Educational Use Only — Interactive walk-through of the Bühlmann/Haldane equation for on-gassing and off-gassing.
Haldane Equation
Interactive walk-through of how tissue inert-gas pressure changes during on-gassing and off-gassing.
Two targets, two equations
Haldane — fixed target
At t = 0, alveolar pressure is set to a new value.
It then remains constant.
The tissue approaches it exponentially.
Schreiner — moving target
Alveolar pressure rises or falls linearly during a depth change.
The tissue follows the moving target with a delay.
It stays below a rising target and above a falling target.
Situation: the target changes once, then stays fixed
The alveolar target pressure is constant throughout the modeled segment. The tissue pressure smoothly catches up according to its half-time.
The vertical step is an input idealization. It is not the physical course of a descent or ascent; a depth change that takes time is modeled by the Schreiner equation.
pt(t) =
pt,0 +
(palv − pt,0) ·
(1 − e−kt)
current pressure =
starting pressure +
total change toward equilibrium ×
completed fraction of the change
… bar =
… +
(… − …) ·
(1 − …)
Equation term breakdown
pt,0 | tissue pressure at t = 0
…
…
… bar
Where this compartment started, assumed equilibrated at start depth.
palv | equilibrium target pressure
…
…
… bar
The pressure the compartment approaches at the selected depth and gas.
Δp | total change toward equilibrium
= palv − pt,0
…
… bar
Positive while on-gassing, negative while off-gassing.
1 − e−kt | completed fraction of the change
k = ln(2) / t1/2
…
…
… (… %)
Zero at the start, approaching one over time. Each half-time completes half of what still remains.
At t = 0
1 − e0 = 0, so pt(0) = pt,0.
After a long time
e−kt approaches 0, so the completed fraction approaches 1 and pt approaches palv.
Equivalent canonical form
pt(t) =
palv +
(pt,0 − palv) ·
e−kt
pt(t) − palv =
(pt,0 − palv) ·
e−kt
This standard form follows the remaining deviation from equilibrium: the initial deviation is multiplied by an exponentially decreasing factor.