Parallel circulation: Qp:Qs from saturations, and systemic O₂ delivery versus Qp:Qs (after Barnea 1994)
Experimental prototype. Built from published sources but not yet independently verified. Do not use for clinical decisions; check results against the primary references.
Educational model. A steady-state, complete-mixing teaching model. It ignores dissolved O₂, streaming, collateral flow and measurement error, and the assumed pulmonary venous saturation drives every result. Use it to understand the physiology, not to manage a patient.
All fields start blank. Enter saturations for the bedside Qp:Qs, or the model inputs to draw the curves.
Oxygen delivery and saturations across Qp:Qs
Enter haemoglobin, VO₂, total output and pulmonary venous saturation to draw the curves.
Systemic DO₂ / SaO₂
SvO₂
VO₂ (top) / SpvO₂ (bottom)
Qp:Qs range where O₂ supply exceeds VO₂ (SvO₂ > 0)
Highest DO₂
Qp:Qs from the saturations
Table view
Qp:Qs
Qs L/min/m²
Qp L/min/m²
SaO₂ %
SvO₂ %
DO₂ mL/min/m²
Model, equations and sources
Bedside Qp:Qs (Fick principle, same as the Cath page)
Qp:Qs = (SaO2 - SvO2) / (SpvO2 - SpaO2)
With complete mixing in the single ventricle, SpaO2 = SaO2:
Qp:Qs = (SaO2 - SvO2) / (SpvO2 - SaO2)
O2 capacity C = 1.36 x Hb x 10 mL O2 / L blood
Qs = VO2 / (C x (SaO2 - SvO2)) Qp = VO2 / (C x (SpvO2 - SaO2))
Systemic DO2 = Qs x C x SaO2
Parallel-circulation model (after Barnea 1994)
Inputs: Qt = Qp + Qs (L/min/m2), VO2 (mL/min/m2), Hb (g/dL), SpvO2 (fraction)
r = Qp:Qs Qs = Qt / (1 + r) Qp = Qt x r / (1 + r)
Lungs: VO2 = Qp x C x (SpvO2 - SaO2) -> SaO2 = SpvO2 - VO2 / (Qp x C)
Body: VO2 = Qs x C x (SaO2 - SvO2) -> SvO2 = SaO2 - VO2 / (Qs x C)
DO2(r) = Qs x C x SaO2 = C x SpvO2 x Qt / (1 + r) - VO2 / r
Highest DO2: r/(1+r) = sqrt( VO2 / (C x SpvO2 x Qt) )
SvO2 > 0 between the roots of VO2 r^2 - (A - 2 VO2) r + VO2 = 0, A = C x SpvO2 x Qt
(exists only if A > 4 VO2; the two roots multiply to 1)
[unverified] Barnea and colleagues describe “a mathematical model that was based on the simple flow of oxygen uptake in the lungs and whole-body oxygen consumption” and “an equation … that related the key variables of cardiac output, pulmonary venous oxygen saturation and the QP/QS ratio to systemic oxygen availability” (abstract). The full paper could not be retrieved for this prototype, so the equations above are re-derived from the Fick principle and have not been checked against the paper's printed equation, symbols or constants.
The re-derived model reproduces all six findings listed in the abstract (checked in the test suite): oxygen availability rises with Qp:Qs, peaks and falls; the optimal Qp:Qs is ≤1; the optimum falls as cardiac output or pulmonary venous saturation rises; the critical range narrows as both fall; the curve is steep near the critical value; and systemic venous saturation is very low outside the range and high inside it.
Assumptions: steady state; complete mixing; pulmonary O2 uptake equals whole-body VO2; dissolved O2 ignored; O2 capacity 1.36 mL/g; cardiac output held constant as Qp:Qs changes (in a real patient it does not).
“SvO2 > 0” is the mathematical edge of the model, not a safe range: tissue hypoxia and lactic acidosis start well above it.
Input checks
Saturations 20–100% (pulmonary venous 70–100%), with SvO2 < SaO2 < SpvO2; haemoglobin 5–25 g/dL; VO2 50–300 mL/min/m2 (a value like 30 is probably mL/min, not indexed); total output 1–20 L/min/m2. Curves are drawn for Qp:Qs 0.2 to 5.
References
Barnea O, Austin EH, Richman B, Santamore WP. Balancing the circulation: theoretic optimization of pulmonary/systemic flow ratio in hypoplastic left heart syndrome. J Am Coll Cardiol 1994;24(5):1376-1381. doi:10.1016/0735-1097(94)90123-6 (abstract read via PubMed; full text not retrieved).
Fick-principle flows and Qp:Qs: see the Cath Hemodynamics page, which uses the same O2 capacity (13.6 × Hb mL/L).
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Teaching tool. It does not replace measured haemodynamics or clinical judgement.