VPMechanics5

Copyright 1999-2001 Eric Maiken

M-Values and the VPM

This notebook discusses the VPM M-values, with a focus on the requirements that:
  1) Boundary conditions are met at zero absolute pressure.
  2) There is a smooth transition from hypobaric to hyperbaric forms.
  3) The hyperbaric boundary values are determined by the dive depth and time according to the Y&H algorithm.

M-Values

As with the preceding notebooks, the Buhlmann ZHL16 set of compartments is used to parameterize the time-scales of the body's response to inert gasses.
Although the VPM is formulated in terms of gradients, which are intimately connected to gas diffusion rates, the equivalent VPM M-values can be calculated for comparison purposes. M-values and supersaturation ratios R can be related to gradients as: M = G + p, or R = M/p, however, neither of these conventional measures of decompression stress is based on fundamental gas transport phenomena. While a part of the traditional decompression modeler's lore, use of M and R usually seems a bit odd to chemists and physicists.
The two VPM parameters: equilibrium bubble radius and the surface tension can be adjusted to modify the minimum allowed supersaturation PssMin, and the converged supersaturation gradients Pss .
In the following plot, two extreme sets of the VPM parameters were used to  model a 100 fsw dive,  using Mvpm = ΔMvpm p + = Gvpm + p. In both cases, ΔMvpm =1. The liberal settings: = 0.7 μm, and = 17.9 dyne/cm produced a range of  converged Pss lying between the red lines, and an NDL of 21 min. The conservative settings = 1.2 μm, and γ = 14 dyne/cm produced a range of values lying between the blue lines, with an 11 min NDL.
For reference, the Buhlmann ZHL16 set of M-values, calculated using Erik Baker's compilation of equivalent s and ΔM s, is shown in gray, with the surface pressure = 1 Ata. The VPM M-values are less than Buhlmann's for the deeper depths, resulting in the characteristic deep VPM stops. On the other hand, the VPM M-values for the longer half-time compartments are greater than Buhlmann's, which results in shorter shallow stop times. The long half-time Buhlmann compartments were intended to control multi-day diving, and should therefore be compared with the RGBM multidiving extension of the VPM rather than the single exposure forms shown below.

M-values at altitude

All conventional M-value sets extrapolate to finite, positive values at zero absolute pressure. Wienke argues against this form and proposes an alternative in: B.R. Wienke, Int. J. Biomed. Comput. 29, 215 (1991).
For saturation diving, Wienke's expressions for the allowed supersaturation gradient and M value are:

In the hypobaric range, Wienke finds that decompression data fit ξ =6.5 fsw, and ζ= 1.45. Hyperbaric exposures are fit by ξ=14.4 fsw, and ζ=1.28.
The following plot shows Wienke's expression for the saturation M as the purple curve, which was obtained by joining the hypobaric and hyperbaric forms at sea-level (p=1 ata). This curve is overlaid on Buhlmann's ZHL-16 set and the black-colored PssMins calculated by the VPM for a 100 ft dive, with = 1 μm.

Two Extremes

Following Wienke's expression for the convergence of saturation M-values to zero at zero ambient pressure, we would like to extend the result to non-saturation diving. The VPM M-Values should therefore merge smoothly with a hypobaric set that follow the form of  Wienke's expression.
To generate the following plots, a dive to the 100 ft was modeled using the VPM to generate a set of allowed PssNew. These values were used to construct a set of hyperbaric M-values as described above. The hypobaric forms were generated ad-hoc by matching the set of VPM M-values at the pressure Pmatch using:

This form results from forcing the set of ξ in the high-pressure limit of Wienke's equations to reduce to the set of .

The following two plots illustrate the results of matching the VPM M-values to Mmatch at two different Pmatch. As previously, the gray-colored set illustrates the Buhlmann ZHL16 M-values. The purple-colored curves in the figures below correspond to the slowest half-time (635 min), and are nearly identical to Wienke's result in the hypobaric region.

The hypobaric boundary: Pmatch = 1 Ata
The sharp transitions of the faster compartments are kind of ugly (though probably irrelevant anyway), but the rest match smoothly.

The hyperbaric boundary: Pmatch = 4 Ata
This plot illustrates the result of extending the hypobaric form to the 4 ata dive pressure. The diagonal black lines are the extremes of the VPM M-value set shown in the previous figure.