### Multi-Exponential washout models.

**How to measure risk?**

DCS risk is intrinsically probabilistic and, not deterministic, in other words, there is no mathematical critical point in which you **“can or can’t”** get DCS. The risk changes continuously and gradually with decompression stress.

Deterministic models work with compartments and base the decompression on a leading compartment in each step or section of the profile. Each compartment is mathematically independent (like Bühlmann). Therefore, these models are named **independent parallel models**.

These models will keep you safe, has been proven in countless man dives. These models are calibrated to work within a certain range and there are procedures to make it safer if you are out of the spectrum, but this is only a concern for the commercial diver or the military.

In other words, if the “dive-profile” is outside the dataset the chance of getting DCS will be higher.

What the diving operator should look for is the **optimal path**, the shortest possible decompression time within a giving probability risk of DCS (decompression sickness)/ VGE (venous gas emboli). Why include VGE, because the likelihood is different.

Likelihood measures the governance between the observed DCS incidence and the DCS probability calculated by the model. For example, the tissue Halftimes & ratios would be chosen so **predicted DCS probabilities** agreed closely with observed DCS incidences.

All probabilistic models must be calibrated to actual dives with having known DCS incidences and by maximizing the likelihood. This means you change the distribution (curve) to match your risk by choosing the values of the halftimes and the ratios.

Halftime refers to the time required to reach 50% saturation in a theoretical tissue, so the rate is different in all tissues and ratio refers to how much the tissue would hold nitrogen in solution before it get out in gas form. This is expressed as critical tissue pressure to ambient pressure ratio.

Optimal paths are useful for generating decompression procedures, they’re useful for exploring the relationship between VGE and DCS, and they’re also useful for investigating the depths for the **first stops**.

The same model may have two versions, probability and deterministic. This is the case with the Thalmann EL algorithm (current US NAVY). In its deterministic version Vval-18, which was implemented in the NAVY DC (dive computer).

- Exponential-linear (EL)
- Published by Thalmann et al. (1997)
- Nitrogen exchange: 3 parallel well-stirred tissues
- Ascent criteria: critical pressure ratios
- Bubbles form if critical ratios are exceeded

- Differences with Haldane model:
- nitrogen exchange treated as if bubbles formed when the critical ratios are exceeded
- nitrogen elimination linear rather than exponential
- nitrogen washes out more slowly
- inert gas elimination is linear rather than exponential until the bubbles resolve
- after bubble resolution, elimination becomes exponential again

- Traditional model / Exponential-exponential (EE- HBG) model
- Thalmann (1997)
- Nitrogen exchange: 3 well-stirred tissues
- Ascent criteria: critical pressure ratios
- Bubbles not assumed to form

*(The only difference between these models was that bubble formation was not assumed to occur in the EE model)*

Important note:

It's known that dive conditions (example: cold water) affect greatly the outcome, hence the conditions of the calibration dives can affect the DCS risk as much as the dives themselves.

** **The optimal path, data calibration section (presented by Dr. Richard Vann):

Models must be calibrated to actual dives having known DCS incidences by maximizing a quantity known as the likelihood. Likelihood measures the agreement between the observed DCS incidence and the DCS probability calculated by the model.

- For example, for a Haldane model, the tissue Halftimes & ratios would be chosen so predicted DCS probabilities agreed closely with observed DCS incidences.

Question:

1.- The parameter values of a decompression model are determined from observed dive data (dive profiles and DCS outcomes) so that the model will describe the actual dive outcomes as closely as possible.

True or False?

All probabilistic models must be calibrated to actual dives with having known DCS incidences and by maximizing the likelihood, and what that means is that you choose the values of the halftimes and the ratios so that they predict the DCS probabilities as agree as closely as possible with the observed DCS incidences. So, the answer to the question is true!

**How interconnected tissue model works on theory:**

**DAN sources: **

“The interconnected tissue model is nondeterministic, it’s a probabilistic model. The Saul model works with 3 tissues linked and calculates the probability of DCS based on a dive profile. The thing is, that normally any given “pure” compartment algorithm is within 2% chance of getting DCS, which could sound a lot, but this is mitigated by standard procedures based on observation, like safety stop, deep stops, super- slow- ascent or tweak the algorithm like gradient factors.

Improvements may be possible for decompression models.

According to Goldman in a 2007 paper published in JAP (Goldman, 2007), parallel tissue models extrapolate poorly to dives not in the calibration data. Series tissue models extrapolate better.

Series tissue models differ in their rates of inert gas exchange and each compartment contributes inert gas to a central DCS risk compartment.

This is Goldman’s model with interconnected tissue compartments. Inert gas diffuses between compartments and DCS risk accumulates only in the central compartment.

Interconnected compartments simulate the blood-tissue exchange of dissolved substances better than parallel tissue compartments. Intercompartmental exchange represents diffusion between adjacent tissues or arterial and venous vessels.

Now, Saul Goldman published a paper last year in JAP in which he investigated and compared another kind of model with the parallel model, and that was the interconnected compartments and this is known as pharmacokinetic, and it’s a class of models totally aside from diving and decompression, and in this, you’ll notice that there are connections between a fast central compartment and peripheral slow and medium compartments. And you’ll also notice that only the central compartment is DCS active, as opposed to all 3 over here. Now, a couple of the interesting things about it that he determined when he, these are probabilistic models, he found that the pharmacokinetic models would extrapolate beyond the calibration data much better than would parallel compartments. He also found that there was, seemed to be more sensitivity to the steeper phases of the dive, say short stops, and to slower ascent rates. So, this is a possibility for, I think that we want to look at in the future.”

### Saul Goldman Presentation

The following article is based on a conversation that I had with Prof. Saul Goldman.

### Decompression: Why it is so much more than meets the eye

Designing decompression models is tricky because it deals with biology, the behavior of the human body per se has unforeseen variables, think how challenging is working with variables like temperature, breathing rate, heartbeat frequency and level of exercise. But what if we add the immune system, how could this affect decompression? As Prof. Saul mentioned, working in decompression is like modelling epidemiology.

Not much has changed since the introduction of bubble models, although scientists have made empirical modifications to dissolved phase models due to the introduction of ultrasound technology. Researchers used ultrasound to detect venous bubbles in divers after surfacing. But none of the changes made affect directly the root of the theory. However, Prof. Saul Goldman brings a new idea to modelling with his algorithm based on interconnectedness, which he calls the modern decompression or Safe Advanced Underwater “a.L.g.o.r.i.t.h.m,” or S.A.U.L.

The reader should carefully take this article for what it is, an idea that has not been implemented yet.

Follow me up to discover the prof. Goldman’s steps to become a scuba diver and how he got the idea for his algorithm.

#### How everything started

Saul was born and raised in Montreal, Canada. Although he claims that nothing extraordinary has happened during his early years, he remembers fondly his math teacher. He taught him there is a world of ideas out there.

Prof. Saul reflects that his teacher encouraged him to appreciate the fascination of the physical world, the beauty of math and the excitement of chemistry. People think that sciences are only number, but behind all their work there is a **creative process**, not only in looking for an answer but also in formulating questions. You see, not everything is right or wrong, sometimes there is no answer at all, or we simply do not know yet. **Science is a moving frontier.**

This forthgoing thinking lead him to become a renowned **Chemistry professor at the University of Guelph in Canada**. His research interests are wide-ranging from Statistical mechanics up to biological models for preventing decompression illness.

#### How did prof. Saul become a diver

He remembers that he always was good in the water, it was part of the high school swimming team, not the best, but he felt comfortable in the water. This is important, because after he established tenure at the university, his wife told him that it was time to take a break, was on this trip to Guadalupe that his life changed, taking his first plunge into the water as a scuba diver.

He describes, in his own words this experience: “**this is a good cool thing. When I get back, I am going to take a real course and make diving part of my life.”**

For him, diving has an extra that makes it different to other recreational activities, the diver needs to learn how to manage anxiety and it feels good when you are prepared and achieved challenging situation. **“I Like that very much”** he said.

#### How did he start researching decompression

Professor Saul began his interest in decompression when he read the manual of his first dive computer, what he came across in the manual was a description of the Bühlmann decompression algorithm. For him, having a model that represents the body in compartments where each compartments intake and off gas at different rates is a little bit off, but what especially concerned him was that the compartments were independent, in other words, they each off gas on their own, they weren’t influenced by the gas content and by anything that might be around them. Why represent the body by independent compartments? There is not such a thing as an organ that does not feel the effect of surrounding tissues.

After he continue researching, he came across an article by Richard D. “Dick” Vann [1] where it was a picture of a central compartment and two others at each side, illustrating the connection and affects (gas exchange) between the central compartment and the other two. According to Prof. Saul that made sense and it was a step in the right direction. There was no math associated with it, at that moment he did not grasp it, but eventually he dug up a book from Dr. John Alfred Jacquez called Compartmental Analysis in Biology and Medicine, and as Prof. Saul put it:

“Right there, it was the math I was looking for! It was not so complicated.” Although, the model refers to what happens to organs when it is injected drug into the body. How drugs dispersed throughout the body and eventually discreet and excreted. However, math is the same, the idea is the same. It was all linear algebra, so everything was analytical solved. So, calculations are like kinetics for epidemiology.

#### Understanding the basics

The basic idea is if nitrogen can get into and out of the tissue, not just by perfusion, but also by their secondary exchange with tissues that it’s connected to on either side, because of its permeability membranes characteristics, then you will influence the concentration of nitrogen in the central compartment, in part by the existence and behavior of those tissues on the sides. So, it is not only a particular half-life rates constant, but you also consider other constants that you must put in the central compartment; how much went back and forth between the central tissue and the surroundings. This is a more reasonable representation of the body.

The real conundrum was, if it is better, how can he prove it? Remember that with algorithms you can add more parameters to fit the data and it does not mean it is better if the model is not degraded.

By that point Prof. Saul got access to NAVY data on incidence (on humans) of decompression sickness and categorized it by saturation diving profiles. Caught his attention an “S shape graphic” where the vertical axis was the probability of DCS, and the horizontal was depth. Remember that these were for saturation dive in which time is no longer a factor. The idea was bringing the diver to the surfaces after being saturated. If you compare a Haldane model; a rate and half-life constant model, and apply the same treatment, you get a quasi-linear behavior and slightly gradually increase curve, no matter how much you play with parameters. But if you used a three-interconnect-compartmental model with a reasonable set of parameters with a central compartment holding the risk and the others feeding it, you get a sigmoid curve. And that is how he proves that using interconnectedness has a proper foundation and a better representation of biological behavior.

Prof. Saul also proved with his interconnected theory that doing **safety stops** is beneficial. Why? Because if you add the safety stop to the sum of the exponential sum of gas pressure equation the shallow stop shows its benefits compared to not having added. The probabilistic graphic shows the S curve, rather than not if you used independent compartments.

#### The recreational decompression planner

The recreational planner is based on a 3D interconnected compartment that has been calibrated in several ways for low-risk diving using recreational diving profiles from DAN. The flexibility of my model is that it used separate set of sensible parameters for the type of diving need it. In this why the user may choose what level of decompression risk is willing to accept. Thus, the diving profile generated will be adjusted. This means there is a continuous probability risk during the dive.

The differences between Prof. Saul’s model and the probabilistic U.S. NAVY model are that the NAVY still used independent parallel compartments, with this kind of model each compartment off gas at different rate according to **first order kinetics**.

“You do not keep working on a structure because you understand it, you work it because it’s good. If it is not good and you understand it, you keep looking. That always has been my attitude.”

#### The future of the Dive Computer

When diving computers were in their infancy, it was impossible that any small processor to manage a probabilistic model. Today microprocessors and microcontrollers are more than enough to calculate in less than a second this kind of model, where it is constantly projecting the hit rate and calculating the level of intake and off gassing of the tissues, that is, because both algorithms run at the same time.

But this story is not finished, still the algorithm isn’t implemented in a DC and we don’t know its capabilities in the field. Only time will tell.

**The more knowledge a diver has about the physiology of the body, the better the diver can assess their own individual risks, hydrate, and plan dive times and depths to reduce the incidence of dive accidents. Planning dives carefully, understanding the decompression model, and bringing guidance of seasoned divers can help a diver safely and successfully enjoy.** AI>> conclusion

##### Expand your knowledge

Compartmental Analysis in Biology and Medicine

[1] A U.S. Navy SEAL, a respected researcher of decompression theory and an expert in hyperbaric and dive medicine, Richard D. “Dick” Vann (1941–2020) had an extensive career spanning more than 60 years. His work contributed to the implementation of safer pressure-exposure protocols in diving, mountaineering and space exploration.