= Fick principle
{c}
The \b[Fick principle] infers <blood flow> from the rate at which a transported substance is added to or removed from <blood> and the corresponding concentration difference. For whole-body <oxygen consumption> in a steady state, <conservation of mass> gives
$$
\dot V_{\mathrm O_2}=Q(C_a-C_{\bar v}),\qquad
Q=\frac{\dot V_{\mathrm O_2}}{C_a-C_{\bar v}},
$$
where $Q$ is <cardiac output>, $C_a$ is <arterial oxygen content> and $C_{\bar v}$ is <mixed venous oxygen content>. The <arteriovenous oxygen content difference> must match the tissue bed whose <oxygen consumption> is measured. The equation follows by subtracting outgoing oxygen transport $QC_{\bar v}$ from incoming transport $QC_a$; it assumes no continuing accumulation of <oxygen> within that system.
For a constructed example, <oxygen consumption> of $250$ mL/min and an <arteriovenous oxygen content difference> of $5$ mL/dL imply $Q=50$ dL/min, or $5$ L/min. Keeping the decilitre in the denominator prevents a factor-of-ten unit error. Using only a difference of <oxygen saturation> would omit the <hemoglobin> concentration needed to obtain oxygen content.
Direct implementation measures <oxygen consumption>, together with arterial and appropriately mixed venous <blood> samples. An estimated implementation substitutes a model for <oxygen consumption>. These are different measurement procedures even though they use the same balance equation. A 2017 comparison explicitly studied <thermodilution> against estimated-oxygen-uptake output, rather than treating estimated uptake as an exact measured input.<Thermodilution vs Estimated Fick Cardiac Output Measurement in Clinical Practice by Alexander R. Opotowsky and others>[Abstract, Importance and Exposures]{ref}
At fixed measured content difference, a $20\%$ error in assumed <oxygen consumption> produces a $20\%$ error in inferred <cardiac output>. More generally, differentiating the <Fick principle> gives $\delta Q/Q\simeq\delta\dot V_{\mathrm O_2}/\dot V_{\mathrm O_2}-\delta(C_a-C_{\bar v})/(C_a-C_{\bar v})$. Thus a small content difference magnifies a fixed absolute sampling error.
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