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Essay on Fast and Superfast Compartments
Essay on fast and super-fast compartments
1) Rationale .................................................................................................................................... 1
2) Background: What is a compartment, anyway??? ..................................................................... 2
3) Experiment with goats: Haldane ................................................................................................ 5
4) Submarine escape: Schaefer ...................................................................................................... 6
5) The pragmatic Schreiner Matrix ................................................................................................. 7
6) United States Navy Method: Workman ..................................................................................... 8
7) Swiss Altitude Diving: Bühlmann ................................................................................................ 9
8) PBPK: Mapleson, Nishi, Flook et al. .......................................................................................... 10
9) Mixing 2 models: Egi & Gürmen ............................................................................................... 11
10) Breath-hold and DCS Type II: Goldman et al. ........................................................................... 12
11) A Fit to the Paulev data ............................................................................................................ 13
12) Take-home Messages ............................................................................................................... 14
The following is an essay on fast and super-fast compartments. So this is not a strict
scientific paper, neither in form nor in contents but a couple of preliminary thoughts on the
topic, intended to raise awareness or for further discussion.
If you are new to the TDM magazine, new to TEC diving or even new to diving, you may
enjoy some basic information on deterministic decompression models and algorithms in
chapter 2, the „Background“; the seasoned diver may skip this safely. Readers not intending
to go into the mathematical details may then proceed as well directly to „Take-home
Messages“ in chapter 12.
During my first course on breathhoId diving some 20 years ago, i stumbled on the inability of
standard decompression tables and algorithms to cope with breath-hold diving profiles. My
then instructor on this topic, Andy Anlauf, who was at times an elite apnea diver, asked me if
i could make a decompression tabIe for the record profiles: for eg. in 2 min down to 130 m
and then up to the surface. If you now look at a compartment, say with a halftime ( ½ ) of
12.5 min (compartment #3 in the standard ZH-L parlance), it will change its initial inertgas
load from ca. 0.8 to only 1.1 Bar after 1 min @ 90 m. The supersaturation of ca. 0.3 Bar is
not enough to yield any basic decompression; even on return to the surface it is still taking up
inertgas and the supersaturation is raised to ca. 0.5 Bar, still not sufficient for a substantial
decompression time. The other compartments from # 8 on will not even take note on this
pressure excursion.
Further on, there is a phenomenon called „Taravana“: these are the many anecdotal reports
on unexplained DCS cases during breath-hold dives, especially for commercial indigenious
As well Paulev (cf. chpt. 11) observed cases of DCS type II during (breath-hold) submarine
escape training; Schaefer (cf. chpt. 4) observed N2-bubbles in blood samples from breath-
hold divers, quickly disappearing after 10 sec.
In the TEC community there is since long a sometimes overheated discussion around the
effectiveness of short, 1 to 2 min, deep stops during decompression from mixgas dives.
The time-domain of all these phenomenon is in the sub-5 min region. Basically a
phenomenological description needs thus an exponential halftime (½ ) in the order of a
fraction of the maximal time-frame.Thus approx. 5 min divided by 6 half-times would allow for
a clean description to cope mathematically with the quick pressure changes: 6 half-times
being the rule-of-thumb for complete saturation or desaturation of any compartment (at
constant pressure). We end up thus with ½ of approx. 60 sec.
After a snappy introduction to decompression models and algorithms in the next chapter,
there will be a short and limited literature overview which reveals if and how other (selected)
researchers have been dealing with the spectrum of used half-times.
Background: What is a compartment, anyway???
The following is a boldfaced copy from a book of Carl Edmonds, another chap of mine (Ref.:
Edmonds, Carl. Diving and Subaquatic Medicine, Fifth Edition, 5th Edition. CRC Press,
20150713. VitalBook file), the graphs used here have been drawed originally by David
Doolette, working now for the NEDU, the Naval Experimental Diving Unit of the USN, the
United States Navy:
(with a friendly permission by Carl Edmonds & Dr. David Doolette, USN)
The box depicted above is a model for the limited volume of some region in a mamalian
body: one compartment is showed here. It is a model for a well-stirred tissue (thus the
symbol with the little mixer) with a defined, perfusion-limited blood supply: the arrows from
left, the arterial part to the right, the venous part.
Then we will look at a dive scenario with more compartments: we see the Nitrogen uptake in
five hypothetical perfusion-limited tissue compartments during a dive to 30 metres (4 ATA)
using air. Pamb is the ambient pressure in atmospheres (atm). The inspired pressure of
nitrogen and the alveolar pressure of nitrogen rise to ~3.1 atm (not depicted in the figure),
and the arterial pressure of nitrogen (PaN2) immediately equilibrates. The tissue pressures of
nitrogen are slower to equilibrate, due to the final capacities of the blood, lung and circulation
carrying the inertgases. Only tissues 1 and 2 approaching saturation within the duration of
the exposure depicted. From the lines in the graph and with the rule-of-thumb cited above
you can derive the half-times of the compartments. For eg. P1 reaches its 50% saturation
after 5 min, so after 6 * 5 = 30 min it is supposed to be saturated; P2 after 6 * 10 min:
(with a friendly permission by Dr. David Doolette, USN)
The lines of saturation follow an exponential curve, typical for many natural phenomena, the
math behind (a simple linear differential equation) described already elsewhere, for eg. there: In this model here we have P1 to P5 in a parallel circuit
(pls. cf. graph below, the lower part), other models with a serial circuit are possible as well.
The most prominent decompression models like the ones from Haldane, Workman (USN
tables), Schreiner and Bühlmann (ZH-L) are using the parallel perfused setup. The serial
circuit showed below (upper part of the graph) is used by Kidd, Stubbs, Nishi et al for the
DCIEM tables and Canadian military and commercial procedures. We see 4 compartments
designated # 1 to # 4, with half-times ½ : HT 1 to HT 4. In the serial set-up these need not to
be different values.
Serial versus parallel coupling of compartments
All these models are called „deterministic“: they try to predict a safe decompression, that is
safe stop depths and stop times, based on the pressure/time profile and the inertgas content
of the breathed gases.
A completely other game is a „statistical“ decompression model: there the outcome of
thousands of dives is analysed after surfacing. The outcomes (DCS: YES or NO) being fitted
to a model and then a decompression table with a defined probability of getting DCS is
Physiologic definition of the compartment halftime:
As was described earlier, the halftimes ( ½ ) are related to the change in the moved blood
volume, i.e. the volume per time (ml per min) per ml of compartment volume; thus the
physiologic definition looks like that:
½ = 0,693 * αti / (αbl * dQ/dt) (0)
The definition of the other variables:
αti: solubility of the inert gas per compartment (tissue = ti), ml(S)gas * mlti -1 * (100 kPa) -1
αbl: solubility of the inert gas in blood (blood = bl), ml(S)gas * mlblood -1 * (100 kPa) -1
dQ/dt: perfusion rate, mlblood * mlti -1 * min -1
The ratio of the solubilities blood / tissue ( αbl / αti ) has a well-known name: the „partition
coefficient“; it could be looked up in tables (pls. cf. the remarks on PBPK in chpt. 8). If you do
not have the partition coefficient of your compartment in question and you do not have a clue
about its perfusion rate, you collapse everything into a single value. This approach leads
directly to the pragmatic Schreiner matrix (pls. cf. chpt. 5).
A compartment as a „low pass“!
The exponential functions to describe the on-/off gasing of the compartments are nearly the
same for an electronic circuit, consisting of a capacitor and a resistor. It is used for eg. to
rectify the current from AC to DC: the high frequency parts of the AC are filtered, allowing
only the lower frequencies to pass the electronic circuit; thus the name „low pass“.
Now, if you have a part of your dive profile with a „high frequency“ behavior, i.e. noticeable
changes of the diving depth versus short times as in yo-yo diving, the decompression
algorithm is „blind“ for it: the dive computer may log the depth changes over time but the
slower compartments will never notice it.
(Ref.: Hahn MH (1989): Reponses of decompression computers, tables and models to „yo-
yo“ diving, Undersea Biomed Res 16 (Suppl.:): 26.)
Experiment with goats: Haldane
The set of halftimes for his 5 compartments was generated by just doubling the 5 min
halftime 3 times, with the longest halftime being 75 min due to a hypothetical saturation of
nitrogen-uptake at around 5 to 7,5 h (p. 349 & 350) for the goats he used for his
experiments: 5, 10, 20, 40 & 75 min. Then there could be as well a compartment with a
halftime of 2.5 or 1.25 min. On page 348 he gave a hint to a faster saturation process
within max. 10 min which would yield a halftime of: 10 min / 6 ca. 1.6 min.
We could easily exploit this with Haldanes rule for safe ascent, the famous „2:1“ rule to
generate a new“ haldanian-type decompression table, but with deep stops! These stops
being noticeably deeper than in the original tables, in the 1 min region and not altering
the shallow stops by much [Rem.: an easy procedure on how to do that and an
appreciation of the work of Haldane and his colleagues you will find in this magazine, pls.
cf. TDM, Issue 25 of December 2016, on pages 13 20.].
Submarine escape: Schaefer
In his 1955 contribution to the 1st. Underwater Physiology Symposion;
he describes on p. 135 that during breath-hold dives in the 90 feet submarine escape training
tank there have been bubbles observed in alveolar and venous blood samples which have
been attributed to N2 and not to CO2. The blood samples were drawn from the divers
immediately on surfacing after a breath-hold dive. The foam due to these bubbles may have
been disappearing 10 sec after surfacing or 40 sec after start of ascent, the duration of these
dives being ca. 1 to max. 2 min. An allowable super-saturation ratio of 3:1 seems to be
This in turn would imply a de-saturation with a half-time of approx. 10 + 40 / 6 ca. 10 sec
and a saturation process with a halftime from 1/6 min up to 2/6 min.
The pragmatic Schreiner Matrix
In this contribution to the 4th. Symposion in 1971:
we see the pragmatic 4 by 4 matrix of the 16 compartments, compartment # 0 never used.
That is: we (*) could easily extract a super-fast compartment with a half-time of 2.5 or 1.25
min by exploiting his scheme on p. 210 with dQ/dt * R = 0.2772 min-1 resp. 0.5544 (fat
fraction X = 0.0)
United States Navy Method: Workman
(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))
Here we have compartment halftimes for N2 from 5 to 240 min (p. 5) and the corresponding
allowed inertgas supersaturations, called M-Values. The M-value follows a simple linear
relationship, based on empirical dive data (Eq. 1):
M = M0 + ΔM * d (1)
where M0 is the maximum inertgas partialpressure in the compartment for surfacing and ΔM
is the change with the diving depth (in feet).
By fitting separately the ΔM (Delta M) and M0 over the halftimes we (*) could as well extract
faster compartments and the corresponding allowed supersaturations.
fit for M0
Our generator function yields with a correlation coefficient of nearly 1, for eg. for the halftimes
1.25, 2 & 2.5 min these values for M0: 156, 134 & 126 fswa respectively.
fit for ΔM
The above generator polynom gives here, as well with a very high correlation coefficient for
the same choosen halftimes of 1.25, 2 & 2.5 min these ΔM Values: 37.5, 8.4 & 4.5
Swiss Altitude Diving: Bühlmann
(Ref.: "Tauchmedizin", Albert A. Bühlmann, Ernst B. Völlm (Mitarbeiter), P. Nussberger; 5.
edition in 2002, Springer, ISBN 3-540-42979-4)
Here we have already a simple relationship between the halftime ½ of a compartment and
the allowed supersaturation for Nitrogen (N2), if we combine the 2 empirical relationships for
the coeffcients a & b from p. 129 (Eq. 2) with the linear equation for the tolerated ambient
pressure (p. 117) (Eq. 3) into one:
a = 2,0 bar * (½ N 2 [min]) -1/3
b = 1,005 - 1 * (½ N 2 [min]) -1/2
Pcompartment = (Pambient,tolerated / b) + a (3)
This yields the following generator function (Eq. 4) by setting the tolerated ambient pressure
to 1 Bar (for a direct ascent to the surface for breath-hold diving or submarine escape
Pcompartment = (1 Bar / (1,005 τ -1/2)) + (2 Bar * τ -1/3) (4)
Thus we could extract here as well faster compartments and the corresponding compartment
overpressures. Here around a halftime of ½ = 1.005 min is a divergence in (Eq. 4) and thus
this is the smallest allowed value.
Our choosen halftimes of 1.25, 2 & 2.5 min are yielding the compartment overpressures of
ca. 11, 4.95 & 4.1 Bar, respectively. These we could compare directly with the M0-values
from the Workman set above, i.e. for d = 0 feet in (Eq. 1): 4.8, 4 & 3.9 Bar respectively.
PBPK: Mapleson, Nishi, Flook et al.
One of the first PBPK (Physiologically Based Pharmaco-Kinetic Model) which has been
solved via a simulation with an electric analog circuit was the one from Mapleson, intended to
simulate the uptake of inhaled narcotic gases like halothane in the human body:
Mapleson , W.W. An electrical analogue for uptake and exchange of inert gases and other
agents. J. Appl. Physiol. 18: 197 204, 1963;
Others, like: Morales, M.F. and R.E. Smith, 1944, 1945 & 1948 in: Bulletin of Mathematical
Biophysics, have not been successfully solved that time due to a lack of fast-enough
Since then the PBPKs are used to simulate as well drugs and other environmental influences
on the human body: by the same token we could designate the Haldane model as one of the
first PBPKs.
Mapleson‘s parameters have been used for operational diving by: Flook, V., R. Nishi, A.
Khan. Modelling and Validation of Treatment Tables for Severe Decompression Accidents;
in: Operational Medical Issues in Hypo-and Hyperbaric Conditions [les Questions medicales
a caractere operationel liees aux conditions hypobares ou hyperbares] ADA395680, DCIEM,
Oct. 2000:
Here we find as well super-fast compartments, i.e. # 1 & 2 in the following table:
Reference values for resting blood flow to organs of man:
L R Williams* and R W Leggett; Metabolism and Dosimetry Research Group, Health and
Safety Research Division, Oak Ridge; National Laboratory, Oak Ridge, Tennessee 37831-
6383, USA, 21 February 1989. On p. 188 we have a compilation of the relevant perfusion
values :
The perfusion rates vary not only with a factor of 250 from ca. 20 (bones) to 5000, but as well
over time course and authors. This variance should be reflected as well in the spectrum of
used half-times for a decompression algorithm. As well there are data for just 14
compartments, meaning that using a lot more, as some of dive computers do, would
probably not give any further clues. The only argument of using more being philosophically,
that „Nature does not make leaps“ (Gottfried Wilhelm Leibniz: La nature ne fait jamais de
Mixing 2 models: Egi & Gürmen
There is a nice method in this paper: Egi SM, Gürmen NM: Computation of decompression
tables using continous compartment half-lives. Undersea Hyper Med 2000; 27(3): 143 153.
The authors were considering the Workman- and as well the Bühlmann framework: but
instead of fitting each M values to the appropriate half-times within the corresponding
framework they fitted all M-values to all halftimes in a hybrid manner and such combining the
Workman and Bühlmann values. The result is a smoothed M versus halftime function with
high correlation coefficients. The plot of ln(M) versus ln(halftime) yields a straight line (Fig. 7
on p. 149):
If we exploit this function with x = 0.25 (i.e.: halftime = 1.28 min) there results a M0 = 117
fswa; with x = 0.1 (halftime = 1.1 min) yields M0 = 126 fswa.
Breath-hold and DCS Type II: Goldman et al.
Ref.: Decompressionsickness in breath-holddiving, and its probable connection to the growth
and dissolution of small arterial gasemboli; Saul Goldman, J.M.Solano-Altamirano,
Mathematical Biosciences 262 (2015): 19.
In this paper we find a super-fast compartment (brain) with the halftime of 72 sec.
(Source: l.c., p. 5)
A Fit to the Paulev data
To be completely honest with my sources, i recieved the Paulev papers from Karl Huggins,
with whom i started to discuss this topic around the turn of the millenium. Karl created his
version of a USN deco table („HUGI table“) as well he was fundamental for the ORCA EDGE
dive computer in the 80s (The ORCA EDGE being one of the first diver carried computers
not only interpolating stored table values but instead using a full-blown decompression
model). Paulev, as described in the „Rationale“, observed on himself a case of neurological
DCS during submarine escape training (ref. 1.) which has been treated successfully in a
deco chamber. Subsequently he made measurements of exhaled gases during breath-hold
diving (refs. 2. & 3.):
1. PAULEV, P. Decompression sickness following repeated breathhold dives. J. Appl.
Physiol. 20(5) : 1028-1031. 1965
2. PAULEV, POUL-ERIK, AND NOE NAERAA. Hypoxia and carbon dioxide retention
following breath-hold diving. J. Appl. Physiol. 22(3) : 436-440. 1967.
3. PAULEV, POUL-ERIK. Nitrogen tissue tensions following repeated breath-hold dives.
J. Appl. Physiol. 22(4): 714-718. 1967:
From this published curve (Fig. 1 on p. 715 in paper 3.; as well the Fig. 3 on p. 438 in the
paper 2.) we (*) extracted graphically the raw data in order to simulate the N2 uptake of
one super-fast compartment. A fit to a mono-exponential saturation function like:
Y = 1- a * EXP(- b * X) (5)
with Y = N2 Saturation, alveolar [%]
and X = dive time [seconds]
yields the following: a= 0.24
b= 0.01
with a relatively high correlation coefficient around 0.97; the mathematical details too specific
for an essay like that. But anyway there is:
Error propagation
we end at an error of approx. +/- 12 % of the fitted values due to uncertainities of the
published graphical data, which is not available in digital form.
Halftime of the super-fast compartment
Thus the halftime is, by definition, ½ = ln 2 / b = ca. 70 sec +/- 12 to 15 %.
with a stunning coincidence with Sauli’s value (chpt. 10). This one would give, in return to
the a-& b coefficients of eq. (2), a maximal inert gas partial pressure (4) in this „fast
compartmentof 8 up to ca. 20 Bar within the Bühlmann framework. One could question
the sheer size of this value derived from the model directly, but presently there are not
enough data at hand. On the other hand, there are no arguments for not keeping the
maximal tolerated overpressure from the fastest compartment as well for the super-fast
compartments. Thus we could designate the ca. 3.5 Bar overpressure from the traditional
2,5 to 5 min compartment to the faster ones.
Take-home Messages
A compartment is not a single physiological site in the body,
instead, it is a group of various tissues, sharing some common properties, like:
the perfusion rate;
this is basically the invers of the half-time used in the exponential curves.
If you use more compartments, say in your dive computer or a decompression
model, you do not get closer to the truth, instead
you just get closer to the data points at hand
For fast processes, like yo-yo diving or breath-hold profiles, the usually used half-
times are by far too slow, i.e.:
the dive computer (resp. the decompression model) acts like a „low pass“.
To simulate processes like that, you need faster and/or super-fast compartments,
namely in the sub-min region, like a halftime ½ from 30 sec to 1.5 min.
(*): SubMarineConsulting:
... Theoretical loci of very fast inertgas absorption and high perfusion rates [11] Haldane & "2:1" One of the first PBPK used to tabulate dcs-free air-dives for surface-supplied hard hat/helmet divers [12] Gradientfactors On the numerical evaluation of calculating stop-times for trimix dives and the use of a linear scaling to hide implementation weaknesses in commercial off-the-shelf desktop decompression-software. ...
Full-text available
Synopsis: some collateral aspects of DCS A collection of papers / essays / presentations and their URLs at, related to DCS (decompression sickness), PBPK (physiologically based pharmaco-kinetic models), diving and their somewhat remote, unusal or at least, unorthodox aspects.
... We extracted fast-compartment data [8] with a half-time of ca. 70 sec from [9]  [11] which are in-line with a recently developped new model [7]. ...
Full-text available
On the theoretical evaluation of one yo-yo diving profile on air for fish-farming Abstract: A yo-yo diving profile is one with very rapid and repeated depth changes. Due to the speed of depth changes in excess of 20 m/min and the quickly repeated ascents and descents within 1 to 5 min, a standard decompression model based on perfusion or a dive computer or a logging device can no longer track the changes in the inertgasload in the diver’s body properly. One form of ubiquitious yo-yo diving is done in fish-farming, clearly needed to change air-tanks, tools, debris and locations within a multiple array of the fish-nets. The already available historic sources ([1], [2], [4], [5] & [6]) address this topic clearly and the connected risks but without hints of complete mitigation. We propose simple & straightforward modifications of an existing perfusion model [12] to mitigate the risk of decompression sickness and/or arterial/ cerebral air embolism.
A report is given of a case of apparent decompression sickness after repetitive breath-hold dives to depths of 50–66 ft (15–20 m). Three similar cases in Norwegian Navy escape-training-tank instructors are also discussed. A parallel is drawn between the Scandinavian cases and the“pearl diver disease”(taravana), found in the Tuamotu Archipelago in the South Pacific. Symptoms and signs in these conditions are consistent with the diagnosis of decompression sickness. It is emphasized that in such cases immediate recompression is the treatment of choice. Consideration of various depths and patterns of breath-hold diving in terms of nitrogen uptake and elimination permits the relative risk of decompression sickness to be predicted with the help of decompression tables. skin diving; recompression; repetitive diving; nitrogen uptake; taravana; nitrogen elimination Submitted on September 25, 1964
  • J M Saul Goldman
  • Solano-Altamirano
Saul Goldman, J.M.Solano-Altamirano, Mathematical Biosciences 262 (2015): 1-9.
) we (*) extracted graphically the raw data in order to simulate the N 2 uptake of one super-fast compartment
  • Poul-Erik Paulev
PAULEV, POUL-ERIK. Nitrogen tissue tensions following repeated breath-hold dives. J. Appl. Physiol. 22(4): 714-718. 1967: From this published curve (Fig. 1 on p. 715 in paper 3.; as well the Fig. 3 on p. 438 in the paper 2.) we (*) extracted graphically the raw data in order to simulate the N 2 uptake of one super-fast compartment. A fit to a mono-exponential saturation function like: