Bühlmann decompression algorithm

Building on the previous work of John Scott Haldane^[2] (The Haldane model, Royal Navy, 1908) and Robert Workman^[5] (M-Values, US-Navy, 1965) and working off funding from Shell Oil Company,^[6] Bühlmann designed studies to establish the longest half-times of nitrogen and helium in human tissues.^[1] These studies were confirmed by the Capshell experiments in the Mediterranean Sea in 1966.^[6][7]

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

${\displaystyle P_{alv}=[P_{amb}-P_{H_{2}0}+{\frac {1-RQ}{RQ}}P_{CO_{2}}]\cdot Q}$ 

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

${\displaystyle P_{alv}=[P_{amb}-P_{H_{2}0}]\cdot Q}$ 

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

${\displaystyle {\dfrac {\mathrm {d} P_{t}}{\mathrm {d} t}}=k(P_{alv}-P_{t})}$ 

This equation can be solved for constant ${\displaystyle P_{alv}}$  to give the Haldane equation:

${\displaystyle P_{t}(t)=P_{alv}+(P_{t}(0)-P_{alv})\cdot e^{-kt}}$ 

and for constant rate of change of alveolar gas pressure ${\displaystyle R}$  to give the Schreiner equation:

${\displaystyle P_{t}(t)=P_{alv}(0)+R(t-{\dfrac {1}{k}})-(P_{alv}(0)-P_{t}(0)-{\dfrac {R}{k}})e^{-kt}}$ 

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

${\displaystyle P_{igtol}=a+{\frac {P_{amb}}{b}}}$ 

Where ${\displaystyle P_{igtol}}$  is the inert gas saturation limit for a given tissue and ${\displaystyle a}$  and ${\displaystyle b}$  constants for that tissue and inert gas.

The constants ${\displaystyle a}$  and ${\displaystyle b}$ , were originally derived from the saturation half-time using the following expressions:

${\displaystyle a={\frac {2\,{\text{bar}}}{\sqrt[{3}]{t_{1/2}}}}}$  ${\displaystyle b=1.005-{\frac {1}{\sqrt[{2}]{t_{1/2}}}}}$ 

The ${\displaystyle b}$  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).^[8]

Versions B and C have manually modified^[8] the coefficient ${\displaystyle a}$ .

In addition to this formulation, the Bühlmann model also specifies how the constants for multiple inert gas saturation combine when both Nitrogen and Helium are present in a given tissue.

${\displaystyle a=a_{N_{2}}(1-R)+a_{He}R}$ 

${\displaystyle b=b_{N_{2}}(1-R)+b_{He}R}$ 

where ${\displaystyle a_{N_{2}}}$  and ${\displaystyle a_{He}}$  are the tissue's ${\displaystyle a}$  Nitrogen and Helium coefficients and ${\displaystyle R}$  the ratio of dissolved Helium to total dissolved inert gas.

Ascent rate is intrinsically a variable, and may be selected by the programmer or user for table generation or simulations, and measured as real-time input in dive computer applications.

The rate of ascent to the first stop is limited to 3 bar per minute for compartments 1 to 5, 2 bar per minute for compartments 6 and 7, and 1 bar per minute for compartments 8 to 16. Chamber decompression may be continuous, or if stops are preferred they may be done at intervals of 1 or 3 m.^[9]

The Buhlmann model has been used within dive computers and to create tables.

Since precomputed tables cannot take into account the actual diving conditions, Buhlmann specifies a number of initial values and recommendations.

• Atmospheric pressure
• Water density
• Initial tissue loadings
• Descent rate
• Breathing gas
• Ascent rate

In addition, Buhlmann recommended that the calculations be based on a slightly deeper bottom depth.

Buhlmann assumes no initial values and makes no other recommendations for the application of the model within dive computers, hence all pressures and depths and gas fractions are either read from the computer sensors or specified by the diver and grouped dives do not require any special treatment.

Several versions and extensions of the Bühlmann model have been developed, both by Bühlmann and by later workers. The naming convention used to identify the set of parameters is a code starting ZH-L, from Zürich (ZH), Linear (L) followed by the number of different (a,b) couples (ZH-L 12 and ZH-L 16)^[10]) or the number of tissue compartments (ZH-L 6, ZH-L 8), and other unique identifiers.

ZH-L 12 (1983)

• ZH-L 12: The set of parameters published in 1983 with "Twelve Pairs of Coefficients for Sixteen Half-Value Times"^[10]

ZH-L 16 (1986)^[11]

• ZH-L 16 or ZH-L 16 A (air, nitrox): The experimental set of parameters published in 1986.
• ZH-L 16 B (air, nitrox): The set of parameters modified for printed dive table production, using slightly more conservative “a” values for tissue compartments #6, 7, 8 and 13.
• ZH-L 16 C (air, nitrox): The set of parameters with more conservative “a” values for tissue compartments #5 to 15. For use in dive computers.
• ZH-L 16 (helium): The set of parameters for use with helium.
• ZH-L 16 ADT MB: set of parameters and specific algorithm used by Uwatec for their trimix-enabled computers. Modified in the middle compartments from the original ZHL-C, is adaptive to diver workload and includes Profile-Determined Intermediate Stops. Profile modification is by means of "MB Levels", personal option conservatism settings, which are not defined in the manual.^[12]

ZH-L 6 (1988)

• ZH-L 6 is an adaptation^[13] (Albert Bühlmann, Ernst B.Völlm and Markus Mock) of the ZH-L16 set of parameters, implemented in Aladin Pro computers (Uwatec, Beuchat), with 6 tissue compartments (half-time : 6 mn / 14 mn / 34 mn / 64 mn / 124 mn / 320 mn).

ZH-L 8 ADT (1992)

• ZH-L 8 ADT: A new approach with variable half-times and supersaturation tolerance depending on risk factors.^[13] The set of parameters and the algorithm are not public (Uwatec property, implemented in Aladin Air-X in 1992 and presented at BOOT in 1994). This algorithm may reduce the no-stop limit or require the diver to complete a compensatory decompression stop after an ascent rate violation, high work level during the dive, or low water temperature. This algorithm may also take into account the specific nature of repetitive dives.

• ZH-L 8 ADT MB: A version of the ZHL-8 ADT claimed to suppress MicroBubble formation.^[14]
• ZH-L 8 ADT MB PDIS: Profile-Determined Intermediate Stops.^[15]
• ZH-L 8 ADT MB PMG: Predictive Multi-Gas.^[16]

Many articles on the Bühlmann tables are available on the web.

• Chapman, Paul (November 1999). "An Explanation of Professor A.A. Buehlmann's ZH-L16 Algorithm". New Jersey Scuba Diver. Archived from the original on 2010-02-15. Retrieved 20 January 2010. – Detailed background and worked examples
• Decompression Theory: Robert Workman and Prof A Bühlmann. An overview of the history of Bühlmann tables
• Stuart Morrison: DIY Decompression (2000). Works through the steps involved in using Bühlmann's ZH-L16 algorithm to write a decompression program.