Schreiner Equation — Deco Theory

p_t(t) = p_alv,0 + R · (t − 1/k) − (p_alv,0 − p_t,0 − R/k) · e^−kt

… bar = … + … · (… − 1/…) − (… − … − …/…) · …

p_alv,0 | alveolar N₂ at start depth

…

… bar

Starting alveolar pressure. Slope-zero point of the moving alveolar source.

p_t,0 | tissue pressure at t=0

…

… bar

Tissue assumed equilibrated at start depth → equals p_alv,0.

R | alveolar pressure rate

…

… bar/min

Positive = descent (alveolar climbing), negative = ascent. R = 0 → equation collapses to Haldane.

k | rate constant

= ln(2) / t_1/2

…

… min⁻¹

Faster compartments → bigger k → catch up sooner.

e^−kt | exponential decay factor

= e^{−k · t}

…

1.0 at t=0, decays toward 0. Multiplied by the initial-disequilibrium term.

Three-term decomposition · the formula's structural pieces

Moving alveolar source

= p_alv,0 + R · t

…

… bar

Where the alveolar source is right now (linear in t).

Phase lag (R/k)

= R / k

…

… bar

How far below (ascent) or above (descent) the source a perfectly-tracking tissue would settle. Equivalent to 1/k minutes of slope.

Decaying initial gap

= (p_alv,0 − p_t,0 − R/k) · e^−kt

…

… bar

The initial offset between actual Pt and the tracking line — settles exponentially toward zero.

p_t(t) = [moving] − [phase lag] − [decaying gap]

M-value check

p_alv

p_amb

p_t

p_t = 0.7510 bar | M = 2.50 bar within

M = a + p_amb / b