0} +a (\frac{1 - RQ}{RQ} P_{CO_{2}}]\cdot Q</math>

Where <math>P_{H_{2}0}</math> is the water vapour pressure at 37 degrees centigrade (conventionally defined as 0.0627 bar), <math>P_{CO_{2}}</math> the carbon dioxide pressure (conventionally defined as 0.0534 bar), <math>Q</math> the inspired inert gas fraction, and <math>RQ</math> the respiratory coefficient: the ratio of carbon dioxide production to oxygen consumption. The Buhlmann model sets <math>RQ</math> to 1, simplifying the equation to

<math>P_{alv} = [P_{amb} - P_{H_{2}0}]\cdot Q</math>

===Tissue inert gas limits===

Similarly to Workman, the Bühlmann model specifies an affine relationship between ambient pressure and inert gas saturation limits. However, the Buhlmann model expresses this relationship in terms of absolute pressure

<math>P_{igtol} = a + \frac{P_{amb}}{b}</math>

Where <math>P_{igtol}</math> is the inert gas saturation limit for a given tissue and <math>a</math> and <math>b</math> constants for that tissue.

To calculate the maximum tolerable pressure <math>P_{t.tol}i.g.</math>, theThe constants <math>a</math> and <math>b</math>, whichwere areorignally derived from the saturation half-time asusing follows (ZH-Lthe 16following A)expressions:

<math>a = \frac{2\,\text{atm}}{\sqrt[3]{t_{1/2}}}</math>

<math>b = 1.005 - \frac{1}{\sqrt[2]{t_{1/2}}}</math>

The <math>b</math> values calculated do not precisely correspond to those used by Bühlmann for tissue compartments 4 (0.7825 instead of 0.7725) and 5 (0.8126 instead of 0.8125).<ref name="Buhlmann-a-b-2002" />