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General discussion:

First of all, I believe that before someone engages in the actual programming of a decompression model, they need to do quite a bit of research into the fundamentals behind that model. Usually this means the dissolved gas (Haldanian) model as implemented by Bühlmann and/or others. Unfortunately, the relevant information is not conveniently compiled or located in one handy reference. The books by Bühlmann have been the closest thing to an all-in-one reference, but the information is incomplete, especially if you want to program the model.

Bühlmann's work has to be taken in the historical context from which it was derived. Bühlmann did not "invent" most of the concepts that he presents in his books. He took the work done by others in the field before him and refined the model (slightly). The major elements of the dissolved gas model were developed by John S. Haldane, Robert D. Workman (U.S. Navy), and Heinz R. Schreiner (American researcher). Bühlmann relied heavily on the work of Robert Workman and communicated frequently with Schreiner as a colleague in the late 60's and early 70's. Bill Hamilton was a co-worker of Schreiner's in the early years.

Workman, Schreiner, and Bühlmann are deceased now. Bill Hamilton is still very active in the field and is probably one of the best ongoing sources for information on decompression topics.The key elements of the present day dissolved gas model, however, were laid down in a few research papers many years ago. These papers contain the core fundamentals about the model and its assumptions. This is information that every decompression programmer needs to read and to know. The references are listed as follows:

1. Boycott, A.E., Damant, G.C.C., & Haldane, J.S. "The Prevention of Compressed Air Illness," Journal of Hygiene, Volume 8, (1908), pp. 342-443. [This is the classic paper by Haldane and associates which started the field of decompression science. Haldane offers many insights (far ahead of his time) and a few misguided assumptions. There is a lot of talk about decompressing goats! It is well worth the time to read this paper. Much of it is still applicable today. An old 1908 copy of the Journal of Hygiene, Volume 8, can be found in the library of most major universities, especially those involved in medical science].

2. Workman, Robert D. "Calculation of Decompression Schedules for Nitrogen-Oxygen and Helium-Oxygen Dives," Research Report 6-65, U.S. Navy Experimental Diving Unit, Washington, D.C. (26 May 1965). [This paper is available through the National Technical Information Service (NTIS) or a photocopy can be ordered from the Undersea & Hyperbaric Medical Society (UHMS)].

3. Schreiner, H.R., and Kelley, P.L. "A Pragmatic View of Decompression," Underwater Physiology: Proceedings of the Fourth Symposium on Underwater Physiology, edited byC.J. Lambertsen. Academic Press, New York, (1971) pp. 205-219. [This paper is available by finding the book in a major university library or a photocopy can be ordered from the UHMS].

The reason you should read the above papers is to understand the historical context and development of the dissolved gas model, which is necessary to understand Bühlmann's implementation of the model. Key points are as follows:

Haldane established the concept of various "tissue" compartments within the body in which the gas loading behaves according to the law of exponential decay found throughout nature. Haldane also established the concept of "ascent limiting criteria," in his case it was through supersaturation ratios.

Workman used the research data of the U.S. Navy to establish the the concept of "M-values" for the ascent limiting criteria. These are expressed as a linear relationship between tolerated supersaturation in the "tissue" compartments and ambient pressure. Workman's M-values are based on the partial pressure of the inert gas in question, not on the total pressure of the breathing gas. Workman explained the concept that fast half-time compartments tolerate a greater supersaturation than slow half-time compartments. Workman also developed a detailed calculation procedure which is the foundation of those used today. A colleague of Workman's, William R. Braithwaite, later modified Workman's procedure to include the calculation of "tolerated ambient pressure" as a means to determine a "trial first stop."

Schreiner explained the decompression model in terms of actual physiological elements such as gas transport in the blood to the tissues, solubility of gases in body fluids, fat fractions and composition of "tissue" compartments, and alveolar partial pressures of gases. He established the very important concept that the total inert gas partial pressure in a compartment is the sum of the partial pressures of all inert gases in that compartment, even if they have different half-times. Another major contribution that Schreiner made was to solve the differential equation for gas exchange when the ambient pressure changes at a constant rate. This is the general solution to the differential equation, of which the familiar instantaneous equation is only a subset. The general solution makes it possible to directly calculate the inert gas partial pressure of a compartment, as a function of time, for any linear (constant depth) or stepwise ascent or descent (at a constant rate). There are many other insights into decompression physiology given in this paper including a basis for the half-time constant, k.

As you can see from the above, many of the key elements used in the "Bühlmann Algorithm" were really developed by others and carried forward by Bühlmann. Of course, Bühlmann made a number of contributions to the science and practice of decompression calculations as well. His greatest contribution was to publish his book, in four editions from 1983 to 1995, as a nearly complete reference on making decompression calculations. Because this was the only "one-stop shopping" book widely available, it became the basis for most of the world's decompression computers and do-it-yourself programs. Key Bühlmann concepts, many explained in the 4th Edition (1995) of his book, are as follows:

The variation in half-times between two gases is inversely proportional to the square-roots of their molecular weights. This is a well-known relationship from chemistry
called Graham's Law. It is particularly applicable when gases pass through a finely-pored membrane, a process called effusion which is a subset of diffusion.

The overpressure or supersaturation tolerance in a compartment is based upon the excess volume of gas tolerated by the body in that compartment. The tolerated partial pressures between two different gases in the same compartment will vary according to their solubilities in the transport medium that delivered those gases to that compartment (blood plasma, in this case).

The two key concepts above can be used to derive complete sets of half-times and M-values for other gases such as argon and neon (although due to their higher solubilities when compared to nitrogen and helium, respectively, they offer no substantial benefit for decompression under most scenarios).

The overall M-value for a compartment with multiple gases, each gas having different M-values, will vary in according to the proportion of each gas present in the compartment.

An explanation about Bühlmann "M-values" needs to be given. First of all, they are traditional M-values just as Workman defined them. Bühlmann simply modified the linear equation to suit his application. He started out with the traditional equation for an M-value in the form y = mx + b and he solved it for x. This gives x = (y - b)/m. To get rid of m, the slope, in the denominator he just took the reciprocal and called it "Coefficient b." Traditional M-values are given as P = m(Pamb) + Mo [y = mx + b form], where P = tolerated inert gas partial pressure, m = slope, Pamb = ambient pressure, and Mo = intercept at sea level. Bühlmann expressed the same thing in absolute pressure coordinates. Bühlmann's Coefficient a is the intercept at Pamb = 0 and Bühlmann's Coefficient b is the reciprocal of the slope. It is easy to convert back and forth between Bühlmann M-values and traditional Workman-style M-values. This is something that I think a lot of people don't understand.

In the 1995 Edition of his book, Tauchmedizin or "Diving Medicine," Bühlmann gives a lot of insight into diving physiology and talks about much of his experience in the field over the years. He presents many of the results from his experimental research. In so doing, he also points out some of the shortcomings in the model. For example, he gives compartment partial pressures calculated at the end of dive series and expresses them in percent of the theoretical values. One thing is clear from his data. This is that, in every test series where incidences of DCS are shown, the affected divers are at a certain percentage less than the theoretical M-values in terms of compartment gas loading upon surfacing. This is usually in the range from 90% to 97%. The situation gets worse for repetitive dives which Bühlmann acknowledges and he cautions that reduction factors must be applied for repetitive diving calculations. One interpretation of Bühlmann's data is that his M-values do not represent a reliable line between NO SYMPTOMS and MASSIVE SYMPTOMS, but rather they represent a line between a LIMITED NUMBER OF SYMPTOMS and a MASSIVE NUMBER OF SYMPTOMS. This is consistent with most other decompression model experience which acknowledges that an M-value line is a solid line drawn through "a fuzzy gray area."

This kind of information should encourage decompression modelers who use the Bühlmann M-values to incorporate an M-value reduction mechanism that is consistent across the entire ambient pressure range. One such mechanism is by reduction of the M-value Gradient. This is simply the difference between the M-value and ambient pressure. Another good decompression reference, which I haven't mentioned previously, is Dr. Bruce Wienke. He has published several books which are available from Best Publishing Company. He is the author of the Reduced Gradient Bubble Model (RGBM). His excellent discussion about gradients is primarily focused on bubble models, but it is just as applicable to the dissolved gas model.

Well, I hope I have given you some encouragement to do some further reading into the fundamentals of decompression modeling. It is more than just programming a set of equations out of a book. There are a lot of complex considerations involved. I don't have all the answers, but collectively, we in the diving community can arrive at a lot of answers by exchanging information.

Gas loading calculations and the Schreiner Equation:

The fundamental relationship of the dissolved gas model is described by a differential equation:

dP/dt = k(Pi - P)

which states that the instantaneous rate of change of inert gas pressure (dP/dt) in a hypothetical tissue compartment is proportional to a time constant (k) multiplied by the gradient between the inspired inert gas pressure (Pi) and the present (or initial) compartment inert gas pressure (P). This kind of relationship in common in the natural sciences; Newton's Law of Cooling, for example. The main principle is that a GRADIENT is the driving force behind the rate of change in the amount of something.

In order to solve this differential equation for P, compartment inert gas pressure, as a function of time, we must use integral calculus. Before we do this, however, there are two conditions we must consider. The first is when the inspired inert gas pressure, Pi, remains constant such as during a constant depth dive profile. The second is when Pi changes with respect to time such as during ascents and descents. In order to simplify the integration in the second case, we will stipulate that the inspired inert gas pressure changes at a constant rate such as with a constant rate of ascent or descent, i.e. 10 msw/min.

Rather than writing out the integrals here, we will go right to the solutions! In the first case (constant depth), the solution is:

P = Po + (Pi - Po)(1 - e^-kt)

This is the "Haldane" equation or the "instantaneous" equation.

This same equation can also be written as:

P = Po + (Pi - Po)(1 - e^(-ln2t/half-time)) or

P = Po + (Pi - Po)(1 - e^(-0.693t/half-time)) or

P = Po + (Pi - Po)(1 - 2^(-t/half-time))

(the latter form appears in Bühlmann's Tauchmedizin book). In the above equations:

P = compartment inert gas pressure (final)
Po = initial compartment inert gas pressure
Pi = inspired compartment inert gas pressure
t = time (of exposure or interval)
k = time constant (in this case, half-time constant)
e = base of natural logarithms
ln2 = natural logarithm of 2

In the second case, ascent or descent at a constant rate, the solution is:

P = Pio + c(t - 1/k) - [Pio - Po - (c/k)]e^-kt

This is the general solution or "Schreiner" equation. It can also be written as:

P = Pio + R(t - 1/k) - [Pio - Po - (R/k)]e^-kt

In the above equations:

Pio = initial inspired (alveolar) inert gas pressure
(Pio = initial ambient pressure minus water vapor pressure)
Po = initial compartment inert gas pressure
c = rate of change in inspired gas pressure with change in ambient pressure
(this is simply rate of ascent/descent times the fraction of inert gas)
R = same as c
t = time (of exposure or interval)
k = half-time constant = ln2/half-time (same as instantaneous equation)

Note that when c (or R) = 0 in the above equation, it is reduced to the more familiar instantaneous form, P = Po + (Pi - Po)(1 - e^-kt).

Page 1 of 5

This article was contributed by Erik C. Baker, P.E. The articles presented here do not necessarily reflect the beliefs and/or opinions of SDTechDiving; they are the sole written opinion/expression of the authors. SDTechDiving is not responsible for content contained within this article, including links which may take the reader to websites outside of our control

General discussion:

First of all, I believe that before someone engages in the actual programming of a decompression model, they need to do quite a bit of research into the fundamentals behind that model. Usually this means the dissolved gas (Haldanian) model as implemented by Bühlmann and/or others. Unfortunately, the relevant information is not conveniently compiled or located in one handy reference. The books by Bühlmann have been the closest thing to an all-in-one reference, but the information is incomplete, especially if you want to program the model.

Bühlmann's work has to be taken in the historical context from which it was derived. Bühlmann did not "invent" most of the concepts that he presents in his books. He took the work done by others in the field before him and refined the model (slightly). The major elements of the dissolved gas model were developed by John S. Haldane, Robert D. Workman (U.S. Navy), and Heinz R. Schreiner (American researcher). Bühlmann relied heavily on the work of Robert Workman and communicated frequently with Schreiner as a colleague in the late 60's and early 70's. Bill Hamilton was a co-worker of Schreiner's in the early years.

Workman, Schreiner, and Bühlmann are deceased now. Bill Hamilton is still very active in the field and is probably one of the best ongoing sources for information on decompression topics.The key elements of the present day dissolved gas model, however, were laid down in a few research papers many years ago. These papers contain the core fundamentals about the model and its assumptions. This is information that every decompression programmer needs to read and to know. The references are listed as follows:

1. Boycott, A.E., Damant, G.C.C., & Haldane, J.S. "The Prevention of Compressed Air Illness," Journal of Hygiene, Volume 8, (1908), pp. 342-443. [This is the classic paper by Haldane and associates which started the field of decompression science. Haldane offers many insights (far ahead of his time) and a few misguided assumptions. There is a lot of talk about decompressing goats! It is well worth the time to read this paper. Much of it is still applicable today. An old 1908 copy of the Journal of Hygiene, Volume 8, can be found in the library of most major universities, especially those involved in medical science].

2. Workman, Robert D. "Calculation of Decompression Schedules for Nitrogen-Oxygen and Helium-Oxygen Dives," Research Report 6-65, U.S. Navy Experimental Diving Unit, Washington, D.C. (26 May 1965). [This paper is available through the National Technical Information Service (NTIS) or a photocopy can be ordered from the Undersea & Hyperbaric Medical Society (UHMS)].

3. Schreiner, H.R., and Kelley, P.L. "A Pragmatic View of Decompression," Underwater Physiology: Proceedings of the Fourth Symposium on Underwater Physiology, edited byC.J. Lambertsen. Academic Press, New York, (1971) pp. 205-219. [This paper is available by finding the book in a major university library or a photocopy can be ordered from the UHMS].

The reason you should read the above papers is to understand the historical context and development of the dissolved gas model, which is necessary to understand Bühlmann's implementation of the model. Key points are as follows:

Haldane established the concept of various "tissue" compartments within the body in which the gas loading behaves according to the law of exponential decay found throughout nature. Haldane also established the concept of "ascent limiting criteria," in his case it was through supersaturation ratios.
Workman used the research data of the U.S. Navy to establish the the concept of "M-values" for the ascent limiting criteria. These are expressed as a linear relationship between tolerated supersaturation in the "tissue" compartments and ambient pressure. Workman's M-values are based on the partial pressure of the inert gas in question, not on the total pressure of the breathing gas. Workman explained the concept that fast half-time compartments tolerate a greater supersaturation than slow half-time compartments. Workman also developed a detailed calculation procedure which is the foundation of those used today. A colleague of Workman's, William R. Braithwaite, later modified Workman's procedure to include the calculation of "tolerated ambient pressure" as a means to determine a "trial first stop."
Schreiner explained the decompression model in terms of actual physiological elements such as gas transport in the blood to the tissues, solubility of gases in body fluids, fat fractions and composition of "tissue" compartments, and alveolar partial pressures of gases. He established the very important concept that the total inert gas partial pressure in a compartment is the sum of the partial pressures of all inert gases in that compartment, even if they have different half-times. Another major contribution that Schreiner made was to solve the differential equation for gas exchange when the ambient pressure changes at a constant rate. This is the general solution to the differential equation, of which the familiar instantaneous equation is only a subset. The general solution makes it possible to directly calculate the inert gas partial pressure of a compartment, as a function of time, for any linear (constant depth) or stepwise ascent or descent (at a constant rate). There are many other insights into decompression physiology given in this paper including a basis for the half-time constant, k.

As you can see from the above, many of the key elements used in the "Bühlmann Algorithm" were really developed by others and carried forward by Bühlmann. Of course, Bühlmann made a number of contributions to the science and practice of decompression calculations as well. His greatest contribution was to publish his book, in four editions from 1983 to 1995, as a nearly complete reference on making decompression calculations. Because this was the only "one-stop shopping" book widely available, it became the basis for most of the world's decompression computers and do-it-yourself programs. Key Bühlmann concepts, many explained in the 4th Edition (1995) of his book, are as follows:

The variation in half-times between two gases is inversely proportional to the square-roots of their molecular weights. This is a well-known relationship from chemistry
called Graham's Law. It is particularly applicable when gases pass through a finely-pored membrane, a process called effusion which is a subset of diffusion.
The overpressure or supersaturation tolerance in a compartment is based upon the excess volume of gas tolerated by the body in that compartment. The tolerated partial pressures between two different gases in the same compartment will vary according to their solubilities in the transport medium that delivered those gases to that compartment (blood plasma, in this case).

The two key concepts above can be used to derive complete sets of half-times and M-values for other gases such as argon and neon (although due to their higher solubilities when compared to nitrogen and helium, respectively, they offer no substantial benefit for decompression under most scenarios).

The overall M-value for a compartment with multiple gases, each gas having different M-values, will vary in according to the proportion of each gas present in the compartment.

An explanation about Bühlmann "M-values" needs to be given. First of all, they are traditional M-values just as Workman defined them. Bühlmann simply modified the linear equation to suit his application. He started out with the traditional equation for an M-value in the form y = mx + b and he solved it for x. This gives x = (y - b)/m. To get rid of m, the slope, in the denominator he just took the reciprocal and called it "Coefficient b." Traditional M-values are given as P = m(Pamb) + Mo [y = mx + b form], where P = tolerated inert gas partial pressure, m = slope, Pamb = ambient pressure, and Mo = intercept at sea level. Bühlmann expressed the same thing in absolute pressure coordinates. Bühlmann's Coefficient a is the intercept at Pamb = 0 and Bühlmann's Coefficient b is the reciprocal of the slope. It is easy to convert back and forth between Bühlmann M-values and traditional Workman-style M-values. This is something that I think a lot of people don't understand.

In the 1995 Edition of his book, Tauchmedizin or "Diving Medicine," Bühlmann gives a lot of insight into diving physiology and talks about much of his experience in the field over the years. He presents many of the results from his experimental research. In so doing, he also points out some of the shortcomings in the model. For example, he gives compartment partial pressures calculated at the end of dive series and expresses them in percent of the theoretical values. One thing is clear from his data. This is that, in every test series where incidences of DCS are shown, the affected divers are at a certain percentage less than the theoretical M-values in terms of compartment gas loading upon surfacing. This is usually in the range from 90% to 97%. The situation gets worse for repetitive dives which Bühlmann acknowledges and he cautions that reduction factors must be applied for repetitive diving calculations. One interpretation of Bühlmann's data is that his M-values do not represent a reliable line between NO SYMPTOMS and MASSIVE SYMPTOMS, but rather they represent a line between a LIMITED NUMBER OF SYMPTOMS and a MASSIVE NUMBER OF SYMPTOMS. This is consistent with most other decompression model experience which acknowledges that an M-value line is a solid line drawn through "a fuzzy gray area."

This kind of information should encourage decompression modelers who use the Bühlmann M-values to incorporate an M-value reduction mechanism that is consistent across the entire ambient pressure range. One such mechanism is by reduction of the M-value Gradient. This is simply the difference between the M-value and ambient pressure. Another good decompression reference, which I haven't mentioned previously, is Dr. Bruce Wienke. He has published several books which are available from Best Publishing Company. He is the author of the Reduced Gradient Bubble Model (RGBM). His excellent discussion about gradients is primarily focused on bubble models, but it is just as applicable to the dissolved gas model.

Well, I hope I have given you some encouragement to do some further reading into the fundamentals of decompression modeling. It is more than just programming a set of equations out of a book. There are a lot of complex considerations involved. I don't have all the answers, but collectively, we in the diving community can arrive at a lot of answers by exchanging information.

Gas loading calculations and the Schreiner Equation:

The fundamental relationship of the dissolved gas model is described by a differential equation:

dP/dt = k(Pi - P)

which states that the instantaneous rate of change of inert gas pressure (dP/dt) in a hypothetical tissue compartment is proportional to a time constant (k) multiplied by the gradient between the inspired inert gas pressure (Pi) and the present (or initial) compartment inert gas pressure (P). This kind of relationship in common in the natural sciences; Newton's Law of Cooling, for example. The main principle is that a GRADIENT is the driving force behind the rate of change in the amount of something.

In order to solve this differential equation for P, compartment inert gas pressure, as a function of time, we must use integral calculus. Before we do this, however, there are two conditions we must consider. The first is when the inspired inert gas pressure, Pi, remains constant such as during a constant depth dive profile. The second is when Pi changes with respect to time such as during ascents and descents. In order to simplify the integration in the second case, we will stipulate that the inspired inert gas pressure changes at a constant rate such as with a constant rate of ascent or descent, i.e. 10 msw/min.

Rather than writing out the integrals here, we will go right to the solutions! In the first case (constant depth), the solution is:

P = Po + (Pi - Po)(1 - e^-kt)

This is the "Haldane" equation or the "instantaneous" equation.

This same equation can also be written as:

P = Po + (Pi - Po)(1 - e^(-ln2t/half-time)) or

P = Po + (Pi - Po)(1 - e^(-0.693t/half-time)) or

P = Po + (Pi - Po)(1 - 2^(-t/half-time))

(the latter form appears in Bühlmann's Tauchmedizin book). In the above equations:

P = compartment inert gas pressure (final)
Po = initial compartment inert gas pressure
Pi = inspired compartment inert gas pressure
t = time (of exposure or interval)
k = time constant (in this case, half-time constant)
e = base of natural logarithms
ln2 = natural logarithm of 2

In the second case, ascent or descent at a constant rate, the solution is:

P = Pio + c(t - 1/k) - [Pio - Po - (c/k)]e^-kt

This is the general solution or "Schreiner" equation. It can also be written as:

P = Pio + R(t - 1/k) - [Pio - Po - (R/k)]e^-kt

In the above equations:

Pio = initial inspired (alveolar) inert gas pressure
(Pio = initial ambient pressure minus water vapor pressure)
Po = initial compartment inert gas pressure
c = rate of change in inspired gas pressure with change in ambient pressure
(this is simply rate of ascent/descent times the fraction of inert gas)
R = same as c
t = time (of exposure or interval)
k = half-time constant = ln2/half-time (same as instantaneous equation)

Note that when c (or R) = 0 in the above equation, it is reduced to the more familiar instantaneous form, P = Po + (Pi - Po)(1 - e^-kt).