Editing Cardiac output (section)

=== Dye dilution method ===
The dye dilution method is done by rapidly injecting a dye, [[indocyanine green]], into the right atrium of the heart. The dye flows with the blood into the aorta. A probe is inserted into the aorta to measure the concentration of the dye leaving the heart at equal time intervals [0, ''T''] until the dye has cleared. Let ''c''(''t)'' be the concentration of the dye at time ''t''. By dividing the time intervals from [0, ''T''] into subintervals Δ''t'', the amount of dye that flows past the measuring point during the subinterval from <math>t=t_{i-1}</math> to <math>t=t_i</math> is:

<math>(concentration)(volume)=c(t_i)(F\Delta t)</math>

where <math>F</math> is the rate of flow that is being calculated. The total amount of dye is:

<math>\sum_{i=1}^nc(t_i)(F\Delta t)=F \sum_{i=1}^nc(t_i)(\Delta t)</math>

and, letting <math>n\rightarrow\infty</math>, the amount of dye is:

<math>A=F\int_{0}^{T} c(t)dt</math>

Thus, the cardiac output is given by:

<math>F=\frac{A}{\int_{0}^{T} c(t)dt}

</math>

where the amount of dye injected <math>A</math> is known, and the integral can be determined using the concentration readings.<ref>{{Cite book|title=Calculus: Early Transcententals|last=Stewart|first=James | name-list-style = vanc |publisher=Cengage Learning|year=2010|isbn=9780538497909|pages=565–66 }}</ref>

The dye dilution method is one of the most accurate methods of determining cardiac output during exercise. The error of a single calculation of cardiac output values at rest and during exercise is less than 5%. This method does not allow measurement of 'beat to beat' changes, and requires a cardiac output that is stable for approximately 10 s during exercise and 30 s at rest.{{citation needed|date=March 2018}}