By Kaitlin Mogentale

*Originally published at ScientificAmerican.com*

If you asked me what my greatest fear is while scuba diving, I wouldn’t hesitate with my answer– the bends. The bends, or decompression sickness (DCS), is a decompression illness arising from the dangers of breathing compressed air at depth for prolonged periods, coupled with improper decompression or excessively rapid ascents.

The most severe of symptoms are permanent CNS damage, and occasionally death. The dangers of improperly off-gassing after a dive have been (rightfully) drilled into my head during the prerequisite ENST-298 Introduction to Scientific Diving class.

In class we studied Haldane’s 1908 report on compressed-air illness to better understand decompression sickness. From Haldane’s research came an initial 2:1 rule. The rule states that so long as a diver remains within the ratio of 2:1 for the pressure at depth versus ambient pressure on ascent, he is safe to ascend without developing a case of DCS–although it is important to note that due to complex human physiology, there is no definitive point between avoiding and getting DCS.

Later extensive research at the U.S. Navy Experimental Diving Unit (NEDU) by Dwyer, Hawkins, Workman, Yarborough, and others from the 1930s-1960s arrived at more precise ratios using empirical data, focusing on the pressure of the inert gas in air directly involved in DCS, nitrogen. Workman’s model introduced M-Values (or greatest partial pressure of nitrogen a “tissue” compartment can tolerate without the onset of DCS at a given absolute pressure) for each of 6 designated hypothetical tissue compartments with 5, 10, 20, 30, 40, 80, and 120-minute nitrogen half-times. These 6 compartment half-times were calculated based on the capacity of the “tissue” to store nitrogen gas, and the effectiveness of nitrogen transport to and from the tissue, such that:

Half-time= (1/c) x S/C, (1)

where c is a constant of proportionality, C is equivalent to gas transport effectiveness, and S is the solubility coefficient for the gas in a tissue. A fast tissue with a short half-time corresponds to a well-perfused tissue, and perhaps lower fat content, as nitrogen is less soluble in aqueous tissues than in fatty tissues.

Haldane calculated the 5, 10, 20, 40, and 75 half-times used in his model based on their representation of what he theorized happens in the body. Haldane hypothesized men became fully saturated with nitrogen in 5 hours and that additional nitrogen loading after 4 half-times “would scarcely be appreciable”. The final 75 minute “tissue” compartment therefore corresponds to the fact that 4 x 75 minutes = 300 minutes, or 5 hours. The Navy Tables use Haldane’s calculations as the basis for their theoretical compartments, changing the 75-minute compartment to 80 minutes and adding a 120-minute compartment. The Navy assumed 6 half-times for saturation. The changes reflected problems associated with earlier US Navy tables that had used Haldane’s original five compartments in their calculation.

Workman’s M-values are based on a compartment’s nitrogen half-time and the pressure of the breathing gas, which is always dependent on depth. M-values are mathematically represented by the equation

M= Mo + ∆Md, (2)

where M is the nitrogen partial pressure limit for each compartment, Mo is the partial pressure of nitrogen tolerated at sea level, defined for each compartment, and ∆Md is the increase of M with increasing depth, defined for each compartment, multiplied by the depth (in feet of sea water).

One simply divides the given compartment’s M-Value by the ambient pressure at sea-level to arrive at the permissible nitrogen pressure surfacing ratio, which is not constant across all compartments as Haldane assumed.

The studies done at NEDU over the decades discovered that faster compartments tolerate a much higher nitrogen pressure gradient than can the slower compartments. This difference can be accounted for in the greater solubility of nitrogen in slower tissues (or a low transport efficiency in those tissues), resulting in a greater molar concentration of nitrogen than the fast tissues. This is why model compartments representing slower tissues feature more conservative M-values and ratios than those representing faster tissues. Slower tissues “hold onto” their nitrogen longer which places them at an increased risk for bubble formation. The M-values are called “sliding scale M-values”–each compartment has a distinct M-value at any given ambient pressure.

We will be focusing on no-decompression dives during the span of the ENST-480 course, so I will focus on using the Navy Dive Tables with no-decompression. M-values can be used to determine the maximum amount of bottom time that can be allotted at a certain depth without requiring decompression stops. For this type of calculation, the planned depth is converted to absolute pressure in feet seawater (fsw) and the partial pressure of nitrogen is calculated. If the M0-value for a compartment happens to be less than the absolute nitrogen pressure in fsw at that depth, the diver would use the following equation to determine the No-Decompression Limit (NDL on the Navy dive tables, or max bottom time without required decompression) for the dive:

t= [T/ln(2)] x ln[(P0-Pa)/(Pm-Pa)] (3)

where t= a compartment’s maximum time at the planned depth, T= the half-life (in minutes) of the specific compartment, Po= initial nitrogen partial pressure in the compartment (at the surface at the start of the dive), Pa= ambient nitrogen pressure at the planned depth, and Pm= the M-value for the specific compartment allowed at the surface (or M0).

Once the NDL is calculated for each compartment, the one with the shortest time becomes the controlling compartment at that depth. The dive must not exceed the controlling compartment’s NDL at that depth to avoid required decompression stops. The Navy Tables round NDL’s down to the nearest 5 or 10 minutes, for easier memorization.

**Pressure Groups and Repetitive Dives**

After a dive, there is a certain amount of nitrogen left over in the various compartments (called residual nitrogen). With proper off-gassing, remnant nitrogen is not problematic to the surfaced diver. Residual nitrogen becomes important when a diver is conducting a repetitive dive.

Any dive completed within 12 hours of a previous dive is considered a repetitive dive. 12 hours has significance as the elapsed time before the slowest 120-min compartment is 98% de-saturated with excess nitrogen (the equivalent of 6 half-times). The Navy Tables use the 120-min compartment to track residual nitrogen. That is, the 120-min is the controlling compartment for determining the entering pressure group on a repetitive dive. Pressure groups are based on intervals of total air pressure in the controlling 120-min compartment. The total air pressure in the compartment is determined by

A(t)= Aa + (Aa – A0)e-kt, (4)

such that A(t)= total air pressure in the compartment, Aa= total ambient pressure, A0= initial load in the compartment. k is a constant determined by the half-life of the compartment, using the equation

k=ln(2)/T. (5)

In the case of the 120-min compartment, T would be 120, and k subsequently becomes 0.00578.

Using equation 4, the total air pressure in the 120-min compartment after a dive of given length and depth can be calculated. The pressure groups are designated by a letter, A-Z, and are identified by a range of total air pressure determined by the Navy. For example, the pressure group with the letter A corresponds to a range of total pressure in the 120-min compartment equivalent to 33-35 fsw. As the letters get closer to Z, the total pressure in the compartment gets higher. The entering pressure group for a repetitive dive determines the maximum bottom time for that repetitive dive, while taking into account the residual nitrogen time from previous dives.

Let’s use these equations to show standard dive table calculations:

A diver wants to complete a dive to 35 feet, without any required decompression. At 35 fsw, nitrogen partial pressure is 0.79 x (33+35)= 53.72 fsw. Compartments 5, 10, 20, and 40 can withstand pressure loading up to 104, 88, 72, and 58 fsw of nitrogen respectively (these are their M0-values). The 80-min and 120-min compartments have M-values less than 53.72, (52 and 51 fsw respectively) and therefore a diver can remain at that depth for only a limited amount of time until a decompression stop is required. Using equation 3, we can determine the maximum amount of time a diver can remain at that depth without required decompression for each of the 2 slow compartments.

t= [T/ln(2)] x ln[(Po-Pa)/(Pm-Pa)]

For the 80-min tissue compartment, t=320.8 minutes

For the 120-min tissue compartment, t=401.9 minutes.

In this case, the 80-min compartment is the controlling compartment. This means that the diver can stay at 35 feet for a max of 320.8 minutes without any required decompression (the US Navy Tables round this down to 310 minutes).

So let’s say our diver acquires a bottom time of 40 minutes at 35 feet.

Using depth equivalent pressures, A0= 33 fsw, k= ln(2)/120, Aa=33 fsw + the planned depth of 35 fsw=68, and t=40 minutes.

Crunching these numbers, we see that the diver exits the water with A(t)= 40 fsw, placing him or her in pressure group D. After some time on the surface (the diver’s surface interval time or SIT), the diver will continue to off-gas and can enter a repetitive dive at a pressure group lower than D. Using this information, a diver can determine the maximum NDL for the next dive using equation 3, incorporating the residual nitrogen into the initial nitrogen load.

The NAUI dive tables we use are much more conservative than the Navy Dive Tables but are largely based off of the Navy’s basic calculations and conclusions. It is important to understand the mathematics and science behind the Navy Tables in order to understand the NAUI tables.

**Author Bio:** Kaitlin Mogentale is a freshman at USC pursuing a B.A. in Environmental Studies. She also looks to complete minors in Urban Policy & Planning and Spanish. She plans to use her interest and knowledge in the field of environmental science to serve as an advocate for businesses and developers, focusing on the importance and pertinence of environmentally sound practices.

**Sources Cited:**

Acott, C. J. “Testing JS Haldane’s decompression model.” *Journal of the South Pacific Underwater Medicine Society* (2000). Rubicon Foundation.Web. 15 May 2012.

Anderson, Marlow. “The Mathematics of the Navy Dive Tables.” *The Physics of Scuba Diving*. Nottingham, UK: Nottingham University Press, 2011. 137-46. Print.

Baker, Erik C., P.E. Understanding M-values. Scuba Diving- New Jersey and Long Island New York. Web. 13 May 2012.

Huggins, Karl E. *The Dynamics of Decompression Workbook*. Ann Arbor, Michigan: The University of Michigan, 1992. Print.

Workman, R. D. Calculation of Decompression Schedules for Nitrogen-Oxygen and Helium-Oxygen Dives. Washington D.C. : U.S. Navy Experimental Diving Unit, 1965. Defense Technical Information Center. Web. 13 May 2012.

**Editor’s note:** Scientific Research Diving at USC Dornsife is offered as part of an experiential summer program offered to undergraduate students of the USC Dana and David Dornsife College of Letters, Arts and Sciences. This course takes place on location at the USC Wrigley Marine Science Center on Catalina Island and throughout Micronesia. Students investigate important environmental issues such as ecologically sustainable development, fisheries management, protected-area planning and assessment, and human health issues. During the course of the program, the student team will dive and collect data to support conservation and management strategies to protect the fragile coral reefs of Guam and Palau in Micronesia.

*Instructors for the course include Jim Haw, Director of the Environmental Studies Program in USC Dornsife, Assistant Professor of Environmental Studies David Ginsburg,, SCUBA instructor and volunteer in the USC Scientific Diving Program Tom Carr and USC Dive Safety Officer Gerry Smith of the USC Wrigley Institute for Environmental Studies.*