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## X:\web\local\apache\services\xfer\3996d912-0c72-f050\vpm derivation.wpd

*. . . this paper is prepared with fond memory of the late Professor David E. Yount . . .*
The gradient Pmin correlates with the constant bubble number N
volume that can be tolerated indefinitely (i.e. as in a decompression profile from saturation).

number of bubbles regardless of gradient. At a larger gradient Pnew ,
the excess released gas volume is proportional to the excess bubble number N
Some brief notes about the notation used in this paper:
A capital letter P has traditionally been used by physiologists to denote pressure, either totalpressure or partial pressure. Additional description is given by the symbols for individual gases orthe use of subscripts. Examples:
Yount and colleagues used the notation of physicists in their papers. This included a lower caseletter p for pressure with descriptive subscripts and superscripts. The same notation is used in thispresentation except that a capital letter P is used for pressure (to follow the convention used indiving physiology). Specific Yountian notation is as follows:
Pmin = minimum supersaturation pressure [gradient for bubble formation]
Pnew = new [larger] supersaturation pressure [gradient for bubble formation]
= crushing pressure [gradient for reduction in radius of gas nuclei]
rmin = minimum initial radius of gas nuclei [probed for bubble formation]
rnew = new [larger] initial radius of gas nuclei [probed for bubble formation]
A gradient is a difference in pressure which makes it distinct from an absolute pressure. Most ofthe gradients in this paper use the same notation P as that used for absolute pressures (followingthe presentation of the original authors). However, I think Bruce Wienke’s notation “G” forgradients makes more sense (i.e. a gradient G is independent of the absolute pressure scale whereasa pressure P is not). Accordingly, I use the notation G in a few places where I introduce my ownarguments (poetic license!).

The dynamic critical volume hypothesis assumes that the body can eliminate or tolerate N
bubbles and associated volume of released gas for an indefinite period of time. It also assumes that thebody can eliminate or tolerate an even greater “critical volume” of released gas for a limited period of time. This critical volume is proportional to the excess bubble number N
leaving the free-phase throughout decompression, so the situation is dynamic. The rate at which the free-phase gas inflates is assumed to be proportional to the product Pnew (t) ⋅ (N
The total volume of released gas in the body at any time t should never exceed some critical volume V .
The decompression criterion is then given by the critical volume equation:
whereα is a constant of proportionality.

The gradient Pnew is assumed to be held constant during the in-water deco time t
exponentially on the surface as the surface interval time t
→ ∞ . Since gradients to drive bubble
formation and growth can persist for a period of time after the diver has surfaced, the critical volumedecompression criterion includes a phase-volume computation for both the in-water and post-dive(surface) portions of a profile.

Accordingly, the total phase-volume integral is computed in two parts; the in-water portion during thedeco time t
, and the post-dive portion during the surface interval time t
Steps to evaluate the phase-volume integrals in the critical volume equation:
First, factor out constants and separate phase-volume integrals by in-water and surface portions:
Note that at this point the equation has been set equal to α V in order to establish a definite upper limit.

Since Pnew is assumed to be held constant during the deco time t
In the next step, the phase-volume over the post-dive surface interval must be calculated. Thiscomputation is simplified in the original Yount & Hoffman paper by assuming that the diver only breatheda single inert gas such as nitrogen in air during the dive, the diver breathes normal atmospheric air on thesurface, and the gradient for dissolved gas elimination is P(t) − P instead of P(t) − P . This last
assumption is somewhat conservative and dramatically simplifies evaluation of the integral as will bedemonstrated below. In the above notation, P(t) is the partial pressure of dissolved inert gas in ahypothetical tissue compartment as a function of time; P
surface; and P is inspired (alveolar) partial pressure of nitrogen in atmospheric air at the surface.

The gas loading as a function of time, P(t) , for one inert gas during the surface interval is given by
In this case, Yount & Hoffman made the simplifications that P = P and P
Plugging these in and simplifying the gas loading equation:
Since the ambient pressure at the surface is constant, the gradient of interest as a function of time, G(t) , isgiven by P(t) − P . Thus,
The post-dive (surface) portion is evaluated as:
Thus, after evaluating the in-water and surface phase-volume integrals, the critical volume equation yields:
Note that Pnew is factored out of the total phase-volume integration in order to isolate the variable.

The goal of this mathematical exercise is to be able to solve the critical volume equation for Pnew in terms
of calculable parameters. In order to accomplish this, the quantity (N
terms of VPM parameters that can be calculated in a decompression program. This leads to the next stepin the derivation.

Expressing bubble numbers in terms of calculable VPM parameters:
The VPM primordial (pristine) radial distribution (a continuous distribution function) relating bubblenumber versus initial radius of gas nuclei, r , is given by:
where β is a VPM constant, N is a normalization constant, S is the constant area occupied by one
surfactant molecule

*in situ*, k is the Boltzmann constant, and T is the absolute body temperature which isalso assumed to be constant.

The bubble numbers versus initial radii at the gradients Pmin and Pnew , are given by:
It should be noted at this point that the radial distribution of gas nuclei in humans is not exactly known, sosome further assumptions and simplifications are required.

For small values of the exponential argument (linear-small region of the exponential distribution), theabove equations can be expanded to simplify the calculation:
To further simplify the calculation, r min is factored out of the parentheses and a VPM relationship for
rmin is introduced [reference the “Skins” paper by Yount, J. Acoust. Soc. Am. 65(6)1979a]:
At this point, r min and r new must be expressed in terms of Pmin and Pnew . From the “core” VPM
− γ c ⋅ Pcrush − (Pss − γc ⋅ Pcrush)
Arranging the critical volume equation so that it can be solved for Pnew :
Now all the previous solutions and simplifications can be plugged back into the critical volume equation:
To solve this equation for Pnew , the quadratic formula must be used. To do this, further simplification is
required and the equation must be rearranged in the form ax2 + bx + c = 0 .

First, the decompression parameter λ is defined :
α Vcrit ⋅γ ⋅ No ⋅ (γ c − γ )S
This substitution is made into the critical volume equation and it is rearranged and simplified:
Next, to make the value of the constant a in the quadratic formula equal to 1, divide by t
Now the critical volume equation is in the form ax2 + bx + c = 0 and the quadratic formula can be
In this case, the constants are as follows:
So, the critical volume equation is solved for Pnew by:
Note that the + b2 − 4c term yields the largest value for Pnew .

Yount, D.E. and Hoffman, D.C. 1986. On the use of a bubble formation model to calculate diving tables. Aviat. Space Environ. Med. 57:149-156.

Yount, D.E. and Hoffman, D.C. 1983. Decompression theory: a dynamic critical-volume hypothesis. In:Bachrach A.J. and Matzen, M.M. eds. Underwater physiology VIII: Proceedings of the eighth symposiumon underwater physiology. Undersea Medical Society, Bethesda, 131-146.
Varying Permeability Model (VPM) References:
Yount, D.E. 1979a. Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei. J.

Acoust. Soc. Am. 65:1429-1439.

Yount, D.E. 1979b. Application of a bubble formation model to decompression sickness in rats andhumans. Aviat. Space Environ. Med. 50:44-50.

Yount, D.E. 1979c. Multiple inert-gas bubble disease: a review of the theory. In: Lambertsen, C.J. andBornmann, R.C. eds. Isobaric Inert Gas Counterdiffusion. Undersea Medical Society, Bethesda, 90-125.

Yount, D.E. 1982. On the evolution, generation, and regeneration of gas cavitation nuclei. J. Acoust.

Soc. Am. 71:1473-1481.

Yount, D.E., Gillary, E.W., and Hoffman, D.C. 1984. A microscopic investigation of bubble formationnuclei. J. Acoust. Soc. Am. 76:1511-1521.

Yount, D.E. and Hoffman, D.C. 1989. On the use of a bubble formation model to calculate nitrogen andhelium diving tables. In: Paganelli, C.V. and Farhi, L.E. eds. Physiological functions in specialenvironments. Springer-Verlag, New York, 95-108.

Yount, D.E., Kunkle, T.D., D’Arrigo, J.S., Ingle, F.W., Yeung, C.M., and Beckman, E.L. (1977). Stabilization of gas cavitation nuclei by surface-active compounds. Aviat. Space Environ. Med. 48:185-191.

Yount, D.E. and Strauss, R.H. 1976. Bubble formation in gelatin: A model for decompression sickness. J. Appl. Phys. 47:5081-5089.

Yount, D.E. and Yeung, C.M. 1981. Bubble formation in supersaturated gelatin: a further investigation ofgas cavitation nuclei. J. Acoust. Soc. Am. 69:702-708.

Yount, D.E., Yeung, C.M., and Ingle, F.W. 1979. Determination of the radii of gas cavitation nuclei byfiltering gelatin. J. Acoust. Soc. Am. 65:1440-1450.

Yount, D.E. 1997. On the elastic properties of the interfaces that stabilize gas cavitation nuclei. J. Colloidand Interface Science 193:50-59.

Yount, D.E., Maiken, E.B., and Baker, E.C. 2000. Implications of the Varying Permeability Model forReverse Dive Profiles.

*In*: Lang, M.A. and Lehner, C.E. (eds.). Proceedings of the Reverse Dive ProfilesWorkshop. Smithsonian Institution, Washington, D.C. pp. 29-61.

Additional Decompression and Related References:
Albano, G. 1970. Principles and Observations on the Physiology of the Scuba Diver, English translation:ONR Report DR-150, Office of Naval Research, Department of the Navy, Arlington, Virginia.

Baker, E.C. 1998a. Understanding M-values. Immersed. Vol. 3, No. 3, 23-27.

Baker, E.C. 1998b. Clearing up the confusion about deep stops. Immersed. Vol. 3, No. 4, 23-31.

Bennett PB, Elliott DH, eds. 1993. The Physiology and Medicine of Diving. London: WB Saunders.

Bühlmann, AA. 1984. Decompression-Decompression Sickness. Berlin: Springer-Verlag.

Bühlmann, A.A. 1995. Tauchmedizin. Springer-Verlag, Berlin.

Cowley, J.R.M., Allegra, C., and Lambertsen, C.J. 1979. Measurement of subcutaneous tissue pressureduring superficial isobaric gas counter-diffusion. J. Appl. Physiol.:Respir. Environ. Exercise Physiol.

47:224-227.

Gerth, W.A. and Hemmingsen, E.A. 1976. Gas supersaturation thresholds for spontaneous cavitation inwater with gas equilibration pressures up to 570 atm. Z. Naturforsch. 31A:1711-1716.

Hamilton RW, Rogers RE, Powell MR, Vann RD. 1994. Development and validation of no-stopdecompression procedures for recreational diving: The DSAT Recreational Dive Planner. Santa Ana, CA: Diving Science and Technology Corp.

Harvey, E.N., Barnes, D.K., McElroy, W.D., Whiteley, A.H., Pease, D.C., and Cooper, K.W. 1944. Bubble formation in animals. I. Physical factors. J. Cell. Comp. Physiol. 24:1-22.

Hennessy, T.R. and Hempleman H.V. 1977. An examination of the critical released gas volume conceptin decompression sickness. Proc. R. Soc. Lond. B. 197:299-313.

Hills, B.A. 1966. A thermodynamic and kinetic approach to decompression sickness. Doctoral thesis, Theuniversity of Adelaide, Australia.
Hills, B.A. 1977. Decompression Sickness. John Wiley and Sons, Inc., New York.

Jablonski, J. and Baker, E.C. 1999. Taking cave diving to the limit. Immersed. Vol. 4, No. 3, 29-33.

Kidd, D.J., Stubbs, R.A., and Weaver, R.S. 1971. Comparative approaches to prophylactic decompression.

In: Lambertsen, C.J., ed. Underwater physiology: Proceedings of the fourth symposium on underwaterphysiology. Academic Press, New York, 167-177.
Kozlov, M.M. and Markin, V.S. 1990. Elastic properties of membranes: monolayers, bilayers, vesicles. J.

Colloid Interface Sci. 138:332-345.

Kunkle, T.D. 1979. Bubble nucleation in supersaturated fluids. Univ. of Hawaii Sea Grant TechnicalReport UNIHI-SEAGRANT-TR-80-01. Pp. 108.

LeMessurier, D.H. and Hills, B.A. 1965. Decompression Sickness: A Study of Diving Techniques in theTorres Strait. Hvaldradets Skrifter 48:54-84.

Maiken, E.B. 1995. "Bubble Decompression Strategies."

* *tek95 Diving Technology Conference.

Pyle RL. 1996. The importance of deep safety stops: Rethinking ascent patterns from decompressiondives. DeepTech. 5:64; Cave Diving Group Newsletter. 121:2-5.

Schreiner, H.R. and Kelley, P.L. 1971. A pragmatic view of decompression. In: Lambertsen, C.J., ed.

Underwater physiology: Proceedings of the fourth symposium on underwater physiology. Academic Press,New York, 205-219.

Wienke, B.R. 1991. Basic Decompression Theory and Application. Best Publishing Co., Flagstaff,Arizona.

Wienke, B.R. 1994. Basic diving physics and applications. Best Publishing Co., Flagstaff, Arizona.

Workman, R.D. 1965. Calculation of decompression schedules for nitrogen-oxygen and helium-oxygendives. Research Report 6-65. U.S. Navy Experimental Diving Unit, Washington, D.C.

Source: http://www.diverclub.ru/File/VPM%20Critical%20Volume%20Derivation.pdf

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