(PDF) Introduction to Decompression Calculation

Introduction to

Decompression

Calculation

for Scientists, Engineers and Physicians

© SubMarineConsulting                                Introduction to decompression calculation                                             page # 2 
 
Introduction to Decompression Calculation for Scientists, Engineers and Physicians 
Contents: 
 
Definition of abbreviations, acronyms & terms ................................................................... 3 
Scope ................................................................................................................................ 4 
References & further reading ............................................................................................. 6 
Overview of Perfusion Models ........................................................................................... 7 
References & further reading ............................................................................................15 
The Haldane Equation ......................................................................................................17 
References & further reading ............................................................................................23 
The Schreiner Equation ....................................................................................................24 
Further Reading ................................................................................................................34 
Calculation of „No Stop Times“ and decompression stop times ........................................35 
„5 simple steps“ ................................................................................................................36 
The rationale for mix gas diving ........................................................................................49 
Calculation of decompression stop times for mix gases ....................................................50 
Variations in the TTS: where do they come from? .............................................................54 
Further reading .................................................................................................................56 
Short detour on „Bubble“ models ......................................................................................57 
Further Reading ................................................................................................................59 
Gradient Factors (GF) .......................................................................................................60 
Dive Computer Simulation ................................................................................................69 
Tools .................................................................................................................................82 
Index .................................................................................................................................83 
 
 
 
 
 
 
 
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Per: Freitag, 1. März 2024 
Filesize: 11086, # pages: 83   

Definition of abbreviations, acronyms & terms

AAB

Albert Alois Bühlmann (16.05.1923, Berlin – 16.03.1994, Zürich)

deco

decompression, decompression stop time

bottom time

also: dive time, time at bottom depth of the dive, etc. ...

BSAC

British Sub Aqua Club

DAN

Divers Alert Network

DCIEM

Defence and Civil Institute of Environmental Medicine, now:

Defence Research and Development Canada

DRA

Decompression Risk Analysis, a feature from the DSG

DSG

is the DAN portal for the: Diver Safety Guardian

EAN

Enriched Air Nitrox, i.e.: O2 & N2

GF

gradient factor, GF Hi = GF High and GF Lo = GF Low

Heliox

Helium-Oxygen mix

HT

Half-time

JSH

John Scott Haldane

LDE

Linear Differential Equation

NEDU

Naval Experimental Diving Unit

Nitrox

EAN, (oxygen) enriched air nitrox

NOAA

National Oceanic & Atmospheric Administration

Ox-Tox

(coll.) oxygen toxicity

RNPL

Royal Naval Physiological Laboratory

SAT

saturation dive, usually a dive with bottom time > 72 h

SI

surface interval, the pause between dives

SL

Sea Level, i.e.: altitude of 0 m

stage

part of a dive profile, being at the bottom or during ascent

stop time

time required to stop at a certain stage

TDT

Total Decompression Time, usually TTS + Air Breaks

Trimix

Tri = 3, 3 gases in the breathing mix, for e.g.: O2, He, N2

TTS

time-to-surface, by convention:

the sum of all stop times + (bottom depth / ascent speed)

UHMS

Undersea & Hyperbaric Medical Society

USN

United States Navy

VPM

Varying Permeability Model

WOB

Work Of Breathing

ZH-L 16

ZH: Zürich, L : linear, 16 a- & b-coefficients

Scope

First, about the somewhat lengthy title „Introduction to Decompression Calculation for

Scientists, Engineers and Physicians“: it is a short introductory text with simple examples to

get you to the basics of it and then afterwards you could run full speed with your own ideas

and implementations. Thus each chapter ends with a list of references and further readings.

It is intended for scientists & engineers and physicians (which are neither: nor scientists or

engineers). As such it is not a continuation of any outline-series, like „Deco for Dummys“ or

so.

As well the detailed physiology or diving medicine is out of scope: this is already described

elsewhere at great length, the available literature growing. We will have only a superficial

glance at the physiology as far as the basic calculations here are concerned. At the end of

this chapter, you will find a list of books on diving medicine and physiology, along with the

links with more information on these books and sometimes a download-link, free of charge.

Here we are dealing exclusively with the so-called „perfusion models“. I.e.: decompression

models or –algorithms where perfusion, that is: the movement of blood from one part in the

body to another, plays the dominant role. So diffusion or other molecular processes are not

considered. Dominant means here in this context, that the mathematical contribution to a

term in an equation is only from the perfusion, i.e. the processes with the movement of a

certain volume of blood are relevant. Thus other contributions, say from diffusion, are

considered to be negligible. As well the time-scale of diffusion is considered too long in

comparison to a perfusion-process.

One word on „models“: the „models“ covered here are simple inert gas bookkeepers, nothing

more! A real model would include an algorithm which would detail the course of time, locus,

amount and volume of inert gas bubbles: none of these „models“ are able to do that. Not

even the hyped bubble- or the thermodynamic models.

A couple of the chapters start with pure number crunching, i.e: applied calculus; you could

drop that happily on your first reading and go direct to the prose and the examples detailed in

the „blue boxes“. However, a couple of insights are buried in these formula.

History and development of perfusion models, discovering the timeline of the past 120 years,

i.e. from Haldane to Workman, Schreiner, Thalmann, Müller & Ruf, Bühlmann & Hahn.

Improvements in the models in the course of time, i.e. number of compartments, their HT and

their tolerated super-saturations.

And another word here: all these models are of the type „deterministic“. As decompression

sickness is a stochastic phenomenon, there are models which better fit this situation: these

are called „statistical“ models, the tables derived form them are „statistically based

decompression tables“: a topic for another (big) manual … i.e.: out-of-scope here, as the

mathematics for this stuff is a little bit more elaborate ...

Application for practical diving, basically the calculation of relevant dive profile parameters,

i.e.: solutions to the simple LDE like no-stop-times, decompression stop times, time-to-flight,

repetitive diving and diving at altitude. At first, we try this only for one single compartment.

Then we expand these methods for a whole bunch of them: examples for this and how this is

calculated for dive tables and how a dive computer works on these questions.

The rationale for mix gas diving; basically by Meyer-Overton and WOB.

Decompression calculation for mix gases, i.e. breathing mixes with more than two inert

gases like Trimix. The NOAA Ox-Tox doses (or others) are touched upon only slightly, as all

of this is already in EAN- or Trimix Manuals available (our Trimix manual for free download:

pls. cf. one page further down).

Variations in the TTS: where do they come from? I.e.: why do different products, like desktop

decompression software or mix gas dive computers, who all claim that they have

implemented the original AAB perfusion model ZH-L 16, are at variance to each other with

the TTS for the identical dive schedule?

An only short detour on bubble models and why we do not go in-depth because of the

questionable theory of the VPM, the bubble parameters from the agarose gel, which are at

variance to human bubbles and the missing validation of all the 5 basic VPM parameters on

humans.

And, as well, another short detour to one of the most sophisticated 21st. century

decompression models, the „Thermodynamic Model“ from Brian Andrew Hills and why we do

not use it here.

Finally we take a short look at the methods of the so-called „Gradient Factors“, the GF:

where did they come from? And how are they used today? Thus we will discover that the GF-

history dates back to Haldane and all of our champions cited above (Workman, Schreiner,

Bühlmann, Hahn, etc.) had been exploiting them, but just with other methods and with

another wording.

Bottom line is, as there are already a couple of books with pretty much the same / similar

content:

 there is added value in this manual through the internet links to the biggest part of the

here cited papers & books

 there are examples, down to the last bit, calculated with tools,

all of them for a free download

 the calculation of stop times for mix gases is „full blown“ here, i.e.: as much details as

we could give. As far as we presently (03 / 2024) know, there is no other complete

documentation on that one available. If you know otherwise: pls. drop us an e-mail!

We will appreciate, promise!

References & further reading

(and, for the following chapters: as well preliminary reading!) That is: before you go on with

this text, you should digest a couple of other things, like:

 THE PHYSIOLOGICAL BASIS OF DECOMPRESSION, THIRTY-EIGHTH UHMS

WORKSHOP; JUNE 1989

https://www.uhms.org/archived-publications1/the-physiological-basis-of-

decompression/download.html

 [47] Jolie Bookspan, Diving Physiology in Plain English, UHMS 1995, ISBN 0-

930406-13-3

https://www.divetable.eu/BOOKS/47.pdf

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

2004, ISBN 0-7216-9424-1

https://www.divetable.eu/75_rest.pdf

 [62] "Diving & Subaquatic Medicine", Carl Edmonds, Lowry, Pennefather, Walker, 4

th. Ed., Arnold, ISBN 0-340-80630-3

https://www.divetable.eu/62_complete.pdf

 „DMfSD“, i.e: Diving Medicine for Scuba Divers, by the same authors, version of

2010, download for free there:

https://www.divetable.eu/DMfSD_2010.pdf

 Trimix - compact!

https://www.researchgate.net/publication/372775636_Trimix-compact

Overview of Perfusion Models

The fundamental postulations of Haldanes et al. „2:1“ theory:

JSH made experiments on goats, old hens, other critters and human volunteers from the

British Navy, himself and his son Jack, age 13 [1]! The cornerstones of his „2 : 1“ theory and

the fundamentals to calculate the first diving table for air dives with staged decompression

are outlined above.

The broad dissemination due to the striking success of his tables lead directly to the

revelation of their limitations, i.e.: too conservative a TTS for short shallow dives and too

liberal a TTS for deep and long dives. Thus other researchers went to improve this system.

In approximate timely order the names of the principal researchers are (N.B.: this is not an

exhaustive list!):

John Scott Haldane

Robert Dean Workman

Heinz R. Schreiner

Siegfried Ruff & Karl-Gerhard Müller

Robert William Hamilton

Edward Deforest Thalmann

Albert Alois Bühlmann

Max Hahn

Each of them made important contributions and improvements, which we will discuss along

the „2:1“ theory. In due course of time, the limitations of the tables and the theory JSH put

out were, at least sometimes and partially, forgotten. A couple of them were:

➔ the experimental data were mainly from goats, thus they have to be scaled via an

allometric polynom to make them fit for humans

➔ the calculations were with done fN2 = 1.0

➔ only „uneventful“ decompression, i.e.: without inert gas bubbles, was considered

➔ the saturation- and de-saturation processes were considered symmetrical, i.e.: the

exponents with the HT were identical

➔ the TTS should be less then ca. 30 min and the diving depth not deeper than 50 m, as

exceeding these parameters JSH had no data (and, as well that time there was no real

business case to dive deeper and longer, and also the air pumps to deliver the breathing mix

down a hose to the hard-hatted diver, found their limitations)

➔ besides, JSH et al. have been aware on the statistical imponderabilities of their data set:

they wanted more than 30 exposures per profile, which was not possible due to financial /

organisatorial restrictions.

Portrait of JSH. Source: wikipedia.org:

The trailblazing paper [1]:

The „2:1“ looks like that:

The ambient pressure line, i.e. the absolute pressure at the diving depth, is the blue line: it is

the angle bi-sector. Say, you dive to 30 m (X-axis) and hit the ambient pressure line at 4 Bar

(see the Y-axis). And this is the pressure which should prevail in all of your air-filled cavities

in your body, otherwise you have to suffer from a barotrauma. And, b.t.w. this is the

pressure, which your regulator should offer you.

The red line is the „2 : 1“ line for all of the 5 JSH compartments. The surfacing value is at the

intersection of the dotted line, parallel to the Y-axis, from 0 m / 1 bar with the red line: it is @

approx. 2 bar. So the slope of the red line is Y / X = 2 : 1.

The saturation of the so-called „compartments“ should follow an exponential law (blue line);

X-axis: # of half times, Y-axis: saturation in %:

After the 1st. half-time (x = 1) we reach 50 % of the saturation, after the 2nd. (x = 2) we are

at 50 + 50/2 = 75 %, and so on. A little table looks like that:

s

The rule-of-thumbs is: a complete saturation (or, for that matter: de-saturation) at a constant

pressure needs ca. 6 half-times. I.e: for a fast compartment, say with a HT of 5 min, these

processes need ca. 30 min.

As the exponential curves never reach a parallel to the X-axis at Y = 0 % (or Y = 100 %), this

rule-of-thumb is sufficient for regular diving (and humans, b.t.w. ...). The next picture shows

the saturation of the 5 JSH compartments (P1 – P5) at an air-dive to ca. 30 m (4 Bar) during

the first 50 min of bottom-time:

Indicated are the 50 % and 100 % lines of PaN2 (blue lines), the „a“ is for alveolar or arterial.

And this is one of the basic assumptions of perfusion-models: that the partial pressures of

the inert gases in the alveolar and arterial systems are identical. I.e.: if there is a ΔP in

ambient pressure due to ascent or descent or a change in the breathing gas, then this will

affect instantaneously both systems, the alveoli in the lungs and the arterial blood.

The pressure gap in the chart above, between Pamb and PaN2 is basically due to oxygen and

calculated via Dalton’s law (pls. cf. below: there is an example in the „5 simple steps ...“).

P1, with a short HT, is called a „fast“ compartment, whereas P5, with a long HT, is called a

„slow“ compartment. Saturation and desaturation for a constant ambient pressure are

outlined below, now for 3 other compartments with the HT of: 40, 75 and 120 min (picture

taken from „Ehm: Tauchen – noch sicherer“):

Now check, at which time the curves hit the 50 % line: the 3 values from the X-axis should

reflect the 3 indicated HT. N.B.: the saturation and desaturation curves for one HT intersect

this 50 % value at the identical times; this is what „symmetrical“ implies.

The arrangement of the compartments is in a parallel fashion: i.e. all of them are reached by

the same arterial pressure, and, for that matter, alveolar pressure:

There are other arrangements possible, like the Kidd-Stubs model which resulted in the

DCIEM tables: there, the 4 compartments were arranged in a series. Or a completely other

model which used only one compartment, a so-called „slab“. This slab was perfused on one

side, and inside the slab the diffusion of the inert gases were modeled. This resulted in the

RNPL and BSAC tables. You will find in the next „Further Reading“ pages references to all of

these models.

The now following chart is an outline of what we are after at in the next couple of chapters:

We search for a rationale for this LDE and try to solve it. The 2 solutions are the so-called

Haldane- & Schreiner equations. After we have done that, we look into dive tables & dive

computers, i.e.: the practical applications of these 2 solutions.

Modifications to Haldane:

 The „2:1“ which was a constant for all of the 5 compartments, became variable:

 The „M-values“ from Bob W.: these were basically dependant on the HT: the shorter

the HT, the bigger the M0 (surfacing value); more details: two chapters further down

 the spectrum of HT was expanded to short & longer HT: say from 1.25 to 900 min

 and the sheer number of compartments was increased, up to 9, 12, 16 or 20

 the de-saturation process was no longer considered symmetrical to the saturation,

 i.e. the HT have been increased in comparison to the saturation

 as the raw computing-power of PCs and diver carried computers increased, so was

the complexity of additional features, like:

 influence of (skin-) temperature,

 physical workload (oxygen consumption),

 the pO2, if it became higher than 1.3 atm

 and, and, and ...

References & further reading

 Historical dive tables: an overview on ca. 110 years of dive tables development.

https://dx.doi.org/10.13140/RG.2.2.32813.03042

 [1] 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.

 a hard-copy of this original paper along very interesting & topical research stuff from

2008 is there for a free download:

 [262] Brubakk, Alf O.; Lang, Michael A. (Ed.) (2008) The future of Diving, 100 Years

of Haldane and Beyond;

 https://www.divetable.eu/262.pdf

Haldane, J.S.: Respiration. New Haven, Yale University Press, 1922. (here the chapter XII:

“Effects of High Atmospheric Pressures”, p. 334 – 357).

Donald, Kenneth W., Oxygen Poisoning in Man, British Medical Journal, London, Saturday,

May 17, 1947; Part I, p. 667 – 672. Part II, p. 712 – 717, May, 24, 1947. (Description of the

heroic experiments in a pressure-tank with a rebreather-system and pure oxygen at

pressures equivalent to 21.3 and 27.4 m water-depth).

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) (The complete M-value algorithms for N2 and He).

Hempleman, H.V. „British decompression theory and practice“, in: Bennet, P.B., Elliot,

D.H.:“The Physiology and Medicine of Diving and Compressed Air Work“, 1st ed., pp. 291 –

318, Bailliere, Tindall and Cassell, London 1969 (description of the „Slab“ model, basics of

the BSAC table).

Val’s chapter is there for a free download:

https://www.divetable.eu/BOOKS/250_Chpt13_290_317.pdf

Kidd, D.J., R.A. Stubbs and R.S. Weaver „Comparative approaches to prophylactic

decompression“, in: Lambertsen, C.J.: “Underwater Physiology, Proceedings of the Fourth

Symposium on Underwater Physiology“, pp. 167 – 177, Academic Press, New York 1971.

(Outline of the Kidd-Stubbs model which has been used for the DCIEM 1983 tables).

Braithwaite, W.R. „Systematic Guide to Decompression Calculations“ Experimental Diving

Unit Report 11-72 (Comments on Bob’s 6-65 report with worksheets).

Hills, B.A. „Decompression Sickness“, Vol. 1, The Biophysical Basis of Prevention and

Treatment, John Wiley & Sons, Ltd. 1977 (Physiology of decompression and basics of his

thermodynamic model, as well remarks concerning the empirical decompression strategies

of the pearl divers in the Torres Straits).

For the readers from the D-A-CH region, or, if you have an average grasp of the german

language, there is this epic manual for „Dekompression“ with ca. 800 pages. It covers all of

the above cited models in-depth with many formulas, graphs and even more references. An

extract / summary (Leseprobe) is available at the RESEARCHGATE for free download:

https://www.researchgate.net/publication/369196910_Leseprobe_Dekompression

The Haldane Equation

We are looking for a rationale for the LDE from the previous chapter with the help of a simple

mass-balance: the first order mass-balance for an inert gas within a compartment of tissue,

solely feed by blood perfusion simply looks like that:

inert gasstored = inert gasin – inert gasout

here, in our case of diving on air, it is basically nitrogen:

N2stored = N2in - N2out

First-order means ignoring any non-linear, quadratic or other, contributions to the process.

The little drawing (pls. cf. below) from my late friend Dick Vann may help (drop the label N2

for convenience). Now let’s define the needed variables and indices:

t is the time (usually dive time, bottom time or the like)

P is for pressure, with the indices:

a: arterial A: alveolar v: venous t: tissue b: blood 0: zero,

usually at the start of the dive, for e.g.: t0 or like: Pt(t0) = P0

Q is a volume of blood and V is a volume of a tissue (compartment). The index

t is for tissue, but only by convention. We are dealing here with compartments, i.e.:

a whole bunch for various tissues, which share some physiologic properties. One of

these properties is the perfusion.

The inert gas N2, but for that matter all other inertgases, like Helium, being transported via

arterial blood to the tissue and leaving this region with the venous return. Hereby it is

assumed that the arterial (Pa) and the alveolar inert gas partial pressures (PA) are equal, and

that the diffusion between adjacent very closely spaced capillaries is instantaneously, i.e.: as

well the venous (Pv) and the compartment partial pressure (Pt) are also equal for this

particular inert gas.

So with this model we have now the following identities from our simplified physiology:

Pa = PA AND: Pv = Pt

The solubility coefficients α (a greek alpha) for one inert gas for blood (b) and tissue (t) are:

αb and as well αt. 󰇗 is the blood flow, i.e.: the perfusion rate (the usual notation for a variable

changing in time is the little dot above the character: 󰇗, it is exactly the same as the first

derivative with respect to time: 

 ), Vt is the tissue volume (volume of the compartment in

question). Starting at time t = 0 it is assumed that Pa reaches immediately the constant value,

Pa of the pressure step (i.e. when you jump into the water and reach your bottom depth very

quickly ...). The rate of change of the pressure in the compartment dictates the rate of

change for storage in this compartment. We have now the following mass-balance, which

changes in the course of time:

αt * Vt * dPt/dt = αb * dQ/dt * Pa - αb * dQ/dt * Pv

We divide this equation by αt * Vt

and we define now k = αb * dQ/dt / ( αt * Vt ) and thus we get the LDE:

 dPt/dt = k * ( Pa - Pt )

The solution to this LDE is the famous Haldane Equation:

Pt (t) = Pa * [ 1 – exp(-k * t)] + P0 * exp(-k * t)

Later on we derive properly the solution of the LDE: for the time being you could convince

yourself that if you put the frist derivative 

 of the Haldane equation and the Haldane

equation itself both into the equation above, this little LDE is identically satisfied.

Via k we thus have in the end the relationship between the HT (the greek τ) and the tissue

perfusion and the solubilities:

τ = ln 2 / k = 0.693 / k = 0.693 / ( αb * dQ/dt / (αt * Vt ))

The total partial pressure over time in the tissue (the compartment) ( Pt(t) ) is the sum of the

decrease of P0 (the second term in the Haldane solution) and the increase of the alveolar

pressure PA, i.e.: the systemic response to the step change in pressure. The small stirrer in

Dick’s drawing below is symbolizing that we have a so-called "well stirred" compartment:

Source: [75], p.55

Basically the same reasoning with identical results, but with a little bit of other terminology

you will find in: Schreiner and Kelley in [101], p. 205 ff and, as well in the nice synopsis from

Bob Workman in [250], chapter 12; pls. c.f. the references page at the end of this chapter.

As well there is another ref. to Scheiner & Kelley: both of their papers in these UHMS-

symposia are brilliant works on the topic: concise, to the point, and pragmatic! Our opinion:

Worth reading!

Derivation of the LDE and the solution

The on- and off-gasing with inert gases of a system which is not in equilibrium with these

gases is best described with a little differential equation:

dPt(t)/dt = k[Palv(t) - Pt(t)]

(1a)

Descriptively this formula means: the pressure change over time dPt(t)/dt within a

compartment is dependant of the actual pressure gradient Δ P. This is like many processes

in nature: radioactive decay, Newton’s cooling process, ideal population or bacteria growth

etc. So the basic mathematical idea is simply like that:

∂ P / ∂ t  Δ P.

From this proportionality we make a real equation with the help of a simple constant; and

thus the ∂s became real differentials:

dP(t)/dt = k * Δ P

(Later on, sometimes, we drop the „ * “ in the multiplications for convenience; this is also

convention in textbooks for calculus). The central idea of all these perfusion models is an

instantaneous pressure equalization from the alveoli in the lung with the arterias! I.e..:

Palv(t) = Parterial(t) , for all t

Variable

Definition

Pt(t)

Partial Pressure of an inert gas within a compartment at time t, in [Bar]

Palv(t)

Partial Pressure of an inert gas within the alveoli [Bar]

k

a constant, specific for each of the compartments [min-1]

t

time (as well dive time) [min]

Step 1: We are separating the variables in (1a) and find the inhomogeneous differential

equation (1b):

dPt(t)/dt + kPt(t) = kPalv(t)

(1b)

Step 2: The homogeneous equation (2):

dPth(t)/dt + kPth(t) = 0

(2)

For the solution we take, from convention, the following approach: Pth(t) = C0e-λt

The subscript 'h' in Pth means the homogeneous equation. We put this into (2) and get:

-λC0e-λt + kC0e-λt = 0 => (k - λ)C0e-λt = 0,

only if k = λ or C0 = 0 (uninteresting: it means no change in pressure, i.e.: no dive),

and because e-λ*t ≠ 0 for all t, we take k = λ and thus getting the homogeneous solution for

(2):

Pth(t) = C0e-kt.

(3)

Step 3: Find a special solution of the inhomogeneous equation (1b) through the boundary

conditions. We are taking two special situations concerning Palv(t).

1) Palv(t) = const., i.e. the diving-depth as well is constant!

2) Palv(t) varies linearly in time, i.e. the ascent- and descent-ramps are constant, that is the

ascent- and descent velocities remain at a constant value.

Situation 1:

constant depth = constant ambient pressure

In this case the alveolar inertgas pressure will take a constant value: Palv(t)=Palv0 . So the

equation becomes (1b):

dPt(t)/dt + kPt(t) = kPalv0

(4)

We are taking, again as per convention, the 'solution':

Pt(t) = C0e-kt + C1

(5)

we put (5) into (4) and the exponential terms are cancelling themselves:

- kC0e-kt + kC0e-kt + kC1 = kPalv0

thus we recieve: C1=Palv0.

Now we are searching for another boundary condition in order to determine C0.

We assume a certain but constant partial pressure at start in the compartment: Pt(0) = Pt0 for

time t=0. This means for eg. a certain reached saturation pressure or as well the starting

pressure for mountain lake diving. We put his into equation (5):

Pt0=C0e-0 + Palv0

and thus follows: C0=[Pt0 - Palv0].

We will have then the Haldane Equation for the partial pressure of an inert gas in a specific

compartment k:

Pt(t) = Palv0 + [Pt0-Palv0] e-kt

(6a)

Variable

Definition

Pt(t)

partial pressure of an inert gas within a compartment with the constant k [Bar]

Pt0

initial partial pressure of an inert gas within a compartment at time t=0 [Bar]

Palv0

the constant partial pressure of an inert gas within the alveoli [Bar], for t = 0 and

therefore for all times t due to the boundary condition

k

a constant, dependant of compartment [min-1]

t

time [min]

(6a) is the famous Haldane Equation, we re-arrange and arrive at (6b): we simply add at the

left and right side Pt0 and group according Palv0 - Pt0:

Pt(t) = Pt0 + [Palv0-Pt0] [1 - e-kt]

(6b)

This is the equation which is regularly cited, e.g.:

Bühlmann, [4] p. 14, 1983; [5] p. 64, 1993; [65] p. 96, 2002, with slightly changed indices.

For e.g. PI is the inspiratory inert gas pressure, this means the respiratory coefficient Rq (pls.

cf. equation (13)) is implicitly set equal to 1.00, thus we have variances across different

tables/models/researchers (next chapter).

© SubMarineConsulting                                Introduction to decompression calculation                                             page # 23 
 
References & further reading 
 
[75] "Bove and Davis' DIVING MEDICINE", Alfred A. Bove, 4 th. edition, Saunders 2004, 
ISBN 0-7216-9424-1 
more info: Cover  
more info: TOC  
the rest at: manuals_4_free                                                                      
https://www.divetable.eu/75_rest.pdf 
 
Heinz R. Schreiner and P.L. Kelley (on pp. 205): A Pragmatic View of Decompression, in:  
[101] Underwater Physiology: Proceedings of the Fourth Symposium, edited by Christian J. 
Lambertsen, Academic Press Inc.,New York, U.S. (November 12, 1971) ISBN-10: 0-12-
434750-9, ISBN-13: 978-0124347502 (This article is one of the most cited in diving medicine; 
the complete proceedings of the 4th. symposium are there for a free download): 
https://www.divetable.eu/108_Vol_IV.pdf 
 
[250] Bennett, Peter B., Elliott, David H.(eds.) (1969) The Physiology and Medicine of Diving 
and Compressed Air Work, First Edition, Bailliere Tindall and Cassell, London ISBN: - 
Bob Workmans chapter 12 is there for a free download: 
https://www.divetable.eu/BOOKS/250_Chpt12_252_289.pdf 
 
And, more from Heinz R. Schreiner and P.L. Kelley: 
Computation Methods for Decompression from Deep Dives (p. 275) in: 
[107] Underwater Physiology: Proceedings of the Third Symposium, edited by Christian J. 
Lambertsen, (Hardcover, 497 pages) The Williams & Wilkins Company, Baltimore 1967 
Download of the complete proceedings for free there: 
https://www.divetable.eu/108_Vol_III.pdf 
 
from the AAB convolut the latest, the 2002 edition: 
[65] "Tauchmedizin.", Albert A. Bühlmann, Ernst B. Völlm (Mitarbeiter), P. Nussberger; 5. 
edition in 2002, Springer, ISBN 3-540-42979-4 
And an english version of his first german book from 1983 on the topic: 
https://diving-rov-specialists.com/index_htm_files/scient-b_52-
decompression_decompression-sickness.pdf 

The Schreiner Equation

From the previous chapters, the „Situation 1“, we continue now with the next situation:

Situation 2:

the ambient pressure changes linearly with time

During ascents and descents with constant velocity the inspiratory partial pressures of the

inert gases change linearly in time (with an open circuit SCUBA!). With rebreathers (for e.g.:

a CCR = closed circuit rebreather) we have, according to make, other dependencies!). In

equation (1) this implies:

Palv(t) = Palv0 + R * t.

Palv0 is the partial pressure of the inert gases from start, i.e.: at time t=0, and R is the rate of

change (in bar/minute) of the partial pressure of the inert gases in the alveoli. R is positive for

descent (increase of pressure) and negative for ascent (decrease of pressure).

We put it into equation (1b):

dPt(t)/dt + k Pt(t) = k Palv0 + k R t

(7)

We try, once again, per proven conventions, this solution:

Pt(t) = C0e-kt + C1 t + C2

(8)

We plug solution (8) into equation (7) and recieve:

-k C0e-kt + C1 + k C0e-kt + k C1 t + k C2 = k Palv0 + k R t

and thus:

[k C1 - k R] t + [ C1 + k C2 - k Palv0 ] = 0

(9)

In order to find a solution for C1 and C2 which should be valid for all times t, we put the terms

in the squared brackets in (9) equal to 0. Thus we have:

C1 = R and C2 = Palv0 - R/k.

Thus we have the folowing:

Pt(t) = C0e-kt + R t + Palv0 - R/k

(10)

Once more we exploit the boundary condition Pt(0) = Pt0 for t=0 in order to calculate C0. We

put this into (10) and get:

Pt0 = C0e-0 + Palv0 - R/k

and thus: C0 = Pt0 - Palv0 + R/k.

Finally, as one definite solution we find:

Pt(t) = Palv0 + R * t - R/k + [Pt0 - Palv0 + R/k] e-kt

resp.:

Pt(t) = Palv0 + R[t - 1/k] - [Palv0 - Pt0 - R/k] e-kt

(11)

Variable

Definition

Pt(t)

partial pressures of the inert gases within compartment [Bar]

Pt0

initial partial pressures of the inert gases in compartment at time t=0 [Bar]

Palv0

initial (alveolar) partial pressure at time t=0 [Bar]

k

a constant, dependant of compartment

R

rate of change of the partial pressure of an inert gas in the alveoli (Bar/min)

R = f * Ramb,

f is the fraction, i.e.: volume part of inert gas and Ramb is the rate of change of the

ambient pressure

t

time [min]

the so-called Schreiner equation.

By setting the rate of change R = 0 (i.e.: a constant diving depth) the Schreiner equation (11)

contracts herself and becomes again the Haldane equation (6a).

Half Times (HT)

The variable τ (a greek tau, or tav, for that matter) is designated as a 'Half Time' (HT) and is

specific for each compartment:

-k * τ = ln(1/2) = -ln(2)

The relation between k and the HT is:

τ = ln(2) / k

resp.: k = ln(2) / τ

(12)

The stringent physiological relation between the HT and the compartments is through the

solubility and the perfusion (blood supply):

τ = 0,693 * αti / (αbl * dQ/dt)

(12a)

with the following definitions:

αti = solubility of an inert gas in tissue, ml(S)gas * mlti -1 * (100 kPa) -1

αbl = solubility of an inert gas in blood, ml(S)gas * mlblood -1 * (100 kPa) -1

dQ/dt = compartment perfusion, mlblood * mlti -1 * min -1

pls. cf. a little bit further up for a rationale of this relationship.

The partial pressures in the lung alveoli

we want to determine the alveolar partial pressure Palv more exactly. The gas composition

and thus the partial pressures of the component gases are dependant of the ambient

pressure Pamb, water vapour, carbon dioxide, and the other components of the breathing mix,

i.e. the chemical composition; pls. c.f the example in the „5 simple steps“.

The ambient pressure Pamb is the sum of the atmospherical pressure (ca. 1 Bar at SL) + the

hydrostatic pressure of the water column (from the diving depth), thus it is increased ca. 1

bar per every 10 m increase in depth (not considering the differences between fresh and sea

water ...). The ambient pressure and the absolute pressure within the lung have to be

(roughly) equal, only through in- and exhalation there is a small pressure difference

of approx. and up to max. 30 cm of water column.

The partial pressure of an inert gas in the lung Palv can be roughly estimated through:

• the partial pressure (through the fraction f) of the inert gas in the air or in the mix

• the water vapor partial pressure PH2O. The dried air from the compressor has to be

humidified on the way to the lungs.

The nose, voice box, tracheas and various mucus membranes put their water vapor

into the dry air: thus the breathing mix gets diluted.

At 37 degrees of Celsius the PH2O is in saturated environment ca. 0.0627 Bar (47 mm

Hg)

• Oxygen O2 decreases through metabolism and ventilation with used air

• Carbon dioxide CO2 is being added through metabolism / ventilation

• and because CO2 in normal clean breathing air can be neglected, we put CO2 partial

pressure in the lung alveaoli equal the arterial CO2 partial pressure with: 0.0534 Bar

(40 mm Hg).

The relation of the oxygen consumption of the metabolism to its carbon dioxide production is

designated with the respiratory quotient Rq. Rq is the proportion in volumes of

carbondioxide production to oxygen consumption in ml, average values are:

200 ml carbondioxide production / oxygen consumption 250 ml per minute, i.e. ca.:

Rq = 200 / 250 = 0.8

Rq is dependant on nutrition and on the workload: we have Rq = 0.7 for metabolism mainly on

fat, ca. 0.8 for proteins, ca. 0.9 – 1 with carbohydrates. With average workload and nutrition

we have 0.8 -> 0.85, with CO2-retainers < 0.7 and > 1.0 with heavy workloads or

acidosis/enrichment with lactic acid.

The equation of the alveolar ventilation describes this:

Palv=[Pamb - PH2O - PCO2 + ΔPO2] * f

or:

Palv=[Pamb - PH2O + (1 - Rq)/Rq * PCO2 ] * f

(13)

Variable

Definition

Palv

partial pressure of the inert gas in the alveoli [Bar]

Pamb

ambient pressure, i.e. absolute pressure of breathing mix [Bar]

PH2O

water vapor partial pressure, at 37 degrees of Celsius ca. 0.0627 Bar (47 mm Hg)

PCO2

carbon dioxid partial pressure, ca. 0.0534 Bar (40 mm Hg)

ΔPO2

Delta = change of oxygen partial pressure through metabolism / ventilation in the

lung

Rq

the respiratory quotient

F

fraction of inert gases in the breathing mx; e.g.: N2 in dry air = ca. 0.78. Normally

for diving we put f = 0.79xx, i.e.: we take into account the various trace gases

Across the various models/researchers we have as well various settings for the Rq, even if

not explicitly stated:

Schreiner: Rq = 0.8

Workman: Rq = 0.9

Buehlmann / Hahn: Rq = 1.0

critical supersaturation, symptomless tolerated inert gas overpressure, M-Values

Haldane

The tolerated overpressure / super-saturation was postulated as „2 : 1“ for all

5 compartments. These 5 compartments had the HT: 5, 10, 20, 40, 75 min. The 2:1 ratio is

valid only for the absolute pressure! Concerning the inert gas overpressures in relation to the

ambient pressure we have the following:

2 * 0.78 = 1.56, i.e.: 1.56 : 1 (elucidation from Bob Workman)

and is a constant for all compartments.

Workman

The linear Workman equation, is a simple straight line in a pressure coordinate system. It

looks like that for each compartment i (index i):

Mi = M0i + ΔMi * d

(14)

Variable

Definition

M

M-Value, maximal partial pressure of an inert gas in a compartment [fsw]

M0

M0 Value, for sea level or diving depth = 0 ft for each compartment [fsw]; aka:

surfacing value

ΔM

Delta M, increase of M per each foot of diving depth, defined for each compartment

[fsw/ft]

D

diving depth [ft]

Bottom Line: Workman had the M-Values decreasing with increasing HT, thus the fast

compartments (short HT) can tolerate a higher inert gas overpressure than the slower

compartments (longer HT).

From Bob’s NEDU 1965 report we have the following tables for Nitrogen & Helium from the

pages 31 & 32:

There we have the HT in min in the 1st. column and the surfacing values M0 in fsw in the

2nd; below is the M per 10 feet. N.B.: the slope and the Y-axis intersection are different for

the 2 gases, but the HT for N2 and Helium are identical!

With equation (14) we could determine the minimal depth dmin where the diver has to stay

during her deco stops, aka „ceiling“. The calculation we have to do for each of the

compartments and is dependant of the actual inert gas overpressures:

dmin = (Pt - M0) / ΔM

(15)

So each compartment has got its own saturation and thus its own minimal depth.

We simply have to take the biggest value (greatest depth) out of the family of these values.

More to that and how a dive computer takes this will be covered in the examples.

Between the M-values and the HT we have approximately the following empirical relation

according to Workman:

M = 152.7 τ -1/4 + 3.25 d τ -1/4 = M0 + ΔM d

(16)

According to Hempleman we have an approximate empirical relation between the no-

decompression limits tNDL [min] and the diving depth d [fsw]:

d * tNDL1/2 = 475 fsw min1/2

(16.1)

The PADI/DSAT RDP table has been generated with such a formula, but with a couple of

more conservative modifications:

d - A = C * tNDL- x

(16.2)

• A = "shallow" asymptote, 20.13 fsw

• C = 803

• x = 0.7476

• "deep" asymptote = 262 fsw (theoretical diving depth with NDL = 0!)

Bühlmann / Hahn

AAB offers, basically identical like his forerunners Workman, Schreiner, Müller & Ruf, a linear

relationship between the tolerated tissue overpressure and ambient pressure ([65], p. 117):

Pt.tol.ig = Pamb / b + a

(17)

Variable

Definition

Pt.tol.ig

tolerated inert gas pressure, for each compartment, (analogous M) [Bar],

sum of all partial pressures of the present inert gases

a

boundary value at a theoretical ambient pressure of 0 Bar, i.e. the axis intercept

[Bar]

Pamb

ambient pressure, absolute pressure of the breathing mix [Bar]

b

1/b pressure gradient: value of increase per pressure unit depth (dimensionless),

i.e. the slope of the straight line

In principal there is no difference between Workman and AAB & Hahn except that the M-

values are referenced to the ambient pressure at sea level whereas AAB & Hahn are

extrapolating against 0 Bar ambient pressure and thus reaching the region between 1 and 0

bar automatically, i.e. mountain lake / altitude diving. I.e.: as (14) and (17) are both linear

equations, you could convert both systems into eachother.

First, we identify the maximal inertgaspressure M with the allowed / tolerated inertgas

pressure Pt.tol.ig, then we re-write Pamb in terms of diving depth (analogous like in the „5

simple steps“) by writing the diving depth as the geometrical length of the water-column with

h in an SI-unit system and re-grouping terms in a constant and a variable part for

comparison:

Pt.tol.ig = M

Pamb = p0 + ρ * g * h

Pt.tol.ig = (p0 + ρgh)/b + a = (p0/b + a) + ρgh/b = (p0/b + a) + c/b * h

ΔM = c * 1/b

(18)

M0 = a + p0/b

As for the most dive sites ρ*g are usually more or less a constant, we write for convenience

this new constant c = ρ*g. By comparison of the parts which remain constant and the variable

parts of the two equations (14) & (17) we find these 2 identities, the last two lines of (18).

N.B.: the innocent looking set of equations (18) has buried some stumbling blocks if you want

to convert these systems now for actual figures, i.e. if you want to change from the imperial

set of variables to the metric, the SI units, or vice-versa. It is not only the question to convert

feet to meter!

In the constant is hidden the transition from seawater (USN, DCIEM) to freshwater (european

tables) via the ρ:

c = ρ * g

pls. cf. again the example in the „5 simple steps“ on how to calculate an absolute, ambient

pressure for diving.

M and M0 are in fsw: this is a pressure and not a geometrical length in feet. That M fits

dimensionless into the equation, the diving depth d as well has to be in the pressure-units of

fsw (which is happening eventually in some diving tables). If d is in the geometrical

dimension of a length from the Imperial System, which happens to be feet, the dimension of

M has to be fsw / feet (as outlined above in Bob’s tables)!

For the useful conversion factors use the nice table on p. 3 from Wayne A. Gerth, David J.

Doolette: VVal-79 Thalmann Algorithm Metric and Imperial Air Decompression Tables,

NEDU TR 16-05:

According to AAB and his co-workers (Keller, H.; Herrmann, J.; Müller, B.; Völlm, E.; et al ...)

there is the following empirical relationship between the a- and b- coefficients to the HT. This

one is valid only for nitrogen. ([65], p. 129)! For helium it looks a little bit different ... (pls. cf.

[65], S. 131).

a = 2 Bar * τ-1/3

b = 1.005 - τ-1/2

(19)

This is similar to the ideas formulated in (16). One is the so-called set "A" from purely

theoretical contemplations; further on there is a set called "B" (more conservative, for printed

tables) and as well a "C" set for on-line calculations with a dive computer ([65], p. 158). A cut-

out of the complete table with these 3 systems for N2 & Helium is below, it is this particular

page 158:

The ratio of the HT of two immediately sequenced compartmens is roughly 1.2 to 1.4.

(Workman for eg. puts this to another value: 5 * 2n, n being the compartment no. ...) This is

not a strict physiological law but respects the desire to have a narrowly cramped network of

the HT. Only the very short and the extremly long HT are representing compartments which

have been verified through a lot of experiments on humans in the pressure chamber in

Zürich in the 60's. Nevertheless there is a coarse one-to-one mapping of HT with "real" body

tissues from p. 115 in [65]:

• Nr. 1 - 4: brain and spinal cord

• Nr. 5 - 11: skin

• Nr. 9 - 12: muscles

• Nr. 13 - 16: joints, cartilage, bones

• Nr. 7 - 16: internal ear

The cited root to the third from the derivation of the coefficient a pinpoints the tolerated

surplus volume of the inert gas. A little bit more handy is the following, from (17):

Pamb, tol = ( Pt, ig - a ) * b

© SubMarineConsulting                                Introduction to decompression calculation                                             page # 34 
 
Further Reading 
 
A short synopsis on pressure units & the conversion: 
The diving medical detectives: when diving medicine books are 
completely wrong Part V (02.02.2023): On pressure units   
Download for free there: 
https://dx.doi.org/10.13140/RG.2.2.31827.45600 
 
[78] Key Documents of the Biomedical Aspects of Deep-Sea Diving: Selected from the 
World's Literature 1682 - 1982, Undersea & Hyperbaric Medical Society, Inc., 1983 
(3,700 pages, compiled from the worlds leading experts!) 
more info: the complete TOC of the CD:  
Vol. I 
Vol. II 
Vol. III 
Vol. IV 
Vol. V 
https://www.divetable.eu/BOOKS/78_toc.pdf 
https://www.divetable.eu/78_Vol_I.pdf 
https://www.divetable.eu/78_Vol_II.pdf 
https://www.divetable.eu/78_Vol_III.pdf 
https://www.divetable.eu/78_Vol_IV.pdf 
https://www.divetable.eu/78_Vol_V.pdf 
 
  

Calculation of „No Stop Times“ and decompression stop times

We take now the Haldane equation for granted by applying it to questions from real-world

diving, at first for only one single compartment and then later on for the whole bunch of

compartments.: i.e. we want to calculate „ND“-limits (so-called „no decompression limits“ or

„no-stop-times“), real decompression stop times, time-to-flight, resp. desaturation times (if

there is one, at all ...). And in order to do so we take equation (6a) and re-arrange a little bit:

e- k * t = [ (Pt(t) - Palv0) / (Pt0 - Palv0) ]

(20)

The above fraction we put as one expression into the squared brackets [ ... ] . Then we put

the equation to logarithms:

ln [...] = ln ( e- k t ) = - k t

We are solving for t:

t = - 1 / k * ln [ ... ]

and from (12) with k = ln(2) / τ we have now the blue-print for all these times t in question

with:

t = - τ / ln2 * ln[ (Pt(t) - Palv0) / (Pt0 - Palv0) ]

(21)

With (21) and reasonable assumptions for the each of the pressure terms we are able to

calculate now:

• the NDL with (17) and Pamb = 1.0 bar

• if you put Pamb < 1.0 bar, say 0.8 for a mountain lake around 2,000 m above SL, you

could calculate an NDL table for altitude diving (or the required deco-stops, for that

matter)

• the time to flight, if we put in (17) the cabin pressure in the airplane to:

Pcabin = 0.58 Bar = Pamb.

This is the somewhat lowest pressure, normally you have in civilian air planes during

continental flights approximately 0.8 bar.

• the desaturation time, if we introduce a certain mathematical fuzziness:

Pt(t) = PN2 + Delta, Delta being a very small number, because the straight line with

constant P = PN2 and the exponential curve of the desaturation P = Pt(t) will meet only

at infinity, which, btw is a bit too long for a diving vacation...

Völlm suggests in [65], on p. 216, to take a certain part of the daily variation of air

pressure, i.e. 1/3 of the ca. + - 30 mbar variation.

• the required stop-times for the deco stops

• (21) could be used as well for mixed gases but only with reasonable assumptions for

the respective a- and b- coefficients for each inert gas component in the breathing

mix (to be detailed in next chapter).

AAB and others are offering, for eg. in [65], on p. 119:

Pt(t) = Pt, He(t) + Pt, N2(t)

with the a- and b- coefficients being normalized with the respective partial pressures in each

compartment (pls. cf. [54] on p. 86): so for each compartment at any time t and for each

combination of a- & b-values we have:

a (He + N2) = [( Pt, He * aHe ) + ( Pt, N2 * aN2)] / ( Pt, He + Pt, N2 )

b (He + N2) = [( Pt, He * bHe ) + ( Pt, N2 * bN2)] / ( Pt, He + Pt, N2 )

(22)

(pls. cf. the examples in [4], p. 27 and as well in [5], p. 80.). We are going to exploit (22) later

on for the calculation of stop times for mix gas diving.

„5 simple steps“

The next 12 pages will feature the „5 simple steps“ with 24 colourful examples along the

equations (6a) or (6b) and (21) for only one compartment with air.

The rationale for mix gas diving

„Nitrox is as good as air, only better!“

Source: Dr. Morgan Wells, from “USS Monitor”, [169], p. 104, line 7

The more oxygen the less nitrogen: OK! This one reduces the N2-load on your body and thus

you could increase the „NDL“ or truncate your SI in comparison to your air-diving colleague.

The downside is, you increase your Ox-Tox dose, regardless of the method you calculate it!

(NOAA / %CNS and Repex / OTU or the k-values and / or ESOT) ... I.e.: EAN / Nitrox is

good for long but shallow dives.

If you have to dive deeper than ca. 30 m / 100 feet and stay functioning, fully perfect clear-

headed, you have to replace N2 with Helium. If she wants to stay a little bit longer: replace a

little part of Oxygen with Helium. If she wants to dive even deeper, replace more Oxygen with

Helium ...

The idea behind is to reduce the nitrogen-narcosis and the ox-tox doses:

Meyer-Overton theory: reduced narcosis, if N2 is being replaced with Helium

Reduced WOB (work-of-breathing) as the density of the breathing mix is reduced, when

Helium is used instead of N2.

Say, for a deep dive to 90 m, where she has to work for a couple of hours you could prepare

for her the following bottom-mix: Trimix 12 / 68 / 20

Partialpressures P Gas = P amb * f Gas = 10 * f Gas

PO2 = 1.2 Bar

PHe = 6.8 Bar

PN2 = 2.0 Bar

You could make it super-frosty, by dropping the N2 alltogether and you end up with a:

Heliox 12 / 88

If you want to reduce the Ox-Tox doses, decrease the O2 fraction for her bottom-mix and

prepare for her a travel mix, say a EAN50, i.e. 50 % O2 and 50 % N2, good for the descent

from 0 to 30 m and back.

Anyway, the simple formula for the stop-times with one inert gas (21) won’t do it any longer!

We will cover that in the next 5 pages ...

Calculation of decompression stop times for mix gases

1) Mix Gas Theory

If we have more than only one inert gas (as is the case with nitrogen N2 in air/nitrox/EAN or

with Helium in Heliox), for e.g. two as in the case of Trimix (nitrogen N2 and helium He) we

have to expand the Haldane equation (6a).

Basically the law of Henry will be valid for each inert gasi separately:

(well, well, except the region, what specialists call "Poyinting effect", i.e.: very high pressures

and very complex organic molecules ...)

qi = li * pi * Vk

Q = ∑ qi

(50)

The sum encloses all inert gasesi and as well all compartmentsk

Variable

Definition

qi

amount of diluted inert gas i within a compartment with volume Vk

li

solubility coefficient of inert gasi valid for this volume Vk

pi

partial pressure of inert gasi

Vk

volume of compartment k

For the sum of all partial pressures we will have by convention the law of Dalton:

P = ∑ pi = p1 + p2 + p3 + ...

(51)

Thus we will have for a stationary state in a compartment with Trimix the following:

Pt(0) = ∑pt, i(0) = pt, N2(0) + pt, He(0)

(52a)

as well for the alveoli:

Palv(0) = ∑palv, i(0) = palv, N2(0) + palv, He(0)

(52b)

as well for all times t:

Ptiss(t) = Pt(t) = ∑ ptiss, i(t) = ptiss, N2(t) + ptiss, He(t)

(52c)

Thus we expand the Haldane-equation (6a):

Pt(t) = Palv0,N2 + [Pt0,N2 - Palv0,N2] e-kN2t + Palv0,He + [Pt0,He - Palv0,He] e-kHet

(53)

We have to check if this expansion will satisfy the basic differential equation (1a). So we

expand (1a) with an additional term for a second inert gas:

dPt(t)/dt = k1 * [Palv,N2(t) - Pt,N2(t)] + k2 * [Palv,He(t) - Pt,He(t)]

(54)

We determine the proportionality factors k1 and k2, by calculating d Pt(t) / dt from (53) :

dPt(t)/dt = - kN2 * [Pt0,N2 - Palv0,N2] e-kN2t - kHe * [Pt0,He - Palv0,He] e-kHet

(53a)

... and with this terms we replace the left hand side of (54):

- kN2 * [Pt0,N2 - Palv0,N2] e-kN2t - kHe * [Pt0,He - Palv0,He] e-kHet =

k1 * [Palv,N2(t) - Pt,N2(t)] + k2 * [Palv,He(t) - Pt,He(t)] =

k1 * Palv,N2(t) - k1 * Pt,N2(t) + k2 * Palv,He(t) - k2 * Pt,He(t)

(55)

For the pt, i(t) we exploit once again (52c) resp. (53), put this into (55), right hand side, re-

arrange and recieve:

- kN2 * [Pt0,N2 - Palv0,N2] e-kN2t - kHe * [Pt0,He - Palv0,He] e-kHet =

k1 * Palv,N2(t) + k2 * Palv,He(t) - k1 * [Palv0,N2 + [Pt0,N2 - Palv0,N2] e-kN2t - k2 *

[Palv0,He + [Pt0,He - Palv0,He] e-kHet

(55a)

We multiply the terms of the squared brackets [ ... ] of the right hand side and re-arrange

according to constant resp. time-dependant terms:

- kN2 * [Pt0,N2 - Palv0,N2] e-kN2t - kHe * [Pt0,He - Palv0,He] e-kHet =

k1 * Palv,N2(t) + k2 * Palv,He(t) - k1 * Palv0,N2 - k1 * (Pt0,N2 - Palv0,N2)e-kN2t - k2 *

Palv0,He - k2 * (Pt0,He - Palv0,He)e-kHet =

k1 * [Palv,N2(t) - Palv0,N2] + k2 * [Palv,He(t) - Palv0,He] - k1 * (Pt0,N2 - Palv0,N2)e-kN2t -

k2 * (Pt0,He - Palv0,He)e-kHet

(55b)

We take now the identical boundary conditions for the both inert gases which guided us to

the solution of (6a):

Palv,N2(t) = Palv0,N2 for all times t and analogous

Palv,He(t) = Palv0,He

the equation (55a) then will be identically satisfied with:

k1 = kN2 as well: k2 = kHe

(55c)

Thus the approach (53) satisfies the central LDE (1a), if the second gas is applied as purely

additive: this yields for the stationary part, i.e. the part with no varation in time in the alveolar

gas (52b) and as well for the time-dependant part (52c).

2) Calculation of stop times for mixed gases

The extensions for mixed gases with (22) are for the a- und b-coefficients, and as well the

sets for τ resp. λ (equation set 56):

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 )

λ* = λ (He + N2) = [( Pt, He * λHe ) + ( Pt, N2 * λN2)] / ( Pt, He + Pt, N2 )

(56)

and this is for all the k compartments, e.g.: k = 1 - 16! (within the standard ZH-L framework; if

you want to compare other things, like the ANALYST software from COCHRAN, k runs from

1 to 20, that is from approx. 1.25 min to 900 mins.)

As well we have for our two inert gases He and N2 (52a, 52b, 52c) and thus:

• Palv0 = palv0, N2(t0) + palv0, He(t0)

• Pt0 = pN2,t0 + pHe,t0

• for the tolerated tissue pressure ptissue, tolerated we have now analogous:

•

Pt.tol.ig = Pamb / b* + a*

If you take now for the calculation of the deco times td the equation (21) with all of the above

cited extensions for mixed gases, you generate an equation with two more unknowns. But

you have only one equation and now 3 unknowns: i.e.: you can not solve this equation

analytically ...

So we have:

ptiss, N2(t) und ptiss, He(t) for the time t = td because of: Pt(t) = ptiss, N2(t) + ptiss, He(t), and

this for all compartmentsk! As a*, b* and as well λ* have been assesssed for time t = t0 , but

this is only the start of the deco phase! At first, td is unknown and thus as well the respective

compartment saturations ptiss(td) and thus consequently also a*, b*, λ* at time t = td.

• 1) numerical method:

In plain language: in dealing with two (2, or more) inert gases, we have to solve (21)

either approximatively by taking small incremental time-steps like Δt, e.g.:

Δt = 1 sec, or Δt = 1 or 1.5 or 2 min and checking after every time-step the new

Pt(t + Δt) = ptiss, N2(t + Δt) + ptiss, He(t + Δt)

derived from the previous Pt(t), and comparing the tolerated ambient pressure

Pambient, tolerated of the desired depth / deco-stage depth and the Ptissue, tolerated in question:

is after n time-steps

Pt(t = td) = Pt(t0 + n * Δt) < = Ptiss, tol

we have found with:

t = td , i.e.: td =  the required deco-time.

Further Possibilities:

• 2) "quasi"-analytical method:

we accept an error, by taking the well-known t0 from the start of the deco:

Pt(t0) = ptiss, N2(t0) + ptiss, He(t0)

and the respective weighting of the a-, b-coefficients and λ (equations 56) and simply

using all of these for the calculations of Pt(t) for all times t.

This error produces at first an over-estimation of the deco-times!

(in dependancy of: depth, time and He-fraction)

(But it seems, that this approach is used in a couple of desktop deco-softwares and

regular mix gas diving computers: the numerical calculations (method 1) require more

efforts during programming and as well additional variables, which means more

memory. As well it is time-consuming, dependant on the amount of time-steps and

the step-size of time-steps: each compartment has to go through long loops for each

deco-stage.)

Finally, we try also:

• 3) a Taylor-Expansion in Series for equation (21):

we take equation (53) and re-arrange analogous with (21):

Pt(t) = Palv0,N2 + Palv0,He + [Pt0,N2 - Palv0,N2] e-kN2t + [Pt0,He - Palv0,He] e-kHet

= Pamb / b* + a*

Now we are using the taylor expansion for the exponentialterm within the Pt(t)

with x = - ki * t, and for ex :

ex  1 + x + (x2 / 2) + (x3 / 6) + ... + ...

We stop the Taylor expansion after the 3rd or even already after the 2nd term: this is

sufficiently accurrate for small x, that is: short deco stops!

The Pt(t) in dependancy with the b* and a* means that the time t appears as well in

the numerator AND the denominator and thus we get a cubic or bi-quadratic

polynomial for t in its fullest beauty. This one we have to solve as well numerically.

But thus we wouldn't gain anything in comparison to method 1!

All this complicated stuff is avoided easily within the traditional set of M-values from Bob

Workman from his 1965 NEDU report, as there are no different half-times for Helium and N2.

As well the idea to use and calculate Trimix schedules was not a widespread method these

times.

Variations in the TTS: where do they come from?

If you want to compare dive tables or dive computers with each other or a dive computer

simulation with a printed dive table, you have to know the used variables with a precision to

the 4th. or 5th. decimal, if not, you compare apples and oranges.

This set of variables we have been discussing & using here, an exhaustive list with examples

of the comparisons along an air diving table you will find there:

Recovery of selected ZH-86 air-diving schedules via a decompression shareware

https://dx.doi.org/10.13140/RG.2.2.34235.13609

The overview of this list:

One source of ΔTTS are the ZH-L 16 parameters by themselves:

AAB published 17 compartments and it is not defined to use all of them or only 16; and if only

with 16, there is still leeway in using compartment # 1b or not … As well for a GF-

implementation there exists no agreed-upon „gold standard“ … And, well, finally: you could

use the published numbers for the a- & b-coefficients or you could implement the formulas

(19), which are a little bit at variance …

But now we take a couple of examples (*) with Trimix / Heliox gases and check the error

produced with the prevailing method 2 from the previous chapter, by comparing with the

"real" deco-times calculated with method 1; we did so for various square (box) depth-profiles

and various mixes for the bottom-gas. We compared thus the deco times from

two commercially available desktop deco software products and DIVE V3 with the times

assessed by method 1 (source: pls. cf. the „further reading“-section).

(*) a couple here means exactly:

• more than 1,900 profiles from the TEC-recreational set:

• We took 6 depth profiles from 30 - 80 m,

• 5 bottom times from 20 - 60 min,

• and helium- fractions from 5 - 80 % (16 mixes),

• over all 480 profiles,

• and the whole then 4 (four!) times ...

This error increases with the decompression obligation, i.e. with increasing diving-

depth and bottom-time and the dependancy on the helium fractions f is, roughly

speaking like this, only a qualitative, sketch:

We are seeing the deco-times tD sketched as TTS over the helium-fractions fHe:

the deviations are the differences from equation (17) (= method 2) and the average /

mean-values out of all the 3 softwares (the  symbolizes the average out of the 3

products) in comparison to the numerical values according method 1. I.e. as pointed

out above:

 in the beginning we have this overestimation of deco times (error < 0),

 in the region of the regular TEC-REC trimixes, especially at around 35 - 45 % He the

error converges -> 0,

 and, by increasing fHe, the error is growing steadily (error > 0).

 …

   

   







And, also:

Further reading

➔ Additionally we have compiled some in-depth background information on the data pool

used above, to be used in conjuction with the 2011-papers further down:

Background Information on the update 03 / 2021

https://dx.doi.org/10.13140/RG.2.2.22574.02888

• CAISSON 03, 2011: Dekompressionsberechnungen für Trimix-Tauchgänge mit PC-

Software: Reparieren Gradientenfaktoren defekte Algorithmen oder defekte Software-

Implementierungen?

CAISSON 26. Jg./2011/Nr. 3, S. 4 - 12

Or there: https://www.divetable.info/sammelband/Sammelband_CAISSON.pdf

• Tech Diving Mag, Issue 05 / 2011: Decompression calculations for trimix dives with

PC software; gradient factors: do they repair defective algorithms or do they repair

defective implementations?

TDM , Ausgabe 05 / 2011, S. 41 - 53

• Or there, the update in 03 / 2021 on these seasoned papers from 2011:

https://www.researchgate.net/publication/350176530_Update_per_03_2021_on_Dec

ompression-Calculations_for_Trimix_Dives_with_PC-

_Software_Gradient_Factors_do_they_repair_defective_algorithms_or_do_they_repa

ir_defective_implementations

• International Journal of the Society for Underwater Technology, November 2012:

Variations in the TTS: where do they come from?

SUT, Vol. 31, No. 1, pp. 43 - 47, 2012

Short detour on „Bubble“ models

... and why we don’t use them here.

The so-called „Bubble models“ are trying to model the free gas phase, i.e.: gas bubbles. The

seasoned perfusion models are dealing exclusively with dissolved gases. This was one of

the caveats Haldane put out for his tables: only for „uneventful decompression“, i.e.: no

bubbles. His reasoning was that bubbles could block the perfusion and thus hinder saturation

and desaturation processes.

The most popular and, at least among tekkies (tec divers), well-known bubble models are:

 The VPM (Varying Permeability Model)

 The thermodynamic model from Hills, B.A.

For both of these there is also abundant literature (not to be cited or referenced here). Here

we put just a couple of remarks:

VPM:

 The origin of the VPM are optical bubble counts in agarose gel

 There are a handful of free parameters, never to be verified on humans,

 but instead by a “best-fit“ with already existing diving tables

 the authors, Yount & Hoffman pointed out by themselves, that their algorithm is not

suited for and was never intended for trimix or the like, only for breathing gases with

ONE inert gas (i.e.: Air / EAN or Heliox)

Thermodynamic Model:

 Brian put a lot of thermodynamic reasoning and efforts into his model and simulated it

with analogous machines;

 see his epic ph.d. thesis (link in the references, pls. cf. next page)

 however the success of his model was limited, basically to the complexity of all these

Bessel functions he needed and

 the dive profiles from the pearl divers in the Torres Straits where just not enough

 and there was too much statistical fluctuations in the profile data, so the success of

their „deep stops“ could be explained as well easily with standard variations from the

old USN helium tables

There is still another one, the RGBM, the „Reduced Gradient Bubble Model“.

It enjoyed some implementations, for eg. in Mares® and Suunto® dive computers from ca.

2000 to 2018. As well there have been a couple of little software-rat shops, like ABYSS, GAP

and rgbmdiving.com which licenced the source code, but had left the market already since

long. However, their success was also limited as there existed no complete, transparent

documentation: all the papers & books available for this topic have been peppered with

errors. Especially the dive computer implementations found only limited enthusiasm, as

these products behaved erratically, to put it mildly. When dived in a group and compared with

other computers, they displayed sometimes unpredictable dive plans, especially with multi-

level diving and decompression, altitude diving and repetitive diving. One diving instructor,

Mark Ellyatt, tested this stuff around 260 m and thereafter coined his experience in a new

tag: „Really Good Bends Model“. The person, responsible for it (Bruce Wienke) kept his

algorithm and the source code as a company-secret. Bruce died in 2020 and thus there will

be no further development and no error correction resp. bug fixing.

In concluding this chapter, there is an interesting information from one of the (then, 2017)

first examinations from the DSG database:

 out of 10738 dive-profiles, recorded with a dive computer & uploaded into DSG-

portal, there have been:

 165 cases of DCS

 the DRA service from this portal found as well, that:

 ca. 74 % of all DCS cases were in the saturation range of only ca. 70 to 90 %

 all of these DCS cases were distributed evenly between „perfusion model“

computers and „RGBM-like“ computers!

In the end, all VPM-, RGBM- & perfusion algorithms like USN, DSAT, ZH-L, are using the

same haldanean workhorse with very similar sets of compartments with exponential kinetics

for the calculation of the saturation / desaturation processes. The only strategic difference is

one equation (or: just one sub-routine, if you want to have it) to assess the ceiling and how

long you have to stay there.

Once, our boss from the SubMarineConsulting Lab said at a TEC 4.0: update! -conference:

„VPM and RGBM are nebech“. (*)

Or, as it was put already 45 years ago by Bill Hamilton:

“Haldane works if you use it properly!”

R.W. Hamilton, 17th UHMS workshop, p. 135; 1978

If you really want to explore inert gas bubbles in the human body: there is one ultimate

source available! It is the book from Saul Goldman et al. (pls. cf. next page).

As well there is a link to the scientific background of the VPM and a benchmark with the then

(03 / 2018) available VPM implementations.

And there is the link to the books and papers from DAN, dealing with the „Science of Diving“

and the DSG / DRA.

(*) „nebech“ [nä-bä-ch] is yiddish and has a multitude of meanings. Among some other

things, it can imply sheer nonsense. On the proper use of „nebech“ have a look & fun there:

Ulam, S. M. (1991) Adventures of a Mathematician, p.: 194 - 195

© SubMarineConsulting                                Introduction to decompression calculation                                             page # 59 
 
Further Reading 
 
  the original VPM source, i.e.: the ph.d. thesis of Donald Clinton Hoffman from 1985: 
https://www.divetable.eu/PAPERS/Hoffman_1985.pdf 
  Hills trailblazing book and his ph.d. thesis: 
https://www.divetable.eu/BOOKS/102_cover.pdf 
https://www.divetable.eu/BOOKS/102_toc.pdf 
https://www.divetable.eu/BOOKS/102_phd.pdf 
  The ultimate source on bubbles in the human body: 
Goldman, Saul; Solano-Altamiro, J. Manuel; LeDez, Kenneth M (2018) Gas Bubble 
Dynamics in the Human Body, AP Elsevier, ISBN 978-0-12-810519-1 
https://www.divetable.eu/BOOKS/199.pdf 
  AMC, Amsterdam, 03/2018; International Symposion on:                                                          
21.st Century Decompression Theory;                                                                                                  
1.: Dual Phase Decompression Theory and Bubble Dynamics,  
https://www.divetable.info/skripte/Bubble_Dynamics_02.pdf 
  2.: Implementations of Dual Phase Decompression Models in Tables and Meters for 
technical diving 
https://www.divetable.info/skripte/TEC_DIVE_02.pdf 
 DAN: SOD, DSG & DRA: 
[170] Balestra, Constantino; Germonpre, Peter (ed.) The Science of Diving: Things your 
instructor never told you (2015) Lambert Academic Publishing, ISBN: 978-3-659-66233-1, 
more info: cover; https://www.divetable.eu/BOOKS/170_cover.pdf 
more info: TOC; https://www.divetable.eu/BOOKS/170_toc.pdf 
Frontiers in Psychology 09 / 2017: 
post-publication comment from SubMarineConsulting: 
https://dx.doi.org/10.13140/RG.2.2.28029.79847 
 
from Mark Elyatt‘s homepage:  
https://www.inspired-training.com/RGBM%20Really%20Good%20Bends%20Model.htm  
  

Gradient Factors (GF)

Now we have a look at more compartments, i.e.: we have to look for the „leading

compartment“.

The leading compartment is, out of the bunch of all your compartments, the one with the

highest value of the ceiling (= the deepest decompression stop) and/or the longest stop time.

As well we have a look at the GF and why they sometimes come in pairs like a GF High with

her GF Low.

First, a little bit of history and the idea behind the GF. We will not repeat it here, as the whole

story is already detailed elsewhere:

• Gradient Factors: on the rise? Lecture for the annual

meeting of OeGTH in 12/2022 at Vienna

Download for free there:

https://dx.doi.org/10.13140/RG.2.2.20301.20963

The next 13 slides feature 3 compartments and examples with various GF, as there are

many possible methods to do it. At the end of this slide-show we summarize all comments &

explanations, the reference here is the slide # at the lower right corner of the blue boxes. The

graphs are done with Microsoft® Mathematics, the link to this tool you will find in the last

chapter:

Summary:

#1: is just the title and

#2 & #3: are showing in one graph the saturation and desaturation of the 3 compartments

with the HT of 5 (blue & black), 10 (light- & dark-green) & 20 (magenta) min

#4: features again the Haldane equation and outlines the used parameters. The formula and

the parameters are then input to MS® Mathematics to generate these graphs

#5: saturation along the time trajectory from 0 to 20 min

#6: the desaturation process starts with the next pressure-step: the jump to the surface; the

start-parameters are the then reached end-parameters from the saturation-process. The time

starts again from 0 at the next pressure step, #6 combines both processes

#7 features JSH 2:1 for reaching the surface in comparison to the desaturation, i.e.: all

newer perfusion-models allow a higher supersaturation for the fast compartments

#8: overview of the GF formulas in the grey info-box, designated as „Standard“, which, btw.

is no standard at all! Our paradigm will be compartment #1 with the HT of 4 min

#9: features the linear equation for the allowed / tolerated supersaturation for this

compartment and their purely geometric interpretation. The green line for the compartment

with the used parameters in the green box. Depicted is the slope (black values, black

triangle) and the Y-axis intersection (double red arrow), the 2 parameters needed for all

linear equations.

#10: depicts the possible variations of a linear equation, and who did use which one

#11: combines all graphs and parameters with the used formulas, with

#12: the same, but in relation to the bi-sectrix (bold blue)

#13: is a little look-up table for the unmodified a- & b- coefficients and as well the

modifications through the GF and the then resulting modified (lowered) Pt,tol

Dive Computer Simulation

Following is now the final example, where we put everything together what we have learned

the previous 70 pages:

how is this stuff calculated for a dive table resp.: how does a dive computer figure this one

out? No problem: we will have a look at ALL of the 16 compartments of a ZH-L16 C model,

with & without GF (pls. cf. the next 19 slides). At the end of this slide-show, again, we

summarize all comments & explanations, the reference here is the slide # at the lower right

corner of the blue boxes:

Summary:

#2 shows how the DIVE Version 3_11-software output is organized: the bar charts in the

lower part of the display and the upper is full of numbers: just concentrate on the 1st. white

line with the depths and times. Just by far enough for our exercise, forget the other

parameters. These will become interesting for longer & deeper dives and for dives with

Helium. If you are interested, check the manual for the DIVE software, the link is below (but

this is not needed for the gist of the examples here).

#3: the inert gases in your body, here all of the 16 compartments, are in equilibrium with what

mother nature offers to you. Thus the yellow bars have all the same length and hit the green

pN2 line at ca. 0.8 Bar.

#4: you jump to 30 m, the green line to 4 * 0.8 = 3.2 bar. As the saturation process takes

time, your compartments lack behind. The yellow bars go up to the green line in

#5 to #7: the fastest compartments (small HT) on the left side are already saturated

(remember the rule-of-thumbs), the slower ones (bigger HT) are lacking behind. The little red

lines are the calculated allowed / tolerated ambient pressures. If such a pressure is > 1, it

implies a decompression stop. Say, it is 1.3 Bar for a compartment, this calls for a

decompression stop @ 3m. The dive computer does this calculations for all compartments

and all required stops in

#8: you see the various stages from 9 to 3 m, the stop times and the leading compartments.

This is pretty much in line of what is tabulated in the ZH-86 (#19). In order to do so, your dive

computer, or the girl, trying to calculate a dive table, must assume a certain ascent speed.

Btw, the TTS from #19 / ZH-86 and #8, for each stage ca. 1 min, results from:

 The DIVE framework is a dive computer simulator and not a dive table calculator

(even if you could do so by changing the default settings of the software)

 the ZH-86 tables uses the ZH-L 16 B (with the „B“ for taBle :-) ) set of the a- & b-

coefficients

 default in DIVE is ZH-L 16 C, the „C“ for Computer :-)

 so already a little bit of conservativism is built-in here

 also, all the ZH-based tables enjoy a somewhat slightly reduced pamb in direct

comparison to a USN or a DCIEM table

 and, well, the above introduced ascent speed varies across the ZH-framework, and is

not always like the published value of 10 m / min

 for real world diving, this TTS is irrelevant due to the already described

measurement errors of the devices

 and anyway: a regular diver, un-tethered or not-surface-supplied, will fail constantly

to master such a run-time with the required accuracy ...

So your dive computer has now the task to pick the leading compartment: she does so every

3 to 5 seconds by searching for the compartment with the biggest value for the tolerated

ambient pressure: this is the lowest ceiling. Then she has to look for the one which takes

longest to de-saturate. And this one is the winner, i.e.: the „leading compartment“. But just for

this deco stage: when you look closely at #8 to #12, you see, that the leading compartment

shifts slowly, slowly to the higher #, the closer your cyber-diver gets to the surface. The stop

times are assessed by the methods we practised in the „5 simple steps“.

#9 to #12 are featuring the decompression prognosis with and without the GF.

#10 shows a moderate GF with 0.9: the stop times on all 3 stages are increased, just a little

bit; so the TTS goes up by 8 min. In

#11 the TTS went up by 20 min through the GF 0.8; whereas

#12 features a GF sliding from GF Low = 0.7 to GF High = 0.9. The GF Low makes a deeper

stop @ 12 m (instead of 9 m) and the GF High = 0.9 sets the last stop to ½ h, instead of the

37 min from the GF 0.8. Thus the GF slide avoids excessively long shallow stops. But this

mechanism (pls. cf. also slide #7 of the example with the 3 compartments & GF) is without

any physiologic rationale.

#13: after the ascent to 6 m within .3 min you see the varying super-saturations of the

compartments # 1 to 10. From #11 on, the compartments are still under-saturated (below the

green line). These ones will still continue saturating, even during ascents and during shallow

stops.

#14: after 11 min @ 6 m, the super-saturation of the fast compartments dropped, the others

still saturating.

#15 is featuring a „heat map“ with the GF as yellow lines. The color coding is according to

the ideas from DAN, implemented in the DRA from the DSG portal.

#17 shows the situation after 27 min @ 3 m: all these little red indicators are now below 1

bar, so our cyber-diver may ascend to the surface. And this is what she does now in

#18 and displaying the super-saturations (compartments #6 to #16), still there after a surface

interval of 90 min.

Tools

All the here used tools, i.e.: DIVE 3_11 & MS Mathematics, are free of charge for download.

DIVE Version 3_11

The DIVE Version 3 site: https://www.divetable.info/DIVE_V3/V3e/index.htm

The latest BETA software: https://www.divetable.info/beta/D3_11.exe

The handbook / documentation: https://www.divetable.info/DIVE_V3/V3e/DOXV3_0.pdf

Microsoft Mathematics 64-Bit, Version 4.0 of 2011 (standalone)

or Add-In for WORD / 1-Note, Version 2 of 2019; Download for free:

https://www.microsoft.com/de-at/download/details.aspx?id=17786

Handling of DIVE

For our paradigm above, here is a quick start on the software (also in the manual ...):

 after downloading / un-zipping: double-click on the *.exe file

 DIVE asks you always very friendly: what next? for your inputs, without the „“

 „D“ „30.“ „50.“

 „A“ to get the prognosis

 „P“ for the compartment plot

 that’s it, basically ...

 „HELP“ or „?“ is for help

 and the manual also ...

Index

2

2 : 1 7, 10

A

a 31

allometric polynom 8

B

b 31

C

compartment 12

D

deterministic 4

G

GF 60

H

Haldane-Equation 18, 21

Half Time 25

HT 25

L

LDE 14

M

M0 27

Meyer-Overton 49

M-Value 27

N

NDL 35

nebech 58

numerical 52

P

perfusion 4

R

respiratory quotient 26

S

Schreiner equation 25

stochastic 4

symmetrical 8

T

Taylor expansion 53

U

uneventful 8

W

WOB 49

Z

ZH-86 80

ZH-L 16 B 80

ZH-L 16 C 80

Introduction to Decompression Calculation

Abstract

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