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P(DCS) :
on the statistical nature
of decompression sickness –
a short literature review
with limited scope
P(DCS) www.SMC-de.com 1
P(DCS) –
on the statistical nature of decompression sickness;
a short literature review with limited scope.
Contents:
My model is beer than yours! ........................................................................................................... 1
DCS as a Bernoulli trial ........................................................................................................................ 3
A couple of publicaons (16) in chronological order .......................................................................... 4
The NMRI series................................................................................................................................... 6
The UHMS workshops ......................................................................................................................... 8
Tools ..................................................................................................................................................... 9
References ......................................................................................................................................... 10
Appendix: Excerpt from George B. Benedek & Felix M.H. Villars, p. 13 – 24: ................................... 11
My model is beer than yours!
P(DCS) is the statistical probability (P) of contracting a decompression sickness (DCS). This
kind of probability is just a frequency-analysis on how many certain events you may expect in
a long series of similar experiments / repetitions. DCS in divers is relatively seldom: in a
series of ca. 5,500 not-so-similar dives of one guinea-pig it happened only once [4].
Starting with the advent of decompression models, deterministic decompression models to
be precise, there was a, sometimes over-heated, debate on whether which model (which
dive table or dive computer) to prevent this decompression sickness is better than another
one.
This debate, counter-productive how it was, especially around the 90’s of the last
yearthousand, and as well the debate on how to do „deep“ stops, slowly drys out since a
couple of years as the know-how on and the results of statistically based decompression
methods is spreading.
As we promised on p.4 of [1], there would be a follow-up: the (limited) scope of this paper
here is to review some of the basic publications on the topic and un-earth a couple of these
documents from the 70’s and 80’s via direct download links [5].
P(DCS) www.SMC-de.com 2
As an extended foreword we cite one of the great grand-masters of decompression
modeling, Paul W.:
The statistical approach also has problems.
The desirable mathematical properties of the likelihood function
and the propagation of errors formulas
have been documented only for problems „better behaved“
than the decompression situation –
thus we are operating at the fringe of accepted statistical practice.
The danger of misapplication grows with smaller data sets
– and 500 is not a large number in this context. ...
For example, many potential probabilistic models would be satisfactory for estimating
p(DCS) for 40 min dives if the data was rich in 30 and 50 min dives. However,
predictions of p(DCS) for 2 hour dives would probably be much less reliable than the
propagation of errors confidence limits because of the substantial extrapolation with
the model.
Source: Paul K. Weathersby at the 37th. UHMS workshop, proceedings p. 127.
P(DCS) www.SMC-de.com 3
DCS as a Bernoulli trial
First, the serious background on Bernoulli trials from an authoritative source:
George B. Benedek & Felix M.H. Villars:
Physics with illustrative examples from Medicine and Biology: Statistical Physics,
Springer, ISBN 0-387-98754-1
On pp. 1 – 36 we find the basics on Bernoulli trials: the pure statistical considerations on
tossing a coin and recieving heads or tails. You have to read it with this is mind: contracting a
DCS after a dive is like tossing a coin when many (N) divers dive! If one diver dives N times it
will be more of an Markov process, as the outcome of the n th. dive is probably related to the
outcome of the (n-1) dive. This even more so, if the time gap between dive #n-1 and dive #n
is converging in on hours or minutes: thus we we would talk about “repetitive dives”: so the
statistical calculations will be different. Pls. find in the attachment a copy of pp. 13 – 24.
P(DCS) www.SMC-de.com 4
A couple of publicaons (16) in chronological order
1) Basically, everything started 1974 with the 280 mice in the cage of Berghage et al.:
the DCS outcome from explosive decompressions from 15 min at ca. 14 ATA were
compared to a purely binominal distribution: and it worked out as a perfect fit!
Berghage, T. E., J. M. Woolley, and L. K. Keating. 1974. The probabilistic nature of
decompression sickness. Undersea Biomed. Res. 1(2): 189 - 196.
2) The topic gained more momentum 10 years later in the seminal paper from the diving
medicine champions around the United States Navy. The methodology of
(logarithmic) maximum likelihood (max LL) was explained and applied again to rat air
dives and human Heliox dives. As well the notion of P(DCS) appeared seriously here!
The max LL fit to a 2-parameter Hill dose-response function is successfully
demonstrated. 23 further references on the topic. In the key-note of the 51st. UHMS
workshop in 2002 (p. ii), Wayne Gerth of NEDU stated that this here started a whole
new line of papers and reports from the NMRI (pls. cf. next chapter):
WEATHERSBY, P.K., L.D. HOMER, AND E.T. FLYNN. On the likelihood of
decompression sickness. J. Appl. Physiol.: Bespirat.Environ. Exercise Physiol. 57(3):
815-825, 1984.
3) Power-curves, confidence limits and the probability of rejecting a good
decompression procedure are discussed. As well the idea of sequential short trials is
evaluated against large trials:
Homer LD, Weathersby PK. Statistical aspects of the design and and testing of
decompression tables. Undersea Biomed Res 1985; 12(3): 239 - 249.
4) 800 man dives on air in a decompression-chamber with 21 cases of DCS were
analyzed and parameter fits with max LL were obtained:
Tikuisis P, Nishi RY, Weathersby PK. Use of the maximum likelihood method in the
analysis of chamber air dives. Undersea Biomed Res 1988; 15(4): 301 - 313.
5) Doppler scores from 108 man air and 622 helium chamber-dives were used as the
calibration data for a max. LL fit to a 2-compartment model:
Tikuisis P, Gault K, Carrod G. Maximum likelihood analysis of bubble incidence for
mixed gas diving. Undersea Biomed Res 1990; 17(2): 159 - 169.
6) Lin, Yu-Chong; Shida, Kathleen K (eds). Man in the Sea, Volume I, 1990, Best
Publishing Company, ISBN 0-941332-12-8; and there in Chapter 11:
Weathersby, P.K.: Confidence in Decompression Safety
https://www.divetable.eu/BOOKS/123_Chpt_11.pdf
7) The master of science thesis from David with binary logistic regression on single
staged air dives (simple box profiles):
David Southerland (1992): Logistic Regression and DCS
P(DCS) www.SMC-de.com 5
8) 2,383 Air/EAN dives with 131 cases of DCS were analyzed and a max. LL fit to a 3-
compartment model with linear-exponential (LE) kinetics was obtained:
Thalmann ED, Parker EC, Survanshi SS, Weathersby PK. Improved probabilistic
decompression model risk predictions using linear-exponential kinetics. Undersea
Hyper Med 1997; 24(4): 255 - 274.
9) The same database from #8 was used to compare the LE-model with a 3-
compartment bubble volume model (BVM(3)):
Gert WA, Vann RD. Probabilistic gas and bubble dynamics models of decompression
sickness occurrence in air and nitrogen-oxygen diving. Undersea Hyper Med 1997;
24(4): 275 - 292.
10) Hill equation dose-response models were fit, by using maximum likelihood, to 898
air-saturation, direct-ascent dives from humans, pigs, and rats:
R. S. Lillo, J. F. Himm, P. K. Weathersby, D. J. Temple, K. A. Gault, and D. M.
Dromsky. Using animal data to improve prediction of human decompression risk
following air-saturation dives. doi: 10.1152/japplphysiol.00670.2001 Journal of
Applied Physiology July 2002 vol. 93 no. 1 216-226.
11) A logistic regression model was fit to USN database without SAT dives and used to
DCS-predictions for Air dives:
Van Liew, Flynn ET. A simple probabilistic model for standard air dives that is focused
on total decompression time. Undersea Hyperb Med 2005; 32(4):199-213.
12) Introducing a 3-compartment-interconnected model which resulted in better fits to the
calibration data sets:
Saul Goldman. A new class of biophysical models for predicting the probability of
decompression sickness in scuba diving. doi: 10.1152/japplphysiol.00315.2006
Journal of Applied Physiology August 2007 vol. 103 no. 2 484-493.
13) Exakt integrals of risk functions and improved LL through model optimization for DCS
predictions:
Laurens E.Howle, Paul W.Weber, Richard D.Vann. A computationally advantageous
system for fitting probabilistic decompression models to empirical data. Computers in
Biology and Medicine 39(2009) 1117–1129.
14) Application of the probabilistic BVM(3) model to shallow- and deep-stop air diving
schedules. The statistical design of this research is outstanding!!! More than 180
man-dives have been analyzed for each of the 2 candidate profiles!
DAVID J. DOOLETTE, WAYNE A. GERTH, KEITH A. GAULT.
REDISTRIBUTION OF DECOMPRESSION STOP TIME FROM SHALLOW TO
DEEP STOPS INCREASES INCIDENCE OF DECOMPRESSION SICKNESS IN AIR
DECOMPRESSION DIVES. Navy Experimental Diving Unit, NEDU TR 11-06 July
2011
P(DCS) www.SMC-de.com 6
15) 3,322 Air/EAN dives with 190 DCS were analyzed and a fit for a trinominal
hierarchical model was obtained:
Howle LE, Weber PW, Hada EA, Vann RD, Denoble PJ (2017) The probability and
severity of decompression sickness. PLoS ONE 12(3): e0172665.
https://doi.org/10.1371/journal.pone.0172665
16) Basically, this is THE book on gas bubbles in the diver! As well, concerning our topic
here: in chapter 8 „Compartmental decompression models and DCS risk estimation“,
p. 187 – 236 a very clear and concise introduction on deterministic and probabilistic
models, fitting to data, and test of the 3CM model (from #12):
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
The NMRI series
Statistically Based Decompression Tables, a 11-volume (11 parts) series of papers / reports
from the NMRI, the Naval Medical Research Institute, Bethesda, Maryland. On ca. 700
pages state-of-art, methodologies, examples and snappy prose from the outstanding experts!
# NMRI, Part #, title:
NMRI 85-16, Part I: Analysis of Standard Air Dives: 1950 - 1970
NMRI 85-17, Part II: Equal Risk Air Diving Decompression Schedules
NMRI 86-50, Part III: Comparative Risk using U.S. Navy, British, and Canadian Standard Air
Schedules
NMRI 86-51, Part IV: Extension to Air and N2-O2 Saturation Diving
NMRI 89-34, Part V: Haldane-Vann Models for Air Diving
NMRI 91-84, Part VI: Repeat Dives on Oxygen/Nitrogen Mixes
NMRI 92-85, Part VII: Selection and Treatment of Primary Air and N2O2 Data
NMRI 92-73, Part VIII: Linear-Exponential Kinetics
NMRI 96-05, Part IX: Probabilistic Models of the role of Oxygen in Human Decompression
Sickness
NMRI 96-06, Part X: Real-Time Decompression Algorithm using a probabilistic Model
NMRC 99-01, Part XI: Manned Validation of the LE Probabilistic Model for Air and Nitrogen-
Oxygen Diving
P(DCS) www.SMC-de.com 7
Overviews:
Part I: Table 9 (p. 37) features an introduction into risc function and the risk integral. 6 risc
functions are presented and the appropriate LL fits. DCS incindences during operational use
of the USN 1957 Table, depths from 100 to 300 feet, bottom times from 10 to 50 min. From
10.391 dives there are 83 cases of DCS. The reported incidence range within the CI goes
from 0.1 up to 4.6 (eg. at 200 feet). The problem with "operational use" is that there is only a
written log of the dive. So the time & depth recordings in the logs are somewhat "creative"
(i.e. unreproducible).
Part II: Fig. 5 (p. 14) features a graph of the "Risk Surface" for a certain dive. The trough of
the 3-dimensional hyperbola shows the optimum distribution of stop times at various depths,
thus minimizing the calculated P(DCS). Equal risk decompression tables for air dives are
presented with 1 & 5 % DCS risk.
Part III: states on top of p. 1: "... if no cases (of DCS) were seen in a trial with 10 divers, the
95% confidence limits still allows an actual incidence of 31 % DCS. A single case in a 30
man trial could come from 0.1 to 17 % underlying incidence. Hundreds of replicated dives are
needed for greater precision." Ca. 2.000 standard single air dives within these frame works
are analysed.
Part IV: 8 risk functions are used to analyze the 279 saturation exposures (SAT): a SAT table
for air dives with P(DCS) = 1 % is presented.
Part V: on p. 3, Table 1, describes their decompression data sets A, B, C, D & L. These are
covering 1.835 dives with 101 cases of DCS with a range of 1.3 to 45.7 % DCS. It ends with
the conclusion: The risk model aspect of integrating a supersaturation over its elapsed time
appears to capture essential aspects of decompression outcome substantially better than the
choice of a maximum “stress” as used in the Haldane-Vann approach.
Part VI: features a good mathematical overview on the whole subject: 2006 well-documented
repeat dives are analysed with the conclusion that single & repeat dives belong to the same
population (i.e. the same model is used) but not for the 128 multilevel dives.
Part VII: appreciation of the data base for the 4,000 + analyzed dives.
Part VIII: gives a nice overview on the LE models (linear - exponential), on Table 5 (p. 48) is
a summary of the used data sets: 5 risk categories in 2.5 % intervalls, for eg. with 2.383
dives and 139 observed cases for DCS for the 0-model. The 0-model comes with a predicted
# DCS of 139 cases, but unevenly distributed along these categories. On Table 7 (p. 50) the
data sets NOT used for modelling with 1.985 dives and a DCS range from 1.0 --> 21.3 %
DCS.
Part IX: the invention of an additional O2-compartment with HT of 0.4 min to create a
substantially better fit to predict the DCS data at hand.
Part X: an algorithm for computing minimal TTS with fixed P(DCS) in real-time.
Part XI: this algorithm is tested with 730 dives & 36 (20 marginal) cases of DCS, 6 h dives
are incorporated and 158 Combat Swimmer Multi-Level Dives.
The link to ALL of the XI NMRI volumes, i.e. the parts I to XI, you will find @ the D3 archive:
https://www.divetable.eu/D3/index.htm
P(DCS) www.SMC-de.com 8
The UHMS workshops
# year title:
36 1986 Decompression in Surface-Based Diving
37 1987 Validation of Decompression Tables
51 2002 Survival Analysis and Maximum Likelihood Techniques as applied to
Physiological Modeling
Overview on the crucial chapters from these 3 workshops:
36th.: RD. Vann on „The production of decompression procedures @ Duke“, p. 70 – 75;
examples with the Hill equation, max. likelihood, risc functions and confidence levels from
real diving.
37th.: in-depth descriptions and examples from real dive data (for eg. a couple of thousand
NEDU experimental dives, etc.) along the 3 possible approaches to attack the statistical
peculiarities with DCS:
Single replicated trial
Closely related procedures
Large complex trial
51th.: intended also as sort-of basic textbook on the technology and summary of the results
achieved up to now, it features an introduction to FTA (failure time analysis), examples with
survivor- and risc-functions along applications to hypo- and hyperbaric exposures, as well to
CNS Ox-Tox.
The link to (nearly) ALL of these UHMS workshops you will find @ the D3 archive:
https://www.divetable.eu/D3/index.htm
P(DCS) www.SMC-de.com 9
Tools
In the DIVE Version 3 frameworks [2] we have implemented 7 various methods. These work
directly on the calculated compartment saturation data and the dive profile details from the
simulations you keyed in. The following listing is compiled from the software manual ([3], and
all the references therein):
Method I: Southerland, David Graham: Thesis, 1992, p. 78 & p. 9
Method II: PME Model, we have expanded it to 6 compartments for TEC diving w. Helium
Method III: Volume VI, „Statistically Based Decompression Tables“, p. 5 & p. 55; we have
simplified the risk-integral
Method IV: NEDU Report 12/2004: TR 04-41, p.8 & p. 11, no adaptions by us, it uses the
TTS
Method V: NEDU Report TR 09-03 1/2009, p. 9 & 11, as well no adaption, it is very
sensitive on the ascent rate
Method VI: „Model 4“ from Statistically Based Decompression Tables, I; p. 5, 29 & 31
Method VII: „P-NO-STOP Model“ from: NEDU TR 04-42, December 2004: Probability of
Decompression Sickness in No-Stop AIR DIVING; with the parameters on
p.14 and the LOG-IT formula from p. 8.
P(DCS) www.SMC-de.com 10
References
[1] Rosenblat, Miri & Salm, Albi. (2024). Introduction to Decompression Calculation.
available @:
https://www.researchgate.net/publication/378653647_Introduction_to_Decompression_Calcu
lation/
DIVE framework:
[2] Rosenblat, Miri & Vered, Nurit. (2021). Synopsis & Fact Sheet: update per 11/2021 for
DIVE Version 3_11. 10.13140/RG.2.2.17024.56326.
available @:
https://dx.doi.org/10.13140/RG.2.2.17024.56326
[3] DIVE Version 3_11 software manual:
https://www.divetable.info/DIVE_V3/V3e/DOXV3_0.pdf
[4] Salm, Albi & Rosenblat, Miri & Eisenstein, Yael & Vered, Nurit. (2021). DCS Type I during
a sub-saturation dive on air to 8.5 m: a case study. 10.13140/RG.2.2.16504.37127.
available @:
https://dx.doi.org/10.13140/RG.2.2.16504.37127
[5] the 215 MB ZIP-archive with most of these seasoned papers:
P(DCS).zip @ the D3 archive:
https://www.divetable.eu/D3/P(DCS).zip
