Mathematical model of tissue inert gas uptake and release with pressure change
The Bühlmann decompression model is a
Haldanian which models the way
inert gases enter and leave the human body as the
ambient pressure changes.[1] Versions are used to create
decompression tables and in personal dive computers to compute no-decompression limits and decompression schedules for dives in real-time, allowing
divers to plan the depth and duration for dives and the required
decompression stops.
Multiple sets of parameters were developed by Swiss physician Dr.
Albert A. Bühlmann, who did research into decompression theory at the Laboratory of Hyperbaric Physiology at the University Hospital in
Zürich,
Switzerland.[3][4]
The results of Bühlmann's research that began in 1959 were published in a 1983 German book whose English translation was entitled Decompression-Decompression Sickness.[1] The book was regarded as the most complete public reference on decompression calculations and was used soon after in
dive computer algorithms.
Principles
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 basic idea (Haldane, 1908)[2] is to represent the human body by multiple tissues (compartments) of different saturation half-times and to calculate the partial pressure of the inert gases in each of the compartments (Haldane's equation):
with the initial partial pressure , the partial pressure in the breathing gas (minus the vapour pressure of water in the lung of about 60 mbar), the time of exposure and the compartment-specific saturation half-time .
When the gas pressure drops, the compartments start to off-gas.
Nitrogen (air, nitrox) set of parameters
To calculate the maximum tolerable pressure , the constants and , which are derived from the saturation half-time as follows (ZH-L 16 A):
are used to calculate M-Value ():
The values calculated do not 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 .
The modified values of and are shown in bold in the table below.
Helium (heliox) set of parameters
According to Graham's Law, the speed of diffusion (or effusion) of two gases under the same conditions of temperature and pressure is inversely proportional to the square root of their molar mass (28.0184 g/mol for and 4.0026 g/mol for , i.e. ), which means that molecules diffuse 2.645 times faster than molecules.
Bühlmann took this into account and divided all the tissue compartment half-time for air (nitrogen) by 2.645 to obtain a helium-specific set of parameters with the longest compartment set at
The parameters of the M-Values (coefficients a and b) were determined specifically.
Trimix (nitrogen + helium) set of parameters
No model can manage the de-saturation of two inert gases.
Some approaches only take into account the main inert gas (and ignore the other inert gas).
With Bühlmann,[9] a weighted average of the half-times and coefficients and is calculated as a function of the percentage of each inert gas to calculate a specific set of parameters.
Example :
Using a 18/50 trimix (18% , 50% , 32% ), the half-time (or the and coefficients) of compartment #1 is calculated by taking 50% of the half-time and 32% of the half-time divided by 50% + 32% = 82%.
Example, compartment #1:
(instead of with and with )
( with and with )
( with and with )
The same calculations can be made using partial pressures rather than percentages.
This approach is controversial with some authors[10] who feel that this calculation does not reflect what should be achieved. Generally speaking, the fact that desaturation with two neutral gases is not modelled encourages caution. Each trimix dive is specific, with no guarantee.
Constant partial pressure of oxygen (closed-circuit rebreathers - CCR)
There are no specific model for constant dives. The difference lies in the fact that, at all times, the proportion of inert gas is calculated in relation to the chosen (e.g. 0.75 or 1.3 ata (bar)).
Table of ZH-L 16 Half-times with and values for nitrogen () and helium ().[8]
Cpt
ZH-L 16
ZH-L 16 A
(min)
A
Experimental
B
Tables
C
Computers
(min)
01 (1a)
004
1.2599
1.2599
1.2599
0.5050
001.51
1.7424
0.4245
01b
005
1.1696
1.1696
1.1696
0.5578
00
02
008
1.0000
1.0000
1.0000
0.6514
003.02
1.3830
0.5747
03
012.5
0.8618
0.8618
0.8618
0.7222
004.72
1.1919
0.6527
04
018.5
0.7562
0.7562
0.7562
0.7825
006.99
1.0458
0.7223
05
027
0.6667
0.6667
0.6200
0.8126
010.21
0.9220
0.7582
06
038.3
0.5933
0.5600
0.5043
0.8434
014.48
0.8205
0.7957
07
054.3
0.5282
0.4947
0.4410
0.8693
020.53
0.7305
0.8279
08
077
0.4701
0.4500
0.4000
0.8910
029.11
0.6502
0.8553
09
109
0.4187
0.4187
0.3750
0.9092
041.2
0.5950
0.8757
10
146
0.3798
0.3798
0.3500
0.9222
055.19
0.5545
0.8903
11
187
0.3497
0.3497
0.3295
0.9319
070.69
0.5333
0.8997
12
239
0.3223
0.3223
0.3065
0.9403
090.34
0.5189
0.9073
13
305
0.2971
0.2850
0.2835
0.9477
115.29
0.5181
0.9122
14
390
0.2737
0.2737
0.2610
0.9544
147.42
0.5176
0.9171
15
498
0.2523
0.2523
0.2480
0.9602
188.24
0.5172
0.9217
16
635
0.2327
0.2327
0.2327
0.9653
240.03
0.5119
0.9267
Versions
Several versions of the Bühlmann set of parameters 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)[11]) or the number of tissue compartments (ZH-L 6, ZH-L 8), and other unique identifiers.
For example:
ZH-L 12 (1983)
ZH-L 12: The set of parameters published in 1983 with "Twelve Pairs of Coefficients for Sixteen Half-Value Times"[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.[13]
ZH-L 6 (1988)
ZH-L 6 is an adaptation[14] (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.[14] 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.[15]
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.[18]
^Bühlmann, Albert A. (1982). "[Experimental principles of risk-free decompression following hyperbaric exposure. 20 years of applied decompression research in Zurich]". Schweizerische Medizinische Wochenschrift (in German). 112 (2): 48–59.
PMID7071573.
^
abVöllm, T.G. (1994). "Leading diving researcher dies unexpectedly: Albert A Bühlmann, 1923 - 1994". Pressure, Newsletter of the Undersea and Hyperbaric Medical Society. 23 (3): 1–3.
ISSN0889-0242.
^Bühlmann, Albert A.; Frei, P.; Keller, Hannes (October 1967). "Saturation and desaturation with N2 and He at 4 atm". Journal of Applied Physiology. 23 (4): 458–62.
doi:
10.1152/jappl.1967.23.4.458.
PMID6053671.
Keller, Hannes; Bühlmann, Albert A (November 1965). "Deep diving and short decompression by breathing mixed gases". Journal of Applied Physiology. 20 (6): 1267–70.
doi:
10.1152/jappl.1965.20.6.1267.
Bühlmann, Albert A (1992). Tauchmedizin: Barotrauma Gasembolie Dekompression Dekompressionskrankheit (in German). Berlin: Springer-Verlag.
ISBN3-540-55581-1.
Bühlmann, Albert A (1995). Tauchmedizin (in German). Berlin: Springer-Verlag.
ISBN3-540-55581-1.
External links
Many articles on the Bühlmann tables are available on the web.