Fick principle

The 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 giveswhere $Q$ 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 $Q{C}_{\bar{v}}$ from incoming transport $Q{C}_a$; 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.^[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.

Oxygen-uptake assumptions in estimated Fick output arise when a Fick principle calculation substitutes predicted oxygen consumption for a direct measurement. With an arteriovenous oxygen content difference $\Delta C=C_a-C_{\bar{v}}$,The measured blood samples and the predicted numerator are different sources of information. Precise arterial oxygen content and mixed venous oxygen content measurements cannot by themselves validate the assumed oxygen consumption.

A primary comparison explicitly examined estimated-oxygen-uptake Fick principle output against thermodilution. Its clinical observations therefore compare particular measurement procedures, not a conservation identity against an unrelated physical law.^[ref]

An original exact error decomposition isSuppose actual oxygen consumption is $250$ mL/min and $\Delta C=5$ mL/dL. Actual cardiac output is then $50$ dL/min, or $5$ L/min. Assuming $200$ mL/min with the correct content difference produces $4$ L/min. If the measured content difference were also underestimated to $4$ mL/dL, the two errors would cancel numerically and again produce $5$ L/min. Agreement of the final answer would not prove either input accurate.

The numerator must also represent the state during sampling. A prediction for one physiological condition need not describe oxygen consumption after a change in activity or metabolism. The steady-state balance assumes compatible sampling and no continuing oxygen accumulation within the represented system. Near a small content difference, even a modest absolute error in $\Delta C$ can strongly change the inferred blood flow. Reporting estimated uptake as an assumed input keeps these model and sampling limitations visible.