= Mixed venous oxygen content
{title2=$C_{\bar v}$}
\b[Mixed venous oxygen content] is the oxygen content of <blood> after the returning venous streams have mixed, representing whole-body venous return to the right side of the <heart>. It is the venous quantity required with <arterial oxygen content> in the whole-body <Fick principle>. A single regional <vein> need not have this content: different tissues remove different amounts of <oxygen> and return different <blood flow> rates.
The mixing rule follows from <conservation of mass>. If streams have <blood flow> rates $Q_i$ and <venous oxygen content> values $C_i$, with no oxygen exchange during mixing,
$$
C_{\bar v}=\frac{\sum_iQ_iC_i}{\sum_iQ_i}.
$$
For two constructed streams, $100$ mL/min at $0.05$ mL of <oxygen> per mL of <blood> and $50$ mL/min at $0.20$ mL/mL carry $5$ and $10$ mL/min of <oxygen>. Their mixture has content $15/150=0.10$ mL/mL, not the unweighted mean $0.125$ mL/mL. The correct weights are volume <blood flow> rates.
The <right atrium> receives several returns before <blood> passes through the <right ventricle> and <pulmonary trunk>.<How Blood Flows through the Heart by NHLBI>[Adding oxygen to blood]{ref} A downstream sample can represent their mixture only to the extent that the streams have actually mixed and the sampling interval represents the desired state. Replacing that sample with one taken from a particular upstream <vein> introduces a representativeness assumption. Likewise, averaging <oxygen saturation> values is not generally an oxygen-content balance when <hemoglobin> concentrations differ. The whole-body balance is about amounts of transported <oxygen>, not averages of instrument displays.
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