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Update per 03 / 2021 on:

Decompression-Calculations for Trimix Dives with PC-

Software;

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)

Preamble

(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

update?

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

trade.

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.

Details @1:

• 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,

Michael Cochran.

Details @2:

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).

Details @3:

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% …:

Page 2

Chart 1: the Corona-abandoned Aqua Sport, ETH / IL

Added Value:

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.

Conclusion:

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.

Abstract:

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.

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Keywords:

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.

Side Note:

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:

https://www.divetable.de/skripte/CAISSON/Extended_2011_03.pdf

Introduction:

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

section.

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

(1)

Variable

Definition

P

t(t)

inertgas partialpressure within a compartment with the constant k [Bar] at time t

after an instantaneous change in pressure

Pt0

initial partialpressure of the inertgas within the compartment at time t=0 [Bar]

P

alv0

the constant partialpressure of the inertgas in the alveoli [Bar], for t = 0 and thus for

all t due to the boundary conditions

k

a constant, dependent on the compartment [min

-1

], with k = ln 2 / τ

t

time [min]

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

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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

rarely used:

[102] 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 ([4], [5], [65]) 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 [63], 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

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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

Trimix tables:

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)):

Table 1a

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:

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Table 1b

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

bluntly:

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) ]

(2)

The criteria for „safe“ decompression within the perfusion-models is a simple linear (straight

line) equation ([65], p. 117, resp.: [102], p. 119 ff):

Pt.tol.ig = Pamb / b + a

(3)

Variable

Definition

P

t.tol.ig

tolerated inert gas partial pressure, for each compartment, (analog to M) [Bar],

the sum of all inert gas partial pressures

a

limit of a theoretical ambient pressure of 0 Bar, i.e. the axis intercept [Bar]

Pamb

ambient pressure, absolute pressure of all breathing gases [Bar]

b

1/b pressure gradient: increase per unit of depth (dimensionsless), i.e.: the slope of

the straight line

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These a-/b-coefficients are constants, tabulated for look up, e.g.: in [4] p. 27, in [5] p. 108 &

109, as well in [65] 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 [65], p.

119:

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: [102], 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*

(3*)

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

[54] 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 )

(5)

Pls. see as well the examples in [4], p. 27; [5], 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

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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 ...)

Methods:

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.:

https://www.divetable.info/skripte/theory.pdf)

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

decompression.

C) An approximation method: all the exponential terms are approximated via a

polynomial expression, aka „Taylor Expansion“ (Bronstein, Chapter: Expansion in

Series)

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),

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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

inquiries.

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

60 minutes:

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 ([65], 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 ([65], p.

158), whereas deco software or dive computers are using normally the ZH-L 16 C set ([65],

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, [113] 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 ([65], 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

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the dive profiles via ascent- and descent rates to make sure that the bottom times and the

inert gas doses are matching.

Results:

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.

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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.

Discussion:

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“

([65], 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;

and:

• 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

Page 12

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.“

Conclusions:

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:

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Chart (3):

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.

Chart (4):

is a paradigm from another COTS, distributed among TEC-divers; the software development

and bug-fixing ceased around 2006.

Page 14

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):

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)

Acknowledgements:

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.

Page 15

References:

[A] Salm, Albi (2020) On the theoretical evaluation of one yo-yo diving profile on air for fish-

farming

DOI: 10.13140/RG.2.2.15432.75522

[B] Salm, Albi (2021) The mapping of a french air diving table (MT92) to a standard Haldane-

/ Workman- /Schreiner-algorithm

DOI: 10.13140/RG.2.2.34271.38567

[C] Salm, Albi (2020) ZH-L 12 : Validation of an old (1982) experimental

Heliox jump dive (30 m, 120 min)

DOI: 10.13140/RG.2.2.24608.20482/1

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:

https://www.divetable.de/books/index_e.htm:

[4] Dekompression - Dekompressionskrankheit, A. A. Bühlmann, Springer, 1983, ISBN 3-

540-12514-0

[5] Tauchmedizin (Barotrauma, Gasembolie, Dekompression, Dekompressionskrankheit) A.

A. Bühlmann, Springer, 1993, ISBN 3-540-55581-1

[54] Enzyklopädie des Technischen Tauchens, Bernd Aspacher

[62] "Diving & Subaquatic Medicine", Carl Edmonds, Lowry, Pennefather, Walker, 4 th. Ed.,

Arnold, ISBN 0-340-80630-3

[63] "Benett and Elliott's Physiology and Medicine of Diving" Alf Brubakk, Neuman et al., 5 th

Ed. Saunders, ISBN 0-7020-2571-2

Page 16

[64] "Textbook of Hyperbaric Medicine.", Kewal K. Jain; 3rd. Revised Ed., Hogrefe & Huber,

ISBN 0-88937-203-9

[65] Tauchmedizin, Albert A. Bühlmann, Ernst B. Völlm (Mitarbeiter), P. Nussberger; 5.

Auflage in 2002, Springer, ISBN 3-540-42979-4

[75] "Bove and Davis' DIVING MEDICINE", Alfred A. Bove, 4 th. edition, Saunders 2004,

ISBN 0-7216-9424-1

[102] Hills, Brian Andrew (1977), Decompression Sickness, Volume 1, The Biophysical Basis

of Prevention and Treatment, John Wiley & Sons, Ltd.. ISBN 0 471 99457 X.

[113] 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 –

345

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!

Page 17

https://www.divetable.info/TEMP/References.zip

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

you!

Internet Ressources:

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;

www.drdc-rddc.gc.ca

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