Diving with Nitrox – What You Don’t Know

I thought about calling this post “The Pros and Cons of Diving with Nitrox”, but there aren’t really a lot of “cons” other than the ones you’re already familiar with, cost and convenience:  The time and cost associated with getting certified for Nitrox diving, the extra cost of Nitrox, and the inconvenience of Nitrox being unavailable in some locations.  On the “pro” side, you’re already aware of the increased bottom times Nitrox allows, compared to air.

What You Don’t Know About Diving With Nitrox

In getting certified for Nitrox diving, you learned that the Nitrox dive tables were developed by extending air dive tables using the “Equivalent Air Depth” (EAD) rule, the intent being to develop NDL’s for Nitrox that were equivalent in decompression risk to the NDL’s for air.  In that respect, the EAD rule earns a big “FAIL”.

The EAD rule results in Nitrox tables whose NDL’s are vastly different in decompression risk from the NDL’s for air.  (More about that below)

The EAD Rule – What it Does and Doesn’t Do

It’s tempting to say “There’s nothing equivalent about the Equivalent Depth Rule”, but that would be incorrect.  In your certification classes, you learned to calculate equivalent air depths for Nitrox (in case you happen to have your air tables, but not your Nitrox tables handy).  Some of you may even still remember how to do those calculations.  What the calculations actually do is produce equivalence for the nitrogen partial pressure being breathed in from air and Nitrox at their “equivalent” depths. Or, to put it in less technical terms, you’re breathing in the same amount of nitrogen from Nitrox as you would be from air at their equivalent depths.  So, in that respect alone – the amount of nitrogen being inhaled – the depths are in fact equivalent.

The problem with the application of the EAD Rule comes from the idea that, since DCS risk stems from excess nitrogen in the blood and tissues, if the nitrogen partial pressure being breathed is the same for two gases (e.g. air and Nitrox) at two different depths, then the risk is the same for these gases at these two depths, provided the same time is spent at these depths.  Unfortunately, this idea is overly simplistic.  While the rule equalizes the nitrogen entering the system through the lungs at the equivalent depths, the amount of excess nitrogen that builds up in tissues depends additionally on all parts of the dive profile, not just the deepest portion.  As just one example, if both dives include (as they should) a safety stop at 15′ for 3 minutes, less nitrogen is entering the system from breathing Nitrox than from breathing air during the safety stop.

The great advantage of probabilistic decompression theory is that it elevates the discussion of decompression risk to a quantitative level.  By using it, “risk” becomes calculable, with actual numerical values assigned to it for specific profiles.  But, even thinking in terms of whatever model of decompression you are currently most familiar with, you can see that, by focusing solely on the pressure of gas breathed in at depth, and ignoring the full on- and off-gassing of the tissues, the EAD Rule is unlikely to actually result in equivalent DCS risks for profiles at the “equivalent” depths.

Comparing DCS Risks for Air and Nitrox at “Equivalent” Depths

“Equivalent” depths for Air and Nitrox will, of course have identical NDL’s (i.e., allowing the same bottom times at their respective actual depths).  So, for example, a 32 Nitrox dive to 75 fsw is treated the same as an air dive to 60 fsw.  They share the same maximum bottom time of 55 minutes.  What they don’t share is the same risk of DCS.  As shown in the table below, the air dive for 55 minutes will have a risk of .32 percent (or, roughly, almost a 1 in 300 chance) of DCS.  The Nitrox dive for 55 minutes will have a risk of .089 percent (less than a 1 in 1000 chance) of DCS.  On these two “equivalent” dives, the DCS risk on air is more than 3 times the DCS risk on Nitrox.

The table below illustrates the differences in DCS risk between supposedly equivalent dives on air and 32 Nitrox.   All the dives include a stop at 15′ for 3 minutes.  The probability calculations are based on SAUL.  (A link to the SAUL Dive Planner is in the header of this page.)  The Nitrox dives shown in red print (the three deepest) are included solely for the DCS risk comparison.  NEVER DIVE THOSE DEPTHS ON NITROX.  The risk there is not of DCS – which, as you can see, is minimal – but of oxygen toxicity.  Not much is known about the degree of risk for oxygen toxicity or what other factors may affect it.  But two things are clear.  First, in the recreational dive range, the risk is very much greater with Nitrox than with air, where it is almost negligible.  And, second – but more important – the result of oxygen toxicity at depth is almost invariably fatal (by way of convulsions and loss of consciousness, leading to drowning).  For this reason, I would recommend an even more cautious approach of using 32 Nitrox only under 120 fsw.


(DCS risks calculated using SAUL and all dives assume 60 fsw/min ascent and descent rates and include a stop at 15′ for 3 min.)


“equivalent” Depths(fsw)             PADI (NDL)                             Risk of DCS (%)

Air              32 Nitrox                  BT (min)                              Air              32 Nitrox

50                  63                               80                               .21                .028

60                  75                               55                               .32                .089

70                  86                               40                               .40                .124

80                  98                               30                               .41                .133

90                 110                              25                               .47                .173

100                121                             20                               .41                .118

110                133                             16                               .30                .060

120                144                             13                               .20                .016

130                156                             10                               .080             0.0 


As you can see from the above table, the “equivalent” depths for air and Nitrox are far from equivalent in terms of DCS risk.   The good news is that Nitrox is a lot safer in terms of DCS risk than current practices assume.

The Significance of DCS Probability on a Single Dive Profile

If you consider only your risk of getting decompression sickness on any single dive, you might wonder whether it really matters (within reason) what risk level you choose.  What’s wrong with choosing a dive profile where the risk of decompression sickness is, say, .005 (being .5%, or 1 chance in 200) instead one with a more conservative risk of .0005 (being .05% or 1 chance in 2000)?  In either case, while there is some non-zero risk of decompression sickness, the most likely outcome of that dive is you won’t get bent. So, no harm, no foul? Not really, because you’re only looking at the outcome of a single dive.  Since, if you’re reading this, we can presume you intend to continue diving, your dive experience is likely to expand over time to hundreds or thousands of dives.   So it’s important to look at where seemingly small differences in risk level can lead you.

The  graph below illustrates how the probability of getting at least one hit varies both with the risk for a single underlying profile “p(dcs)”, and with the total number of dives in a series of dives undertaken on that profile. The dives here are all on air to a depth of 80 fsw with a 3 min safety stop at 15 fsw. The “risk”, or p(dcs) values  for each of these three profiles were determined (using the dive planner) to be: 0.00896 (.896%), 0.00414 (.414%), and 0.000845 (.0845%) for the dives with 40, 30 and 20 min bottom times respectively.

As with the other calculations, the probability of getting “m” hits in “n” identical dives is obtained from the general expression (from basic probability theory):

In the graph below, the number of dives executed for each profile “n”, ranged between 1 and 10,000. Applying this expression, we find that the probability of getting zero hits ( m=0) over a series of  “n” identical  dives, whose individual risk is “p(dcs)”, is given by:

so that the probability of getting at least one hit “P{}” is obtained from :

As you can see from the graph, by the time you get to even 100 identical dives at a single-dive risk of .00896 (roughly.9% or still less than 1 in 100) your chances of being bent at least once has passed the 10% point (reached after only 10 dives) and is starting to approach 100% which is essentially reached before the 1000th. dive.  Similarly, a single-dive risk of .00414 (just over .4% or still less than 1 in 200), while it doesn’t rise quite as quickly with increased numbers of dives, still reaches essentially 100% by the 1000th. dive.  On the other hand, a single-dive risk of .000845 (just over .08% or less than 1 in 1000), while it still rises with number of dives, remains under 1% risk after 10 dives and, after 100 dives, is still less than 10%.  By 1000 dives, the risk here too has risen significantly, but is still well below 100%.


The Saul Dive Planner : Questions Answered

A comment by Craig on the Saul dive planner raised issues that may be of general interest, and merit a somewhat detailed answer, so here it is.

Thanks for posting the planner, interesting to try out. I could not get 130 feet for 32% to do any calculations.

With 32% O2, out of an abundance of caution (in order to avoid Oxygen toxicity), the depth is often limited to less than 120 feet, which is what I have done here.

 Many algorithms have a slower ascent rate of 30 feet/min, or that rate when at 60 ft or less.

During free ascents, i.e. in the absence of an ascent/descent or anchor line, it is actually pretty difficult to maintain an ascent rate of 30 feet/min (or less). 60 feet/min is more common and easier to do, so the ascent rate calculations in the dive planner are for 60 feet/min.  Planning for an ascent rate of 30 feet/min when you’re unlikely to achieve that slow an ascent would be underestimating your actual level of risk.  A dive computer using Saul would, of course, be able to take actual ascent rates into account.

It is difficult for me to interpret the probability of DCS. I ran all the DSAT NDLs between 60-120 feet for 32%. All probabilities ran between 0.27 and 0.46% 

The result you just found, that the DSAT NDLs are not “iso-risk” (i.e. they don’t all entail the same degree of risk), is something that many people have wondered about, but – in the absence of a probabilistic model – didn’t have a handle on. This is one of the many advantages that a probabilistic model provides.  But I presume that your immediate concern is with what those differences in probabilities mean in real life.

Firstly, the probabilities shown in the planner are stated as percentages, so that a 0.27 risk of decompression sickness means a .27 percent risk (or 2.7 chances in 1000, 27 chances in 10,000, or just a bit above 1 chance in 400).   Similarly, a .46 percent risk means 4.6 chances in 1000, 46 chances in 10,000, or  about 1 chance in 218.

With respect to interpreting the probabilities, it may be helpful to appreciate more fully what these numbers mean or imply, by seeing what they predict for large numbers of dives.  We’ve already looked at that a bit in a previous blog post (Probability in Relation to “The Bends”) where we showed that, for a probability of 1 in 400, over a series of 400 dives, the probability of getting no instance of decompression sickness in any of the dives was .367 while the probability of having one or multiple instances of sickness during the series was .633.  (So – very roughly – the probability of being hit at least once in 400 dives at that level of risk was almost double the probability of completing 400 dives unscathed.)   We looked at a series of 1000 dives at that same risk level, and found the probability of being hit at least once in the series works out to .918, while the probability of completing 1000 dives unscathed has decreased to a measly .082 (or less than 1 in 10).

How do the two P(DCS) extremities of 0.27 % and 0.46 %, obtained for the DSAT NDLs on 32 % O2 (above) compare?   If we do similar calculations on those probabilities, we find that – skipping right to a series of 1000 dives –

For a risk level of .27%,  the probabilities of no hits and at least one hit in a 1000 dive series are, respectively, .067 and .933.  (The chances of completing the series of 1000 dives unscathed is not looking too good – just under 7%.)

For a risk level of .46%, the probabilities of no hits and at least one hit in a 1000 dive series are, respectively, .010 and .990.  (The chances of completing the series of 1000 dives unscathed is highly unlikely – 1% .)

So the probability of getting at least one hit over the series of 1000 dives rises from about 93 % to about 99 %, depending on whether one dives the profile corresponding to the low or the high extremity for the range of  DSAT NDL’s for single-dive profiles.  Even the seemingly negligible difference between the 1 in 400 (.25%) level of risk in the initial example and the .27% risk level, when carried over 1000 dives, moves the probability of getting hit at least once from less than 92% to over 93%.

These and related calculations are described more fully at the end of Chapter 8 of the book I recently co-authored:

Saul Goldman, J.Manuel Solano-Altamirano, and Kenneth M. LeDez: “Gas Bubble Dynamics in the Human Body”. Elsevier/Academic Press (2017).

Hope this helps.


The SAUL Dive Planner is Here

The SAUL dive planner is the first probabilistic dive planner available.  Deciding how to structure it took some time but, in the end, after considering more complex set-ups, I decided to keep the dive planner relatively simple.  There is a choice of breathing gas – air or 32 nitrox.   The dive profiles available are all square profile with specified descent and ascent rates (60 fsw/min) and include a mandatory 3 minute safety stop at 15 fsw.  There is a further choice of doing a “forward” or a “reverse” calculation.

In the “forward” calculation, you will input the maximum depth, and the maximum time you want to spend there.  The planner will output the probability of incurring decompression sickness for that dive (unlike currently available planners, which would either permit or not permit that dive).   In the dive planner’s forward calculation, if you should, in some instance, see the probability of decompression sickness shown as 0%, remember that this does not, of course, represent an absolute zero probability but 0% when the calculated probability is rounded to a finite number of figures.

In the “reverse” calculation, you will input your maximum acceptable P(DCS) along with the maximum depth, and the planner will provide you with the maximum bottom time to keep within your chosen P(DCS).  (So, the “reverse” calculation, after you choose your maximum acceptable P(DCS),  will give results in a style similar to currently available planners.)

Just a few reminders before we start:  There is always some possibility (even when very minimal) of decompression sickness when you dive.  You might want to reread the previous blog post on probability.   Because all the dive profiles included are square profiles with a stop, this is what they would look like in diagram form.



You may already know that, by definition, bottom time begins when you enter the water and ends when you begin your ascent.   Please realize as well that, with SAUL, the safety stop is an essential part of the calculations, as are the descent and ascent rates.  Any changes to these could affect the probabilities given.   In addition, these probabilities are for single dives only. Repetitive dives are not included in this version of the planner.

As you should be aware, a dive planner is only that.  It is not an adequate substitute for a dive computer.  This is true in general for all algorithms.  Because of SAUL’s greater accuracy and complexity, it is particularly true for SAUL.


To use the SAUL Dive Planner: http://moderndecompression.com/?page_id=493

Probability in Relation to “The Bends” – Changing the Way We Think About Dive Safety

One of the features I’ve been emphasizing about the SAUL algorithm (besides its greater accuracy), is that it allows the diver to choose a level of risk that he/she feels is appropriate.   Since this is quite different from the way we initially learned to think about dive safety, now is a good time to review the differences and to go over some basic ideas and facts about probability.

When you were first learning to dive, you would have been told that, in diving, there is always some small probability of decompression sickness (DCS), but that you can minimize your risk by diving safely – all of which is true.   But, once you were introduced to dive tables and/or dive computers, there was no further discussion of probability in relation to DCS.  Instead, you went on to look at dive tables to work out no-decompression limits (NDL’s), and then proceeded to dive computers which, in effect, automated the working out of  NDL’s.

The problem with NDL’s  (and with current dive computers) is that they are deterministic.   That is, they have an absolute dividing line: below it you’re okay, above it you’re in trouble.  As with speed limits on highways, there may be a tendency to continually push the limit – until you get caught or “bent”.  (Although, to be fair, PADI dive tables, do have a disclaimer of sorts on them advising that you shouldn’t actually plan to dive to the limits, that the limits are something of an absolute maximum.  But, then, you’re left wondering – “How close to the limits is too close?”)

While some current dive computers offer riskier or less risky options, how can anyone make a sensible choice without a clear idea of how much risk each option involves?

So let’s look at the probability of DCS in diving.  In the higher ranges, if you were to dive each of the listed profiles on the PADI recreational diving tables, right at their no-decompression limits (on air, including a 3 minute safety stop at 15 fsw), your risk of DCS would average out to .00264, which is 264 chances in 100 thousand or, roughly, just over a 1 in 400 chance of getting bent.  (I averaged over all the listed profiles because the risks at the limits of the different profiles are not equal.)  To see the comparative risks of PADI’s NDL’s for different profiles, check out my post on How Saul Relates to the PADI Dive Tables.

On the other hand, most recreational diving typically involves much lower risks.  Project Dive Exploration, which solicits and collects actual  dive data from large numbers of divers found (in one subset of their data, that roughly corresponds to typical recreational diving) a DCS rate in the neighbourhood of .0007, which is about 70 chances in 100 thousand.  Or, to put it another way, only about 1 chance in 1,400 of getting bent.

What factors should you consider when choosing an acceptable level of risk?  Obviously, less risk is preferable to greater risk.  No one wants to get bent.  (On the other hand, the least risk of all is to stay out of the water.  Not a serious option.)   But less risk generally does mean less bottom time.  So what do you do?

Regardless of the level of risk you choose, there is some possibility (even if very small) of getting DCS on any single dive.   As you increase the number of dives, you’re more likely to experience the consequences of a higher level of risk.  Using the (not recommended) 1 in 400 level of risk mentioned above, if you were to dive 400 times at that level of risk you’re quite likely to get bent.

It’s probably unnecessary to remind you (but I’ll do it anyhow) that a 1 in 400  risk doesn’t mean exactly 1 incident of DCS in every 400 dives, but that, over a large enough number of dives, or a large enough number of divers, the number of cases of DCS will average out to 1 in 400.   Meaning that, while you might manage those 400 dives without getting bent at all, you’re about twice as likely to get bent one or more times.  And, if you were to continue at that same level of risk through 1000 dives….  Well, just don’t. (To see the probability calculations, click here. Continue reading

On Planning a SAUL Dive Planner

Okay, I’m finally ready to start working on the long-delayed SAUL dive planner.

As I mentioned briefly in an earlier post, I had been working on a book with two other authors – one of them a math/physics/chemistry type like myself, and the other an anaesthesiologist/hyperbaric medicine physician. Perhaps not surprisingly, writing the book took much more of my time, and lasted longer than I had anticipated.  The book, “Gas Bubble Dynamics in the Human Body”, by Saul Goldman, Juan Manuel Solano-Altamirano, and Kenneth M. LeDez is published by Elsevier/Academic Press, and you can find it by clicking on the book title, or by entering ISBN 9780128105191 on your web browser. It’s aimed primarily at researchers and graduate students in the sciences, and medical doctors (anesthesiologists, hyperbaric medicine and emergency medicine physicians).  But there is a long chapter on decompression models which details the scientific underpinnings of SAUL that will be of interest to SCUBA divers, and another chapter on arterial gas emboli (AGEs) in breath-hold diving that will be of particular interest to breath-hold divers.


On Planning a SAUL Dive Planner:

Complete details have not been finalized yet, but some initial decisions were made.  Two things it will not be – It won’t be in the form of dive tables, and it won’t be downloadable for use.  Two things it will be – It will be useable online.  And it will be, at least initially, free.

Figuring out the best way to set up a SAUL dive planner is more complicated than for currently used deterministic algorithms, because it’s probability-based.  This means that there are many more questions you could ask.

For example, let’s consider a simple square-profile, such as is done on a wreck dive. Let’s assume the wreck is level, and you know its depth, so the question is about how much time you may have to explore it. Other algorithms will give you a definite maximum time (the “NDL”) for that depth. With SAUL, however, there are (at least) two ways to approach the question. You could first decide on a personal acceptable level of DCS risk (e.g. P(DCS) < 0.001 ), and from that point, using it like any other dive planner, ask how long you can stay at that depth without exceeding that risk. But another approach would be to think about a range of possible bottom times spent at the depth of the wreck, and ask what your risk of DCS would be if you were to stay for different lengths of time at the wreck’s depth.

True, that’s only two different ways – rather than “many” – of asking the question in that simple example. But dives being planned are often not that simple.  When you consider repetitive and/or multilevel dives, forward and reverse dive profiles, multi-day diving, and breathing gas choices (air/nitrox), the number of ways you can approach the dive planning process, as well as the numbers of questions you can ask, increases considerably.

Ideally, it should be possible to use the SAUL dive planner for any type of question or approach in dive planning.    I will try to make its overall use as simple as I can, but, for this to happen, it would help if I had some input from potential users as to how they might want to use it. I’m not going to ask you to fill out a survey, but if any of you would like to comment on how likely you might be to use the SAUL dive planner in specific different diving situations, I’d appreciate it. As a starting point I’m listing a number of obvious scenarios, but I welcome your suggested additions and elaborations.

  1. a) a single recreational low-risk profile – either square or multi-level – with a safety stop, on air or nitrox.
  2. b) repetitive recreational low-risk square profiles with a safety stop, on air or nitrox.
  3. c) forward and reverse multilevel dive profiles with a safety stop, on air or nitrox.
  4. d) multi-day diving based on the above profiles, on air or nitrox.
  5. e) a single high-risk decompression dive on air – at most one dive per the day.


While I’m working on getting the SAUL dive planner ready, I will also try to start posting again on a somewhat regular basis. If you’re familiar with this blog, you’ll know that, for me, a regular basis is still not very frequent, but the most recent very long gap has been largely because of the book.

More About The SAUL Decompression Algorithm

As some of you may be aware, the Inter-connected Compartment Model (ICM) described in my published journal article now goes by the name of “SAUL”.  This is not entirely, or even primarily, an ego trip.  Rather, it’s an attempt to avoid the unnecessary use of jargon – particularly when it results in unpronounceable acronyms.  While ICM is an appropriate and relevant way to describe the model’s properties in a scientific context,   in the context of diving,   it becomes less meaningful, less directly relevant and, of course, less easily remembered.

“SAUL” – for Safe Advanced Underwater algorithm – is a more relevant description from a diver’s perspective.

As I’ve mentioned before in this blog, “SAUL” is completely different from current algorithms, being probability-based rather than based on the Haldane model.  This may raise two different questions in the minds of divers: 1. Is “SAUL” tested to the same extent as other models?  and  2. Would I have to learn a whole new way of interacting with my dive computer with “SAUL”?

Is “SAUL” tested to the same extent as other models?                       

The short answer to this question is “yes”.  The most extensive body of research on dive profiles and decompression sickness was done and compiled by the U.S Navy, together with the (British) Royal Navy and the Canadian Navy (currently “DRDC”).  All diving algorithms have (in a rough sense) been “tested”, either directly or indirectly, against the Navy data. This is because they are usually tested against the PADI recreational dive planner data, which is a more conservative form of the US Navy tables. While new dive computers do undergo testing, this is not a test of the accuracy of the algorithm, but a check primarily of being not too inconsistent with other popular algorithms in current use.  I have access to the Navy data and believe that “SAUL”, for both decompression and no-decompression diving on air or nitrox, for both single and repetitive dives,  provides a better fit to the Navy data than does any other algorithm. This is why SAUL, but not any other algorithm, can account for the beneficial effect of safety stops and slow ascents. This, in turn, explains why SAUL accounts for DAN’s Project Dive Exploration (PDE) data for air and nitrox, far better than any other algorithm.

Would I have to learn a whole new way of interacting with my dive computer with “SAUL”?

The short answer here is “no”.  Even though “SAUL” is probability-based, during a dive, you would still have the familiar “time remaining” to go by and your ascent would include a 3-minute safety stop at 15 feet (for recreational dives).  What would be different would be how that function is calculated, and what additional information you would have.   Before the dive, you would select the maximum probability of decompression sickness that you are prepared to accept.  (Of course, we would all prefer a zero probability of DCS  –  except, unfortunately,  a) zero probability of DCS is not really feasible in diving, and b) the lower the probability of DCS, the more limited your dive will be in length and/or depth.)   The “time remaining” that you see during the dive would be calculated to correspond to your chosen probability of DCS, together with the dive profile you have done so far.  In addition to “time remaining”, you would also see two other functions: the probability of DCS if you were to begin your ascent right then, including a safety stop , and the probability of DCS if you were to begin your ascent right then but omit the safety stop.  These three functions would be continually recalculated during the course of the dive.

The above description is what you would see during recreational diving.  (Before the dive you would have selected either recreational diving or decompression diving.)  During decompression diving, what you would see would be very similar, except that, instead of only one safety stop, you would have one or more decompression stops.  You may be wondering why recreational diving and decompression diving are calculated separately instead of being merged as one.  It is because the model parameters (i.e. the constants in the equations of the model) for the two are different, and can’t be switched on the fly.  In any event, if you’re on a recreational dive but have seriously overstayed your maximum time remaining (a seriously bad idea), there’s no point in directing you to do a proper decompression stop if you don’t have enough air left to do it.  If you do overstay your maximum remaining time, the “time remaining” function will, of course, tell you to ascend immediately, but, unlike with other computers, it will not shut you out.  The functions indicating your probability of DCS on an immediate ascent, with and without a safety stop, will continue to provide you that information.

Dive Planner

I do intend to put a dive planner online, but can’t say exactly when that will happen.   It may take some time because I have a number of other professional commitments that I am working on at this time.

SAUL and Decompression Dives

How does Saul handle decompression dives or dives that overstay the allowable no-decompression limit (for the chosen level of probability of DCS) ?  I haven’t previously been discussing decompression because I’ve been focussing on recreational diving.  Recreational diving and decompression diving are handled somewhat separately by SAUL, for reasons outlined below.  When SAUL goes into dive computers, it will handle both recreational and decompression diving  (unless a manufacturer wishes to limit it to recreational).  The reason they are handled separately, is that the algorithm was calibrated separately for each type.

Before the dive, along with choosing the probability level for DCS, the diver will select “recreational” or “decompression”.   Neither of these choices can be changed during the dive.


SAUL’s recreational algorithm does not specifically allow for decompression diving.  On the other hand, unlike other recreational dive computers, SAUL will not lock you out even if you exceed your chosen probability level.  It will continue to update and show you your probability of DCS.  You will see some improvement in  probability with a longer safety stop.   At all stops, whether safety or decompression, the greatest benefit is achieved in the earliest part of the stop.  While benefit continues to accrue throughout the duration of the stop, it does so at an increasingly slower pace.  Obviously, the real problem with trying to significantly decompress on dives planned as  recreational is that you wouldn’t have sufficient air remaining to manage it.


Decompression dives work much the same way as recreational dives except for the stop.  Thus, before starting the decompression dive, the diver would input their maximum acceptable risk, and would input the fact that the dive is to be a decompression dive. (The latter tells the computer to use a decompression diving-based calibration for the underlying model). Then during the dive, the dive computer, every few seconds, tells the diver how much time is left at the depth the diver is at, so as to ultimately ascend with a total risk (ascent risk + risk at the surface) that doesn’t exceed the allowed risk (again, exactly as in no-decompression diving). When its time to come up, the dive computer tells the diver how much time to spend at specified stop depths, so as to surface within the acceptable inputted risk.

Generally, the stop depths for decompression diving start at a deeper 1st stop (e.g. 25 or 30 fsw, rather than 15 fsw), relative to no-decompression diving.  Obviously, less time is needed when on nitrox than when on air, for the same level of risk. Also, of course, the total time spent at all the decompression stops will, in decompression diving,  be considerably greater than the 2-5 minutes typical for no-decompression diving.


SAUL has been calibrated for use in air decompression diving, assuming a maximum of 1 dive per 24 hour period. It will also become available, hopefully in the not too distant future, for use in trimix decompression diving using rebreathers.


A recent comment on my The Doctor is In Part IV blog post, followed by a more detailed letter to me raised some issues that need to be addressed.  The comment read as follows:

SAUL sounds like a giant leap forward in dive safety and one that will always be attributed to the brain work of Dr Saul Goldman. As far as I understand however a route has been chosen to make the invention a proprietary solution, which is an unbelievable shame. Have you considered to make this algorithm available to the public domain so that *every* dive computer manufacturer can implement it, including those that do not want to be involved in the proprietary IP madness of this world. What if Einstein had taken a patent on all his inventions? What’s more important: A bank account or saving lives?

The writer refers to my making SAUL “a proprietary solution”.  This is not exactly the case.  SAUL – or, more accurately, the model upon which it is based – is patented.  What’s the difference?  A proprietary solution is generally kept secret, because, not being patented, secrecy is often the only way to protect it from being “stolen” by others.  A patent provides its own legal protection, is not secret, and, in fact, is published online by the U.S. Patent Office.  In addition to that, my article in Journal of Physiology (which can be downloaded from the Articles section) contains details of my model.

The two main objections the writer seems to have to what he calls a “proprietary solution” are: first, that he would like every dive computer manufacturer to be able to implement it, and, secondly, some apparent distaste for my wanting to make money from it.

I have no problem with the first objection.  In fact, it is my goal and my expectation that  every dive computer manufacturer will eventually implement it.  I have no intention of selling exclusive rights to a single manufacturer, nor have I ever considered doing so.  SAUL was, and is, my way of giving something back to the diving community for all the enjoyment I have had, and continue to have from diving.

So, why do I not overcome the second objection by putting it in the public domain (or, as one computer manufacture I approached put it, “make it open source, like Buhlman”.)?   Perhaps I would – if computer manufacturers offered their products for free, mechanics repaired cars for free, airlines gave free flights, etc. – but I don’t really see that happening.

I spent years of my life, and a fair amount of my own money, developing SAUL and, while I’m not looking to get rich from it, there’s no way I will allow computer manufacturers to profit from it without my profiting as well.

There’s also a secondary reason I won’t make it open source:  The model underlying SAUL is different from, and more complex mathematically than, Haldane-based models (which includes all the others out there, including so-called “bubble-based” ones).  I have some concerns about the potential for people with insufficient mathematical and scientific skills trying to “adapt” or change the algorithm.

In his longer letter to me, the writer mentions some objections he has heard from dive computer manufacturers he queried: specifically, that they were either “unaware of” SAUL or that it was “untested”.  I can’t speak to whether any of them are actually unaware of SAUL (although I have my doubts).   SAUL is also no less “tested” – in a formal sense – than are any of the algorithms in use today.  While dive computer manufacturers do  test their computers before their release, they do not test the accuracy of the algorithm.  What they test is, essentially, the functioning of the hardware and the software environment – in other words, that the computer is actually doing what the algorithm tells it to do.

As far as being tested, the largest body of test data in existence was amassed in an extensive series of experiments conducted by the U.S. Navy, in conjunction with the Canadian Navy and the (British) Royal Navy.  Real-life testing to that same degree can’t  be done by anyone nowadays –  not even by the Navy  (some of it would never pass an ethics committee).  These provide the data that all algorithms use, either directly or indirectly, as they calibrate their equations to fit, as best they can, the known facts.  (In the case of SAUL, I had access to that entire data set and used it extensively.)   To the best of my knowledge and belief, SAUL’s predictions fit the Navy data better than any other algorithm.  I have also been given access to a large portion of the recreational dive data collected by DAN in their Project Dive Exploration.  SAUL predicted the actual incidence of decompression sickness in those dives very accurately, much more so than typical Haldane based algorithms whose predictions were too high by a factor of 10 or more.  SAUL is also the only model to account for the beneficial effect of safety stops and slow ascents.

It’s also worth mentioning, that I have given about 20 invited talks about on my model and related subjects to audiences composed variously of physicians, scientists, and divers.  I have spoken at meetings of the Undersea Hyperbaric Medicine Society (UHMS), American Academy of Underwater Sciences (AAUS), Canadian Association of Underwater Sciences (CAUS), South Pacific Underwater Medicine Society (SPUMS), and the International Congress on Hyperbaric Medicine (ICHM).   Of all the questions I was asked following these talks, the single most common one was “When will this be available in a dive computer?”  So far, I haven’t been able to provide a definitive answer to that question, but I’m still trying.

Decompression Sickness in Breath-Hold Diving

Decompression sickness is not just a worry for scuba divers – some “free” or “breath-hold” divers also suffer from it occasionally.  This could come as a surprise to you, if you had previously taken for granted that you have to breathe pressurized gas containing nitrogen to get “bent”.

Many of you are aware, at least vaguely, of divers in the South Pacific who free-dive to harvest pearls commercially.  You may also be aware of the competitive sport called breath-hold diving. Possibly, you even participate, or know people who do.   Interestingly, both these types of diving, very different from each other as well as from scuba diving, can sometimes lead to decompression sickness.

First, how are they different?  The most obvious difference between both types of free-diving and scuba is: no scuba, i.e., no underwater breathing gas. This leads to other differences.  With less internal gas pressure in the lungs, the lung volume decreases under the pressure of increased depth. With immersion, and increasing depth, a number of other physiological changes occur, collectively known as the “diving reflex” that makes breath-hold diving feasible. These include a slowing down of the heart rate and decreased peripheral circulation, which allows near normal circulation and perfusion to the heart and brain to be maintained.

The differences between the pearl divers and competitive breath-hold divers, while in one sense quite pronounced, are essentially differences in the nature of the dive profiles.  The pearl divers would dive to a range of depths, sometimes to over 100 fsw, with a bottom time of 30 to 60 seconds for a total underwater time of about one and a half minutes per dive. With approximately one minute of surface interval between dives, they would then repeat the process for perhaps 6 hours daily.   By contrast, competitive breath hold divers generally do a single very deep dive, the actual depth and bottom time achieved being specific to the particular competition, but, with the aid of specialized sleds and buoyancy devices, depths of 150 – 250 meters can be – and are – achieved.

It has long been known that the pearl divers of the Tuamato archipelago (near Tahiti) do suffer some ill effects from their diving practices, a sickness known locally as Taravana which translates – very roughly – as “falling crazily”.  More recently, cases of probable inner ear decompression sickness and probable cerebral decompression sickness have been diagnosed. The number of cases of Taravana or DCS decreased dramatically when the surface intervals were increased from 1 minute to 15 minutes.

In the case of competitive “breath-hold” divers, it used to be thought that decompression sickness could not occur, the idea being that the small amount of nitrogen present in a single free dive was not enough to lead to DCS.  Two reported cases, one of cerebral DCS, one with both cerebral and inner-ear DCS symptoms, have shown that DCS can and does occur, even in single free dives. There also have been relatively frequent descriptions by breath-hold divers, after certain types of dives, of symptoms consistent with inner-ear DCS.

From my research group’s 2014 paper, “The lifetimes of small arterial gas emboli, and their possible connection to Inner Ear Decompression Sickness”, which was based on arterial gas emboli in scuba diving, we were able to extend the calculations to both types of breath-hold diving. In the resulting paper, “Decompression Sickness in Breath-hold diving, and its probable connection to the growth and dissolution of small arterial gas emboli”, Mathematical Biosciences, 2015,  the calculated predictions, were consistent with the observed occurrence of DCS in both types of breath-hold diving.

(Copies of both the 2014 and 2015 papers can be read in the Articles section of this blog.)