Update per 03 / 2021 on:
Decompression-Calculations for Trimix Dives with PC-
Gradient Factors: do they repair defective algorithms or do
they repair defective implementations?
(the then DOI of the original paper: 10.13140/RG.2.2.35405.87527)
(to be skipped by novice readers; for the seasoned ones, especially those, who read
already the old paper, there will be new information and as well, added value):
Miri & I (Miri Rosenblat from our Lab) decided to put out an substantial update on this paper:
since 2011 it has been download a couple of thousand times from our non-commercial
website; just recently from researchgate.net it was downloaded approx. 500 times already
within a very short time frame: the ideas, the methods and the results still valid. So why an
1) The links to nearly all the references have changed
2) One of the companies from our benchmark has ceased to exist, their products no
longer working properly, so we may reveal this name without violating the laws of fair
3) Also, this will be no longer a draft or a pre-print: instead, we presented this material
as a paper to the “TEC 4.0: update” conference in ETH / IL.
• The RUBICON portal (obsolete URL: archive.rubicon-foundation.org), where nearly
all of the used references of the old paper pointed to, stopped to work.
• The COCHRAN company has ceased to exist due to the sudden death of its founder,
The distributor of GAP, the Gas Absorption Programme, has ceased to exist. Their products,
at least the then latest software version we used for this benchmark in 2010 (version 3), are
no longer working properly due to a hard-coded reference to the internet: their license- &
update-server is no longer reachable.
There may be older versions which are still running: for eg. in our lab we still have a MS-
Windows XP-environment in a virtual machine where the version 1.2, Build 298 from 2000
runs. But promise: we won’t check again the 480 profiles to see if these errors will also show
up (in our opinion, they will).
Ok, ok: you may question if this was a conference at all during Corona-shutdown. But
anyway we (Miri, Nurit & I) had fun, sitting at the beach of the Aquasport 50 Bar and
distributing our knowledge and findings and discussing with a selected handful of Tec-
Instructors and a couple of unfiltered Goldstar’s or Macabi’s 7,8% …:
Chart 1: the Corona-abandoned Aqua Sport, ETH / IL
We decided to combine all relevant references into one ZIP archive: the link to download it
from our non-commercial website you will find at the end of this paper, in the reference
section. This will be no violation of the copyrights, as a substantial numbers of the authors
are deceased. And, as well we are in contact with a couple of the bereaved. For eg. Thomas
Bühlmann: he is working on his ph.d thesis on the implications of his fathers work and willing
to support us.
For the ease of reading and comparison with the old version: only the substantial updates
have been receiving a colored display.
Still, after 10 years, our thanks goes to the girls and boys who fiddled with the software.
Especially the bug-ridden graphical interface of the GAP software gave us a big “ שׁ אֹ ר בֵאְכּ ”.
Yes, and still we may not reveal their names: all of them have been active in the IDF then,
some of them even in the הים חיל but without their help, ideas and dedication, this paper
would not have been existing.
If there is more than one inertgas in the breathing mixture, the calculation of the
decompression-time td has to be done numerically. We analyzed 480 square / box dive-
profiles in the TEC/REC range with one freeware, two commercially available software-
packages and via numerical methods (depth range: 30 - 80 m, bottom times: 20 - 60 min,
helium percentage: 5 - 80 %, only normoxic mixes i.e.: no travel- or enriched deco gases,
only ZH-L model, no adaptions with gradient factors). There are significant differences in the
calculation of the decompression-times td with trimix gases, obviously dependent on the
helium percentage. In the present analysis, these differences do not come from variations in
the decompression algorithms.
decompression, decompression algorithm, trimix, mixed gas, perfusion models, technical
diving, ZH-L, VPM, Euler method, GF, gradient factor, Haldane, Workman, Schreiner, Müller,
Ruf, Bühlmann, Hahn.
This is a translated, and, in parts, abbreviated version of a paper which appeared in:
CAISSON 2011, 26(3): 4 – 12. Several parts of this paper I presented during a lecture for
which I was invited to the 12.th scientific meeting of the GTUEM (www.gtuem.org) ,
03/20/2011 in Regensburg, Germany; the abstract is under: CAISSON 2011, 26(1): 61.
The extended german version you will find at:
An „Algorithm“ is just a mathematical rule for inert gas bookkeeping during an exposure to
overpressure. An „Implementation“ is the practical translation of this algorithm into a piece of
software, be it for a dive computer or a desktop deco software. A „Gradient Factor“ is a factor
< 1. It is used to multiply the calculated allowed / tolerated inertgas partialpressures in the
various body tissues, the “compartments”; thus a more conservative decompression method
is forced via pure mathematics. With “ZH-L” a certain group of disolved gas deco models is
denoted, the most prominent researchers names are: Haldane, Workman, Schreiner,
Mueller, Ruf, Buehlmann and Hahn (pls. cf. the references).
As there is still debate about the “L”, once and for all:
ZH: is the ISO abbreviation of the city of Zuerich, Switzerland.
L: is for linear, since the equation for allowed/tolerated inertgas partialpressures is a linear
one, as in all the other simple perfusion models, which came by far earlier than Bühlmann.
16: is the number of a-/b coefficients for one inertgas (so its 32 for trimix …) and 12 for both
N2 & He in the ZH-L12, the pre-decessor of the ZH-L 16, pls. cf. Ref. [C]. in the reference
The classical, perfusion-limited decompression algorithms after Haldane et al. describe the
absorption of inert gases per compartment through a mono-exponential function. Normally
the term „Haldane Equation“ is used:
Pt(t) = Palv0 + [Pt0-Palv0] e-kt
inertgas partialpressure within a compartment with the constant k [Bar] at time t
after an instantaneous change in pressure
initial partialpressure of the inertgas within the compartment at time t=0 [Bar]
the constant partialpressure of the inertgas in the alveoli [Bar], for t = 0 and thus for
all t due to the boundary conditions
a constant, dependent on the compartment [min
], with k = ln 2 / τ
The exponent k is basically the perfusion rate, i.e. the inverse of the half-time τ of a model
tissue. These model tissues are called „compartments“. The adaption of a purely
mathematical algorithm to a physiological system is done via a flock of these compartments,
typically 6, 9 or 12, 16 and sometimes as well 20 (or even more). The variability comes with
the different halt-times into play. A typical spectrum of these half-times is from 1.25 to 900
minutes; for e.g. in a dive computer for professional and navy use, the EMC-20H from
Cochran and the corresponding desktop deco-software Analyst 4 (www.divecochran.com).
The mainstream sources for these perfusion algorithms are wellknown and listed in the
appendix. But now we want to try something new and draw upon a source which is relatively
 Hills, Brian Andrew (1977), Decompression Sickness, Volume 1,
The Biophysical Basis of Prevention and Treatment
Formula (1) is there on page 111, the relationship between the half-times and the
perfusionrate is on page 113.
Limits of the perfusion-models:
The perfusion-models for Air/Nitrox/EAN and Heliox as breathing gases are based
worldwide on a very broad number of well-documented dives. They are mathematically
straightforward and have since the papers of Buehlmann (, , ) enjoyed popular
implementations in many dive computers and PC programs (Desktop-Deco-Software). But
the technical diver as such wants to dive deeper / longer and thus is inclined to forget the
trusted envelope. Nonetheless this envelope is already published at length (e.g. in , p.
449 and 463) and is dealing with a couple of the following points, here just as a short
overview and not limited to:
• only „inertgas-bookkeeping“, only mono-exponential for one compartment
• these compartments are all in a parallel circuit, the linear connections like spleen ->
liver & bowel -> liver are not considered
• inconsistent consideration of the metabolic gases O2, CO2 and H2O
• as well: passive / active capillaries, esp. in the lung during exercise
• „uneventful“ decompression, only the gas in solution is considered and not the free
gas phase (bubbles)
• no allowance is made for short-term pressure changes which are small against the
fastest half-times. This is the “low pass” phenomenon, especially for jo-jo-dives [A].
• the calculation of inert gas saturation and de-saturation is done in a symmetrical
manner, i.e. with the identical coefficient in the exponential terms of (1)
• clientele / biometrics and adaption are not reflected in the algorithms
• as well not these circumstances, which affect tec divers even more due to massive
impact on blood-perfusion: de-hydration, workload, temperature and excessive
oxygen partial pressures
• and: the 2nd. inert gas; the 2nd. (n-th) repetitive dive; and, and, and, …
Just a small choice of sources to these points:
Thalmann, ED; Parker, EC; Survanshi, SS; Weathersby, PK.
Improved probabilistic decompression model risk predictions using linear-exponential
kinetics. Undersea Hyper. Med. 1997; 24(4): 255 – 274
Tikuisis, P; Nishi, RY.
Role of oxygen in a bubble model for predicting decompression illness.
Defence R&D Canada, 1994; DCIEM-94-04
Doolette DJ, Gerth WA, Gault KA.
Probabilistic Decompression Models With Work-Induced Changes In Compartment Gas
Kinetic Time Constants. Navy Experimental Diving Unit, Panama City, FL, USA; in: UHMS
Annual Scientific Meeting, St. Pete Beach, Florida, June 3-5, 2010, Session A6.
Hahn MH. 1995. Workman-Bühlmann algorithm for dive computers: A critical analysis. In:
Hamilton RW, ed. The effectiveness of dive computers in repetitive diving. UHMS workshop
81(DC)6-1-94. Kensington, MD: Undersea and Hyperbaric Medical Soc
For Heliox (oxygen & helium mixtures) there is a great abundance of validated tables: quite
in contrary to Trimix (oxygen, helium and nitrogen). There are none (almost). Surely enough
there is anecdotal evidence of sucessfull trimix-decompressions, but limited to a couple of
custom mixes, with a limited group of test persons and limited in the dive profiles. But
„validated“ here means a completely other league of game. It is a journalled procedure in a
decompression chamber, run for a big number of various depth/time combinations, each of
them with big numbers of dives. The journal is a detailed and reproducible log of the
following parameters: biometrics of test persons, time of the day, depth, time, ascent- and
descent-rates, surface intervall (even multi-day), breathing gas composition and- humidity/ -
temperatures, temperatures in the chamber and wet-pot, type of immersion and work-load.
The outcomes (DCS or # of doppler detected bubbles) have to be checked via double-
blinded operators. And when the number of test-persons exceeds the 3-digit limits and the
number of test-dives is in the 4- or even 5-digit range (as with NEDU, DCIEM and COMEX
tables) then there might be a certain tenacity. But none of the known trimix tables is meeting
these requirements. Maybe a laudable exception is the NOAA Trimix 18/50 Table from
Hamilton Research Ltd., 1993, 1998.
Just for the fun of it we draw from the „Journal of Applied Physiology“ the number and
temporal distribution of research papers concerning “trimix“ (title & keyword) from 1948 to
2020 and compared with other topics (Tables (1a) & (1b)):
The papers concerning „air“ are in brackets and only to compare the absolute numbers since
the relationship to exposure to overpressure is not always the case. The first paper was
around 1976; the graph below shows the last 30 years and features a peak in the year 2007.
This results from short discussion-papers concerning the (in)-validity of Henry’s Laws,
especially with binary (half/half) inertgas-mixtures:
The somewhat singulary paper in 2010 is from Ljubkovic et al. (pls. cf. the references), and
reflects very well our topic here, however with a VPM / bubblemodel and is really interesting
for hyperbaric (-diving) physicians. But generally speaking we have here the tendency that
trimix plays only a somewhat neglected role in serious research on diving humans. To put it
the heavily exposed trimix diver is his own guinea pig.
The decompression time td for un-ary mixes (i.e. only one inertgas like EAN or heliox) can be
calculated directly with the Haldane equation (1). This is documented already and elsewhere
(for e.g.: https://www.divetable.info/skripte/theory.pdf), here is the analytic expression for the
decompression time t = td:
t = - τ / ln2 * ln[ (Pt(t) - Palv0) / (Pt0 - Palv0) ]
The criteria for „safe“ decompression within the perfusion-models is a simple linear (straight
line) equation (, p. 117, resp.: , p. 119 ff):
Pt.tol.ig = Pamb / b + a
tolerated inert gas partial pressure, for each compartment, (analog to M) [Bar],
the sum of all inert gas partial pressures
limit of a theoretical ambient pressure of 0 Bar, i.e. the axis intercept [Bar]
ambient pressure, absolute pressure of all breathing gases [Bar]
1/b pressure gradient: increase per unit of depth (dimensionsless), i.e.: the slope of
the straight line
These a-/b-coefficients are constants, tabulated for look up, e.g.: in  p. 27, in  p. 108 &
109, as well in  on p. 158.
A direkt mapping of equation (3) onto other perfusion models, e.g. the „M-Value“ model of
Workman or Schreiner or a relatively modern air table for commercial diving, the MT92 [B], is
done via a comparison of the parameters and the conversion of the SI-units to imperial;
described elsewhere and, as well, here: https://www.divetable.info/skripte/theory.pdf )
During the course of the century the number and absolute values of the coefficients changed
from author to author: this is mostly the reflection of an increasingly conservative
decompression, that is: longer deco stops (pls. cf. Egi et al.).
The analytical expression (2) is only possible with one inert gas, in this case N2 . With more
than one inert gas the calculation of td has to be done numerically, via an approximation
procedure, that is: by trial-and-error. With Tri-Mix we have 2: N2 (nitrogen) and He (helium);
the 3rd. component of trimix is oxygen (O2), not considered in standard perfusion models.
Thus we have to calculate the inert gas absorption for these 2 inert gases separately. This is
a standard procedure, already described elsewhere, for eg. also by Buehlmann in , p.
Pt(t) = Pt, He(t) + Pt, N2(t) (4)
The differences are in the molecular weights, the solubility coefficients and the diffusion
constants (pls. cf.: Rostain JC, Balon N. Nitrogen Narcosis, the High Pressure Nervous
Syndrome and Trimix. In: Moon RE, Piantadosi CA, Camporesi EM (eds.). Dr. Peter Bennett
Symposium Proceedings. Held May 1, 2004. Durham, N.C.: Divers Alert Network, 2007; as
well: , p. 118)
But now the criteria for „safe“ ascent has to be adapted as well to 2 inert gases, (3) changes
simply to (3*):
Pt.tol.ig = Pamb / b* + a*
Here as well there is a simple procedure to determine these new a* and b* -coefficients. The
old a- and b-coefficients (table look-up) for both of the gases are normalized with the
prevailing inert gas partial pressures for each of the compartments (pls. see the remark in
 on p. 86). Thus we have for any combination of a- and b-values for each compartment at
any time t:
a* = a (He + N2) = [( Pt, He * aHe ) + ( Pt, N2 * aN2)] / ( Pt, He + Pt, N2 )
b* = b (He + N2) = [( Pt, He * bHe ) + ( Pt, N2 * bN2)] / ( Pt, He + Pt, N2 )
Pls. see as well the examples in , p. 27; , p. 80 and Rodchenkov et al, p. 474.
The ascent criteria (3*) are now time-dependent by itself, the a*- & b*-coefficients are via (5)
married with the time-dependent exponential expressions of saturation/desaturation and no
longer any constants as per air/EAN or heliox.
The mapping of the compartment halftimes from N2 to He is normally done according to
Graham‘s law with the square root of the proportion of the molecular weights (i.e.: ca. 2.65).
This factor is now keyed in, uniform to all compartments. And exactly at this point we meet
the criticism of serious researchers in the field: D‘ Aoust et al, p. 119 & 121; as well: Lightfoot
et al, p. 453 and: Voitsekhovich, p. 210. In experiments we see the perfusion rates quite
differently! The pivotal 2.65 is, so it seems, really valid only for saturation exposures
(Berghage et al, p.6). But saturation is a state which even the bold tec-diver does not reach
easily … (Well, there are bold divers and there are old divers. But there are no ... Ok, Ok:
you already know the rest of the story ...)
To put it simply: the deco time td is now on the left and the right hand side of eq. (2), a simple
analytical expression to solve for td is not possible due to the exponential sums. How can we
then evaluate td ?
Basically there are at least 3 simple methods. We look at them only skin-deep because they
are described at length already elsewhere (again, for e.g.:
A) „Trial-and-Error“: for small increments in time, e.g. 1 second or 0.1 minute, we
calculate all relevant terms and check if the ascent criteria is met. This is called a
classical „numerical“ solution.
B) „Quasi-Analytical“: we accept tacitly an error by using eq. (2) without changes. Thus
we consider the a*-/b*-coefficients as constants for each phase of the
C) An approximation method: all the exponential terms are approximated via a
polynomial expression, aka „Taylor Expansion“ (Bronstein, Chapter: Expansion in
For commercially available off-the-shelf (COTS) desktop deco software method A) should be
preferred since the computing power of topical PC hardware does not impose any waiting-
time for the users. Thus quite in contrary to standard mix gas diving computers. Due to the
relatively high cost of development for water-proof hardware and, in comparison to other
mobile electronic devices like SmartPhones, virtually negligible lot sizes, there are regularly
no full-custom ASICs in favour of relatively cheap standard chips. These standard chips are
somewhat “slower” and brilliant in a gigantic energy consumption ...
The numerical solution A) consumes, in comparison to method B) more computing power
and thus time and more variables and memory: all of the 3 we do not have plenty under
water! It is thus self-evident to insinuate method B) where cost are at premium and we need
a result on the spot.
How is this handled with commercial standard products? The crux is that producers of dive
computer hardware and deco software are regularly not willing to answer such inquiries with
hints to company secrets. Or, answers are cryptic and thus give room for conjecture!
But to answer this question halfway satisfactorily, we have developed the following
experimental method: 480 square dive profiles from the TEC- and REC- domain with the
depth range: 30 - 80 m (6 profiles at 10 m distance), and bottom times : 20 - 60 min (5
profiles in 10 min increase), with helium fractions: 5 - 80 % (16 profiles in 5% increments),
only with one normoxic mix (i.e.: no travel gases and no EAN deco mixes) have been
evaluated each with 4 software products and compared:
• two commercially available off-the-shelf deco software products,
• one Freeware/Shareware version of DIVE (obsolete, source:
http://www.divetable.de/dwnld_e.htm , version 2_900), and, as well
• version 3_9 of DIVE.
Version 3_9 has implemented exactly the method A), the (now obsolete, since not running
under Win7 or higher) public version 2_900 is flawed with the “blunder” of method B). For the
2 COTS products there are no reliable statements available despite insistent and repeated
As a first step, these 4 products have been tested against each other with 40 different air-
and 40 different Nitrox/EAN32 profiles. Thus we checked the actual convergence of the
numerical method A with the COTS products. As one paradigm we have the following table
(2) with the TTS values for a square dive to 40 m with the bottom times ranging from 20 to
Table (2): TTS vs. the 4 products; TTS = time-to-surface, i.e. sum of all deco stop times +
time for ascent (i.e.: bottom depth / ascent rate)
As well a sensitivity analysis was made for the numerical solution in order to make sure that
minor variations in the starting parameters do not lead to mathematical artefacts. In the end
we compared the 4 against the „Gold Standard“, the „Zuerich 1986 table for air dives“ (ZH-
86) of A. A. Buehlmann (, p. 228). Here we have deviations of + / - 2 min per deco stage,
as well sometimes the staging begins 3 m deeper in comparison to the table. This comes
mainly from the different sets of coefficients: the ZH-86 table uses the ZH-L 16 B set (, p.
158), whereas deco software or dive computers are using normally the ZH-L 16 C set (,
l.c.). As well printed tables are treating truncations in a completely different way than dive
computers. Even the great ex-champion from the NEDU (the United States Navy
Experimental Diving Unit), Cptn. Dr. Edward Thalmann had to admit, that a published diving
table does not jars with a computer-output:
“I think some were just manually adjusted. They just went in and empirically added
five minutes here and five minutes there, yeah.”
(Source: Edward Thalmann,  Naval Forces under the Sea: The Rest of the Story, p. 63
– 70, 197, 274, 361 and as well, the CD “Individual Interviews”).
Similar things may have been happened as well with OSHA tables for caisson/tunnel work
(until 1979). But these have been coined as „typographical errors“ (Kindwall, p. 342).
To force comparability all the calculations are based solely on the set ZH-L 16 C (, p.
158) and there are no manipulations via gradient factors. As well there are slight adaptions of
40 m, Nitrox/EAN 32 bottom times [min]: 20' 30' 40' 50' 60'
TTS DIVE 2_900 816 28 42 55
TTS DIVE 3_0: numerical solution 717 28 40 57
TTS COTS product 3 515 28 41 53
TTS COTS product 4 716 28 41 54
the dive profiles via ascent- and descent rates to make sure that the bottom times and the
inert gas doses are matching.
Evidently there are significant differences in the calculation of the deco times in dependence
of the helium-fraction and the amount of decompression obligations, vulgo the inert gas
dose, see chart (2). These differences are not due to variations in the decompression
algorithm but rather exclusively through different ways of calculation.
Chart (2) shows the deviation of the TTS in dependence of the helium fraction, here as an
example for a dive to 40 m with a bottom time of 40 min.:
x axis: percentage of helium in the breathing mix: from 10 to 80 %
y axis: Delta TTS is a difference of the numerical solution to an arithmetic mean out of the 3
TTS according to:
Σ (td,1 + td,2 + td,3) / 3
the td,i being the calculated td of the products i = 1 - 3 (DIVE 2_900, COTS product 3, COTS
product 4). The x axis is defined as the zero baseline of the TTS of the numerical
solution. An “error” in [minutes] is coined as the deviation (Delta TTS) of this mean value
against the TTS of the numerical solution.
The calculation of this arithmetic mean was superimposed by the strong closeness of the td
from the 3 products. The absolute errors (see the vertical error margins) are increasing with
the increase of the inert gas dose and with the increase of the helium fraction. The above
represented curve progression is more or less universal for all of the 480 square profiles.
Speaking simplified, qualitatively:
in the region of the helium fractions 5 % up to ca. 25 % the TTS is overrated: positive
error; i.e. the TTS is too long, the decompression is too conservative.
in the region of helium fractions which is relevant to most tec divers, that is ca. 30 –
ca. 40 %, the error vanishes: Delta TTS -> 0, and
increases with increasing helium fraction. In this region the error is negative, i.e. the
TTS is too short, the decompression is too liberal.
The results of the 2 COTS products and DIVE 2_900 came very close to each other thus a
somewhat similar calculation method is supposed. But this „similar“ method means in plain
language: the „blunder“ of DIVE 2_900 could be repeated in the implementations of the 2
COTS products ... To put it even more bluntly: the relative identity of the absolute values and
the prefix leave room for the guesswork that the 2 COTS products are using method B).
Well, there are quite a couple of other factors who could have been responsible for these
deviations. To name just a few:
• undocumented gradient factors
• a respiratory coefficient unequal to 1
• another weighting of other inert resp. metabolic gases
• another weighting of the water density, i.e.:
• salinity & temperature of ambient water
• different ambient air pressure at start of dive / end of decompression
• „empirically“ adapted a-/b coefficients, especially for helium and as a consequence:
• small deviations from the original helium ZH-L spectrum of half-times (i.e. a mismatch
of a and b with the half time)
• utilisation of the so-called „1b“ compartment instead or additive to compartment „1“
(, p. 158);
• different ascent rates, or: ascent rates varying undocumented with depth
• different approach to truncations
• and, connected: mismatch of floating point precision, resp.:
• mismatch of the math libraries linked at compilation for the problem at hand;
• failure to implement a rapid converging Euler (or other) method
• including a mismatch in step size to the problem at hand
„Walking stick“ solutions for software implementations due to restrictions of the hardware
have been quite common in the early days of dive computers: for e.g. there was a product in
europe which could only interpolate linearly between stored values instead of calculating a
full-blown saturation/desaturation. But even today there are implementations which rely on a
modified ZH-L instead of the promised (advertised) RGBM model ...
But it seems that there are implementations taking this topic seriously. Amongst others there
is a shareware with a VPM model
(http://www.decompression.org/maiken/VPM/VPM_Algorithm.htm): „The analytic, logarithmic
expression for stop times ... was replaced with a numerical solution of the restriction on the
sum of He and N2 partial pressures.“
What shall we do with these, admittedly rather theoretical considerations? By no means this
should made be a public example for the developers. And in no case there is ample evidence
to draw any solid conclusions, as described above. These are the reasons not to reveal any
brand names. As well there is to consider, at least in Germany, the fair trade law, especially
the §§ 4, 5 and 6.
But the situation stays very unsatisfying concerning the intransparent status of some
implementations and the lack of open documentation of the „defaults“ and constants. To put
it in tec-lingo:
Is there really a ZH-L inside when the label reads ”ZH-L”???
But the clear message is the following: a decompression time in a digital display, be it on a
dive computer or a PC, is subject to interpretation! And this not so much due to errors in the
measurements (pressure, time, temperature, ...) and other statistical contemplations but
rather due to the method of programming and the choice of a solution for a mathematical
algorithm; i.e.: the software technology, the implementation. The range for these
interpretations is not only in ppm or per mill but rather, dependent on the inert gas dose and
the helium fraction, in the one- or even two digit percent range …
To answer the question posed in the title finally:
1) Yes, with gradient factors we could repair defective perfusion algorithms. But the
perfusion models work by far more satisfying than the topical hype around the bubble
models tells. To underline this one with a historical one-liner:
“Haldane works if you use it properly!”
(R.W. Hamilton, Decompression Theory: 17th UHMS workshop, p. 135; 1978)
2) Yes, we need gradient factors to haul up to the safe side bad or negligent
implementations, i.e.: ill-designed COTS, for mix gases!
In a nutshell we have it here for one paradigm, a dive to depth 42 m, bottom time 25 min, gas
mix: 20 % O2, 80 % He; on chart (3): it is a screen copy of DIVE Version 3_09:
the 1st. block of numbers (according to method B) with the deco stages and the TTS @ ca.
64 min is likely to be found with the COTS programs, as one paradigm: pls. cf. chart (4). The
2nd block (TTS = 77, method A rounded to 82) is the numerical solution, not truncated. For a
printed table the rounding-on at every deco stage would result in a TTS of ca. 82 min.
Application of gradient factors (block 3) with for eg. GF high = 0,9 and GF low = 0,8 yields a
TTS of ca. 87 min, thus feigning a big safety buffer of ca. 87 – 64 = 23 min which we do
NOT have in reality, because the „real“, the numerical solution converges @ ca. 82 min: thus
the faked security of the gradients factors of ca. 36 % TTS dwindels to 87 – 82 = 5 min, i.e.:
nearly nill if you take into account the measurement errors introduced into real life diving
through depth gauges and O2-analyzers.
is a paradigm from another COTS, distributed among TEC-divers; the software development
and bug-fixing ceased around 2006.
Around 2010 there was as well a free-/shareware for a jury-rigged dive computer available,
but since long, this software and their documentation is no longer reachable on the internet:
for our paradigm from above we get here a perfect confirmation of our numerical solution,
i.e.: the TTS of 82 min! Pls. cf chart (5):
Here the deviations are in an order of magnitude where even the differences between the
various deco models / algorithms become blurred: even more so, the higher the
decompression obligation and the higher the Helium fraction in the mix, pls. look at tables A
& B in: https://www.divetable.info/workshop/Vergleich2_e.pdf . The discussions on which
model is „better“ and which became here and there sometimes overheated could now be put
into a somewhat cooler context. To put this one as well into tec-lingo:
„It doesn’t matter which model you use, provided it has a sound
implementation!“ (© Albi, CE 2009)
are for the entire crew of GTUEM for the possibility to give a lecture on this topic at the 12. th
scientific meeting of the GTUEM 03/20/2011 in Regensburg/Germany. Especially to the big
“double double-u” Willi W. (Dr. Willi Welslau, the then president of GTUEM & OeGTH,
Vienna) for a constant peer review and to my “JD”, Jochen D. (the then Prof. Dr. Jochen D.
Schipke, University Medical Center for experimental surgery, Duesseldorf) for the lot of
editorial work and for patience with my oft unorthodox approach. As well to a couple of my
tec-diving students @ Israel.
[A] Salm, Albi (2020) On the theoretical evaluation of one yo-yo diving profile on air for fish-
[B] Salm, Albi (2021) The mapping of a french air diving table (MT92) to a standard Haldane-
/ Workman- /Schreiner-algorithm
[C] Salm, Albi (2020) ZH-L 12 : Validation of an old (1982) experimental
Heliox jump dive (30 m, 120 min)
The sources for the perfusion algorithms are the following, generally well-known and
respected and the already cited famous standard books of diving medicine, pls. cf.:
CAISSON 2010; 25(1): 9;
• Boycott, A.E., Damant, G.C.C., & Haldane, J.S.: The Prevention of Compressed Air
Illness, Journal of Hygiene, Volume 8, (1908), pp. 342-443
• Workman, Robert D. "Calculation of Decompression Tables for
Nitrogen-Oxygen and Helium-Oxygen Dives," Research Report 6-65, U.S. Navy
Experimental Diving Unit, Washington, D.C. (26 May 1965)
• Schreiner, H.R., and Kelley, P.L. "A Pragmatic View of Decompression," Underwater
Physiology Proceedings of the Fourth Symposium on Underwater Physiology, edited
by C.J. Lambertsen. Academic Press, New York, (1971) pp. 205-219
• Müller, K. G.; Ruff, S.:
- Experimentelle und Theoretische Untersuchungen des Druck-Fall Problems, DLR,
Forschungsbericht 71-48, Juli, 1971; as well:
- Theorie der Druckfallbeschwerden und ihre Anwendung auf Tauchtabellen, DVL /
Bericht – 623/ 1966
The numbers in square brackets [yxz] relate to the corresponding entry in a book list at:
 Dekompression - Dekompressionskrankheit, A. A. Bühlmann, Springer, 1983, ISBN 3-
 Tauchmedizin (Barotrauma, Gasembolie, Dekompression, Dekompressionskrankheit) A.
A. Bühlmann, Springer, 1993, ISBN 3-540-55581-1
 Enzyklopädie des Technischen Tauchens, Bernd Aspacher
 "Diving & Subaquatic Medicine", Carl Edmonds, Lowry, Pennefather, Walker, 4 th. Ed.,
Arnold, ISBN 0-340-80630-3
 "Benett and Elliott's Physiology and Medicine of Diving" Alf Brubakk, Neuman et al., 5 th
Ed. Saunders, ISBN 0-7020-2571-2
 "Textbook of Hyperbaric Medicine.", Kewal K. Jain; 3rd. Revised Ed., Hogrefe & Huber,
 Tauchmedizin, Albert A. Bühlmann, Ernst B. Völlm (Mitarbeiter), P. Nussberger; 5.
Auflage in 2002, Springer, ISBN 3-540-42979-4
 "Bove and Davis' DIVING MEDICINE", Alfred A. Bove, 4 th. edition, Saunders 2004,
 Hills, Brian Andrew (1977), Decompression Sickness, Volume 1, The Biophysical Basis
of Prevention and Treatment, John Wiley & Sons, Ltd.. ISBN 0 471 99457 X.
 Naval Forces Under the Sea: The Rest of the Story; 2007, Best Publishing Company,
ISBN-13: 978-1-930536-30-2, ISBN-10: 1-930536-30-5
Berghage TE (ed). Decompression Theory. 17th Undersea and Hyperbaric Medical Society
Workshop. UHMS Publication Number 29WS(DT)6-25-80. Bethesda: Undersea and
Hyperbaric Medical Society; 1978; 180 pages.
Berghage, T.E., T.D. David and C.V. Dyson. 1979, Species differences in decompression.
Undersea Biomed. Res. 6(1): 1 – 13
Bronstein – Semendjajew: Taschenbuch der Mathematik, Verlag Harri Deutsch
D’Aoust, B.G., K. H. Smith, H.T. Swanson, R. White, L. Stayton, and J. Moore. 1979,
Prolonged bubble production by transient isobaric counter-equilibration of helium against
nitrogen. Undersea Biomed. Res. 6(2): 109 -125
Egi SM, Gürmen NM. Computation of decompression tables using continuous compartment
half-lives. Undersea Hyperb. Med. 2000; 27(3): 143 – 153.
Kindwall, EP. Compressed air tunneling and caisson work decompression procedures:
development, problems, and solutions. Undersea Hyperb. Med. 1997 Winter; 24(4): 337 –
Lightfoot EN, Baz A, Lanphier EH, Kindwall EP, Seireg A. Role of bubble growth kinetics in
decompression. In: Shilling CW, Beckett MW, eds. Underwater physiology VI. Proceedings of
the sixth symposium on underwater physiology. Bethesda, MD, 1978: 449 – 457
Ljubkovic M, Marinovic J, Obad A, Breskovic T, Gaustad SE, Dujic Z. High incidence of
venous and arterial gas emboli at rest after trimix diving without protocol violations. J Appl
Physiol 109: 1670 - 1674, 2010
Moon RE, Piantadosi CA, Camporesi EM (eds.). Dr. Peter Bennett Symposium Proceedings.
Held May 1, 2004. Durham, N.C.: Divers Alert Network, 2007
Rodchenkov SV, Skudin VK. Saturation decompression schedules based on a critical tissue
supersaturation criterion. Undersea Biomed. Res. 1992; 19(6): 472 – 481
Voitsekhovich, I. A mathematical decompression model based on biophysical and
physiologic laws. Undersea Hyperb. Med. 1994 Jun; 21(2): 209 - 13.
A ZIP archive of nearly all the references cited above, except the german sources, you will
find for free download at our non-commercial website. We will keep this URL up & running for
the next 36 months: promise!
It contains 14 references, it is approx. 175 MB in size.
If you need one of the german references, just drop us an e-mail: we will be happy to help
COCHRAN Consulting Inc.; www.divecochran.com
COMEX: Compagnie Maritime d'Expertises; www.comex.fr
DAN: Divers Alert Network; www.dan.org
DCIEM (old label, now): Defence Research and Development Canada;
Journal of Applied Physiology: http://jap.physiology.org/
NEDU: Navy Experimental Diving Unit; www.supsalv.org/nedu/nedu.htm
NOAA: National Oceanic and Atmospheric Administration; www.noaa.gov
resp. NOAA diving: http://www.ndc.noaa.gov/
OSHA: Occupational Safety and Health Administration; http://www.osha.gov/
(the topical caisson tables are at: Part Number 1926.)
UHMS: Undersea & Hyperbaric Medical Society; www.uhms.org
Version of: 19.03.2021 11:59 # words: 5903