I'm having trouble understanding/deriving Bühlmann coefficients "a" and "b".

[David Carron]

I am trying to derive a and b coefficients for ZH-L16 using the Nitrogen M values in "Understanding M-values" by Erik Baker. I can find already computed coefficients published here and in An Explanation of Buehlmann's ZH-L16 Algorithm by Paul Chapman, so I am able to check my work.

I will take tissue compartment 2 for my examples (just to avoid the potential compartment 1/1b confusion and because M_0 is the same for ZH-L16A, B, and C).

Although Buhlmann originally derived the a and b constants from the tissue half-times, as an implementer you need to view these as parameters and not something to derive yourself.

In general, you're going to find it a lot easier to work with Buhlmann's formulation, as modified by Baker, rather than messing around with (fairly ambiguous) conversions to the earlier "M" formulation

p_igtol(p_amb)_i = p_amb × (gf / b_i - gf + 1) + gf × a_i

Where p_amb is absolute pressure, a_i and b_i the constants for tissue i and p_igtol(p_amb)_i the maximum tolerated inert gas loading for tissue i at p_amb

 

[dmaziuk]

I was thinking actual clinically measurable parameters directly related to gas science, like:

Yes, I too would like a magical gizmo to keep me happy ever after and bring warmed-up slippers to bed in the morning. Instead of this pesky dieting and exercising because gas solubility in lipids.

One can always buy a Shearwater: it'll let one get bent and keep diving too -- next best thing.

 

[atdotde]

The a and b coefficients for the various compartments are not computed but determined empirically (i.e. you put divers of goats in a chamber conduct some thousands of simulated dives at different times to different depths and then see what are the limits for the ascent not to bend too many subjects). These limits are then expressed in terms of a's and b's.

It could still be that there is a fit for these different values that lets you approximate them for a given tissue half-time, but there is no a priori reason there should be a specific functional form for this relation.

For another aspect (the difference between nitrogen and helium, see also N2 vs. He, what’s the difference? – The Theoretical Diver )

This is very different from the different half-times for the tissues: Those you can determine without any experiments: You make an assumption of what are the relevant time scales relevant for diving/decompression and end up with several minutes to several hours. And then you evenly divide this range of time scales on a logarithmic scale (or such that the ratio between consecutive half-times is about constant) into 16 (or whatever number) parts.

Furthermore, it is actually very hard to collect enough empirical data to get a good validation of any model (see Statistics empirics – The Theoretical Diver) so the fourth significant digit of a's and b's is very very likely pure noise and not determined anyways.

 

[dmaziuk]

The a and b coefficients for the various compartments are not computed but determined empirically (i.e. you put divers of goats in a chamber conduct some thousands of simulated dives at different times to different depths and then see what are the limits for the ascent not to bend too many subjects). These limits are then expressed in terms of a's and b's.

It could still be that there is a fit for these different values that lets you approximate them for a given tissue half-time, but there is no a priori reason there should be a specific functional form for this relation.

All English-language sources quote `a = 2 * T ^ (-1/3)` and `b = 1.005 - T ^ (-1/2)` for 'A' set -- I assume Herr Doktor ran a curve fit to the empirical values from ZH-L12 -- and then adjusted the result into 'B' and 'C' sets after trials.

Which lets us derive M-values for any half-time TC we want, we'd just need to back-fit them to the 'C' set curve to add the necessary conservatism.