# Understanding M-values By Erik C. Baker, P.E.

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1 Understanding -values B Erik C. Baker, P.E. In conjunction with an arra of hpothetical "tissue" compartments, gas loading calculations and -values compose the major elements of the dissolved gas or "Haldanian" decompression model. Through the use of widel-available desktop computer programs, technical divers rel on this model for their decompression safet. A good understanding of -values can help divers to determine appropriate conservatism factors and evaluate the adequac of various decompression profiles for a particular dive. hat are -values? The term "-value" was coined b Robert D. Workman in the mid-0's when he was doing decompression research for the U.S. Nav Eperimental Diving Unit (NEDU). Workman was a medical doctor with the rank of Captain in the edical Corps of the U.S. Nav. The "" in -value stands for "aimum." For a given ambient pressure, an -value is defined as the maimum value of inert gas pressure (absolute) that a hpothetical "tissue" compartment can "tolerate" without presenting overt smptoms of decompression sickness (DCS). - values are representative limits for the tolerated gradient between inert gas pressure and ambient pressure in each compartment. Other terms used for -values are "limits for tolerated overpressure," "critical tensions," and "supersaturation limits." The term - value is commonl used b decompression modelers. HISTORICAL BACKGROUND In the dissolved gas or "Haldanian" decompression model, gas loading calculations for each hpothetical "tissue" compartment are compared against "ascent limiting criteria" to determine the safe profile for ascent. In the earl ears of the model, including the method developed b John S. Haldane in 0, the ascent limiting criteria was in the form of "supersaturation ratios." For eample, Haldane found that a diver whose "tissues" were saturated b breathing air at a depth of fsw could ascend directl to the surface (sea level) without eperiencing smptoms of DCS. Because the ambient pressure at fsw depth is twice that at sea level, Haldane concluded that a ratio of : for tolerated overpressure above ambient could be used as the ascent limiting criteria. This approimate ratio was used b Haldane to develop the first decompression tables. In later ears, and up until the 0's, other ratios were used b various modelers for the different half-time compartments. ost of the U.S. Nav decompression tables were calculated using this supersaturation ratio method. However, there was a problem. an of the tables produced b this method were deficient when it came to deeper and longer dives. Robert Workman began a sstematic review of the decompression model including previous research that had been performed b the U.S. Nav. He arrived at some important conclusions. First of all, he recognized that Haldane's original ratio of : (based on air) was reall a ratio of.: if ou considered onl the partial pressure of the inert gas in air - nitrogen. [B that time in decompression research it was known that ogen was not a significant factor in DCS; it was the inert gases like nitrogen and helium that were the culprits.] In his review of the research data, Workman found that the "tissue ratios" for tolerated overpressure varied b half-time compartment and b depth. The data showed that the faster half-time compartments tolerated a greater overpressure ratio than the slower compartments, and that for all compartments the tolerated ratios became less with increasing depth. Then, instead of using ratios, Workman described the maimum tolerated partial pressure of nitrogen and helium for each compartment at each depth as the "value." Net, he made a "linear projection" of these -values as a function of depth and found that it was a reasonabl close match to the actual data. He made the observation that "a linear projection of -values is useful for computer programming as well." THE WORKAN -VALUES Workman's presentation of -values in the form of a linear equation was a significant step in the evolution of the dissolved gas decompression model. His -values established the concept of a linear relationship between depth pressure [or ambient pressure] and the tolerated inert gas pressure in each "tissue" compartment. This concept is an important element of the present-da dissolved gas model as applied b a variet of modelers. Workman epressed his -values in the -intercept form of a linear equation (see Figure ). His surfacing value was designated O [pronounced " naught"]. This was the intercept value in the linear equation at zero depth pressure (gauge) at sea level. The in the linear equation was designated [pronounced "delta "] and represented the change in -value with change in depth pressure. THE BÜHLANN -VALUES Professor Albert A. Bühlmann,.D., began doing decompression research in in the Laborator of Hperbaric Phsiolog at the Universit Hospital in Zürich, Switzerland. Bühlmann

2 continued his research for over thirt ears and made a number of important contributions to decompression science. In he published the first edition (in German) of a successful book entitled Decompression - Decompression Sickness. An English translation of the book was published in. Bühlmann s book was the first nearl complete reference on making decompression calculations that was widel-available to the diving public. As a result, the "Bühlmann algorithm" became the basis for most of the world s in-water decompression computers and do-it-ourself desktop computer programs. Three more editions of the book were published in German in 0,, and under the name Tauchmedizin or "Diving edicine." [An English translation of the th Edition of the book () is in preparation for publication]. Bühlmann s method for decompression calculations was similar to the one that Workman had prescribed. This included -values which epressed a linear relationship between ambient pressure and tolerated inert gas pressure in the hpothetical "tissue" compartments. The major difference between the two approaches was that Workman s -values were based on depth pressure (i.e. diving from sea level) and Bühlmann s -values were based on absolute pressure (i.e. for diving at altitude). This makes sense, of course, since Workman was concerned with the diving activities of the U.S. Nav (presumabl performed at sea level) while Bühlmann was concerned with diving activities in the high mountain lakes of Switzerland. Bühlmann published two sets of - values which have become well-known in diving circles; the ZH-L set from the book, and the ZH-L set(s) from the 0 book (and later editions). The "ZH" in these designations stands for "Zürich" (named after his hometown), the "L" stands for "linear," and the "" or "" represents the number of pairs of coefficients (-values) for the arra of half-time compartments for helium and nitrogen. The ZH-L set has twelve pairs of coefficients for siteen half-time compartments and these -values were determined empiricall (i.e. with actual Compartment Inert Gas Pressure, absolute 0 0 Figure Pressure Graph: Workman-stle -values versus Bühlmann-stle -values Surface Pressure at Sea Level Workman -value Line decompression trials). The ZH-LA set has siteen pairs of coefficients for siteen half-time compartments and these -values were mathematicall-derived from the half-times based on the tolerated surplus volumes and solubilities of the inert gases. The ZH-LA set of - values for nitrogen is further divided into subsets B and C because the mathematicall-derived set A was found empiricall not to be conservative enough in the middle compartments. The modified set B (slightl more conservative) is suggested for table calculations and the modified set C (somewhat more conservative) is suggested for use with in-water decompression computers which calculate in real-time. Similar to the Workman -values, the Bühlmann -values are epressed in the -intercept form of a linear equation (see Figure ). The Coefficient a is the intercept at zero ambient pressure (absolute) and the Coefficient b is the Bühlmann -value Line Bühlmann Coefficient b = reciprocal of (/b = ) Bühlmann Coefficient a = intercept at zero ambient pressure (absolute) Ambient Pressure Line Workman = Workman O = intercept at zero depth pressure (gauge) Ambient Pressure, absolute = reciprocal of the. [Note: the Coefficient a does not impl that humans can withstand zero absolute pressure! This is simpl a mathematical requirement for the equation. The lower limit for ambient pressure in the application of the Bühlmann -values is on the order of 0. atm/bar.] DCAP AND DSAT -VALUES an technical divers will recognize the F set of -values used b Hamilton Research s Decompression Computation and Analsis Program (DCAP). This set or "matri" of -values was determined b Dr. Bill Hamilton and colleagues during development of new air decompression tables for the Swedish Nav. In addition to air diving, the F -values have worked well for trimi diving and are the basis for man custom decompression tables in use b technical divers. an sport divers are familiar with

3 the Recreational Dive Planner (RDP) distributed b the Professional Association of Diving Instructors (PADI). The -values used for the RDP were developed and tested b Dr. Ramond E. Rogers, Dr. ichael R. Powell, and colleagues with Diving Science and Technolog Corp. (DSAT), a corporate affiliate of PADI. The DSAT -values were empiricall verified with etensive in-water diver testing and Doppler monitoring. COPARISON OF -VALUES Tables thru present a comparison of -values for nitrogen and helium between the various Haldanian decompression algorithms discussed in this article. All -values are presented in Workman-stle format. An evolution or refinement in the -values is evident from Workman () to Bühlmann (0). The general trend has been to become slightl more conservative. This trend reflects a more intensive validation process (empirical testing) and includes the use of Doppler ultrasound monitoring for the presence and quantit of "silent bubbles" (bubbles which are detectable in the circulation but are not associated with overt smptoms of decompression sickness). CONSISTENCY OF -VALUES One observation that can be made about the comparison between the -values of the various algorithms is that there is not a great difference between them. In other words, there appears to be a certain consistenc between the values determined b various independent researchers around the globe. This is a good sign as it indicates that the science has determined a relativel consistent threshold for smptoms of decompression sickness across the human population. FORAT FOR -VALUES -values are often epressed in the form of a linear equation as in the Workmanstle or the Bühlmann-stle. This format is ideal for computer programming since it allows the -values to be calculated "on-the-fl" as the are needed. The linear format permits the displa of - value lines on the pressure graph as well. -values can also be epressed in the form of a "matri" or table. This is simpl where the -values for each halftime compartment and each stop depth are pre-calculated and arranged in columns and rows. This format is useful for detailed comparisons and analsis. Some of the earl dive computers and desktop computer programs used the table format to "look up" -values for each stop during the calculation process. -VALUE CHARACTERISTICS -value sets can be classified into two categories, no-decompression sets and decompression sets. No-decompression -values are surfacing values onl. The DSAT RDP -values are an eample. No-stop dive profiles are designed so that the calculated gas loadings in the compartments do not eceed the surfacing -values. This allows for direct ascent to the surface at an time during the dive. Some no-decompression Workman Definitions: = tolerated inert gas pressure (absolute) in hpothetical "tissue" compartment Depth = depth pressure (gauge) measured from surface at sea level Tolerated Depth = tolerated depth pressure (gauge) measured from surface at sea level O = intercept at zero depth pressure (gauge); surfacing -value = of -value line Bühlmann Definitions: P t.tol. i.g. = tolerated inert gas pressure (absolute) in hpothetical "tissue" compartment P t. i.g. = inert gas pressure (absolute) in hpothetical "tissue" compartment P amb. = ambient pressure (absolute) P amb.tol. = tolerated ambient pressure (absolute) a = intercept at zero ambient pressure (absolute) b = reciprocal of of -value line algorithms account for ascent and descent rates in the calculations.

4 Cpt No. Table : Comparison of -values for Nitrogen Between Various Haldanian Decompression Algorithms American Sstem of Pressure Units - feet of sea water (fsw) Workman -values () HT min fsw O Cpt HT No. min. Bühlmann ZH-L -values () fsw O DSAT RDP -values () Cpt No. HT min fsw O Cpt HT No. min DCAP FF -values () 00 0 fsw O Cpt No. b HT min Bühlmann ZH-L -values (0) A fsw O B fsw O C O fsw Cpt = Compartment HT = Half-time O = Surfacing -value (sea level = atm = fsw = bar) = of -value line Table : Comparison of -values for Nitrogen Between Various Haldanian Decompression Algorithms European Sstem of Pressure Units - meters of sea water (msw) Workman Bühlmann ZH-L DSAT RDP DCAP F Bühlmann ZH-L -values () -values () -values () -values () -values (0) A B C Cpt HT O Cpt HT O Cpt HT O Cpt HT O Cpt HT O O O No. min msw No. min msw No. min msw No. min msw No. min msw msw msw b Cpt = Compartment HT = Half-time = Surfacing -value (sea level = msw = bar) = of -value line O

5 Table : Comparison of -values for Helium Between Various Haldanian Decompression Algorithms American Sstem of Pressure Units - feet of sea water (fsw) Workman Bühlmann ZH-L Bühlmann ZH-LA -values () -values () -values (0) Cpt HT No. min fsw O Cpt HT No. min fsw O Cpt HT No. min fsw O b Cpt = Compartment HT = Half-time = of -value line O = Surfacing -value (sea level = atm = fsw = bar) Table : Comparison of -values for Helium Between Various Haldanian Decompression Algorithms European Sstem of Pressure Units - meters of sea water (msw) Workman Bühlmann ZH-L Bühlmann ZH-LA -values () -values () -values (0) Cpt HT O Cpt HT O Cpt HT O No. min msw No. min msw No. min msw.. b Cpt = Compartment HT = Half-time = of -value line = Surfacing -value (sea level = msw = bar) O Decompression -values are characterized b having a parameter which determines the change in -value with change in ambient pressure. The value of the parameter will var depending on the half-time of the hpothetical "tissue" compartment. Generall, faster half-time compartments will have a greater than slower half-time compartments. This reflects the observation that faster compartments tolerate greater overpressure than slower compartments. If the is greater than then the -value line "epands" on the pressure graph and that compartment will tolerate greater overpressure gradients with increasing depth. A fied of means that the compartment will tolerate the same overpressure gradient regardless of depth. In all cases, the value of the can never be less than. Otherwise, the -value line would cross under the ambient pressure line at some point and this would represent an "illogical" situation whereb the compartment could not tolerate even the ambient pressure. THE ABIENT PRESSURE LINE The ambient pressure line is an allimportant reference line on the pressure graph. Passing through the origin, it has a of and simpl represents the collection of points where the compartment inert gas loading will be equal to ambient pressure. This is important because when the inert gas loading in a compartment goes above the ambient pressure line, an overpressure gradient is created. An -value line represents the established limit for tolerated overpressure gradient above the ambient pressure line. THE DECOPRESSION ZONE The "decompression zone" is the region on the pressure graph between the ambient pressure line and the -value line (see Figure ). Within the contet of the dissolved gas model, this zone represents the functional area in which decompression takes place. In theor, a positive gradient above ambient pressure is desireable in order for a compartment to "off-gas" or "decompress." In some

7 however, ma not be entirel acceptable to all divers. an divers would like to be in the range of "no smptoms" and "ver low level of risk" when it comes to their decompression profiles. Fortunatel, it is well understood among decompression modelers and programmers that calculations based on the established -values alone cannot produce sufficientl reliable decompression tables for all individuals and all scenarios. This is wh decompression programs provide for a means of introducing conservatism into the calculations. Some of the methodologies include increasing the inert gas fractions used in the calculations, appling a depth safet factor which calculates for a deeper-thanactual dive depth, calculating for a longer-than-actual bottom time, and adjusting the half-times to be asmmetrical during off-gassing (slower). Some programs use more than one of these methods combined. These methodologies for conservatism are effective when properl applied. The degree of "effectiveness" is usuall gauged b divers in terms of how much longer and deeper the decompression profiles become, and through individual eperience with the outcome of the profiles. -VALUE RELATIONSHIPS Some fundamental relationships involving -values and decompression calculations are indicated on the pressure graph in Figure. The Percent -value calculation has been used b various decompression modelers over the ears. Professor Bühlmann, for eample, evaluated man of his decompression trials on a Percent -value basis and reported the data as such in his book(s). The Percent -value calculation is a measure of how far a decompression profile has entered into the "decompression zone." 0% -value is at the ambient pressure line and represents the bottom of the decompression zone. 0% -value is at the -value line and represents the top of the decompression zone. ANALYSIS OF PROFILES Compartment Inert Gas Pressure, absolute 0 0 Figure Pressure Graph: -value Relationships Safet argin Surface Pressure "Decompression Zone" is between lines -value -value Inert Gas Pressure an divers would like to know precisel what the effect is of the conservatism factors in their desktop decompression program(s). The realize that longer and deeper profiles are generated with increasing conservatism factors, but more fundamental information is desired. Both the Percent -value and Percent -value relationships are useful for the analsis and evaluation of decompression profiles. Using a standard set of reference -values, different profiles can be evaluated on a consistent basis. This includes comparison of profiles generated b entirel different programs, algorithms, and decompression models. UNIVERSAL REFERENCE VALUES The Bühlmann ZH-L -values are emploed in most, if not all, of the desktop decompression programs in use b technical divers. These -values were developed and tested for a broad -value Line Inert Gas Pressure Ambient Pressure, absolute Ambient Pressure Line inert gas loading for tpical deco profile Inert Gas % -value = Pressure -value 0 % -value = Inert Gas Pressure -value 0 range of ambient pressure eposures; from high altitude diving to deep sea diving. When used with appropriate conservatism, the have proven to be "reliable" for technical diving (to the etent that something can be reliable in an ineact science). The have become the de facto world-wide standard that can serve as universal reference values for the comparison and evaluation of decompression profiles. It is a relativel eas task for programmers to include Percent -value and Percent -value calculations in summar form with the decompression profiles. Table is an eample of this and shows the effect of conservatism factors used in a commerciall-available desktop decompression program. At 0% Conservatism Factor, the decompression profile is in the 0% -value range and has entered approimatel 0% into the decompression zone (0% -value ). It is evident that this program emplos a level of baseline conservatism

8 Table : Effect of Conservatism Factors in a Commerciall-Available Program on Decompression Profiles Referenced to Bühlmann ZH-L -values (ZH-LA Helium, ZH-LB Nitrogen) Deco Stop (fsw) /0 Trimi Dive (% O / 0% He) to 0 fsw for 0 min. Deco mies - Nitro % at fsw, 0% O at 0 fsw 0% Conservatism Factor Run (min) aimum * % -value.% () 0.0% () 0.% () 0.% () 0.% () 0.% () 0 0.% () 0 0.% () 0 0.% ().% () 0 0.% () * Upon Arrival at the Stop aimum * % -value.% ().% ().% ().% ().% ().% ().% () 0.% () 0.% ().% () 0.% () Deco Stop (fsw) 0% Conservatism Factor 0% Conservatism Factor Run (min) aimum * % -value.% ().% ().0% ().% ().% ().% ().% ().% ().% ().% ().% ().% ().% () aimum * % -value.0% ().% ().% ().0% ().% ().% ().% ().% ().% ().0% ().% ().% ().% () aimum * Deco Run aimum * % -value Stop (fsw) (min) % -value 0.% ().% () % ().% () % ().% ().% ().% () 0.% ().% () 0.% ().% () 0.% () 0.% () 0.0% ().% () 0.% ().% () 0.% ().% () 0.% ().% () 0 0.% ().% ().% ().% ().% ().% () 0 0.% ().% () since none of the values reaches 0%. At 0% Conservatism Factor (which is recommended in the user s manual), the profile is in the % -value range and has entered approimatel 0-0% into the decompression zone. At 0% Conservatism Factor, the profile is in the % -value range and has entered approimatel 0-% into the decompression zone. Note that the values given in Table are upon arrival the respective stops which is the worstcase condition. This correlates with the edges of the "stair-steps" in the gas loading profile on the pressure graph (see eample in Figure ). The highest values across all profiles are calculated upon arrival at the surface which illustrates wh a ver slow final ascent from the last decompression stop to the surface is alwas prudent. ARGIN OF SAFETY Using the -value relationships and a standard set of reference -values, divers can determine personal decompression limits which are both well-defined and transportable. The margin of safet selected will depend on individual disposition and prior eperience with profiles. An honest assessment of one s fitness for decompression diving is alwas in order. For eample, this author/diver (an office worker) has chosen a personal limit of % -value and 0-0% -value for tpical trimi dives. To ensure a fied margin of safet, a decompression profile can be calculated directl to a predetermined percentage of the -value. The advantage of this approach is complete consistenc across the entire ambient pressure range and precise control over the resultant profile. Z About the Author Erik C. Baker is an electrical engineer with a consulting engineering firm in Florida. He pursues research into decompression and diving phsiolog as a hobb, and has developed several FORTRAN computer programs for decompression calculation and analsis. Erik is a certified cave diver and trimi diver. Decompression References: Bennett PB, Elliott DH, eds.. The Phsiolog and edicine of Diving. London: WB Saunders. Bocott AE, Damant GCC, Haldane JS. 0. The prevention of compressed air illness. J Hg (London) :-. Bühlmann, AA.. Decompression-Decompression Sickness. Berlin: Springer-Verlag. Bühlmann, AA.. Tauchmedizin. Berlin: Springer-Verlag. Hamilton RW, uren A, Röckert H, Örnhagen H.. Proposed new Swedish air decompression tables. In: Shields TG, ed. XIVth Annual eeting of the EUBS. European Undersea Biomedical Societ. Aberdeen: National Hperbaric Center. Hamilton RW, Rogers RE, Powell R, Vann RD.. Development and validation of no-stop decompression procedures for recreational diving: The DSAT Recreational Dive Planner. Santa Ana, CA: Diving Science and Technolog Corp. Schreiner HR, Kelle PL.. A pragmatic view of decompression. In: Lambertsen CJ, ed. Underwater

9 Phsiolog IV. New York: Academic Press. Wienke BR.. Basic decompression theor and application. Flagstaff, AZ: Best. Wienke BR.. Basic diving phsics and applications. Flagstaff, AZ: Best. Workman RD.. Calculation of decompression schedules for nitrogenogen and helium-ogen dives. Research Report -. Washington: Nav Eperimental Diving Unit. Workman RD.. American decompression theor and practice. In: Bennett PB, Elliott DH, eds. The phsiolog and medicine of diving and compressed air work. London: Baillière, Tindall & Cassell.

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