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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.

Rationale

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

sea-harvesters.

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:

https://www.divetable.info/theory.htm. 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

derived.

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

exceeded.

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

respectively.

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:

(2)

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

training):

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

hardware.

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

sauts).

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): 1–9.

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

compartment“ of 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: www.SMC-de.com