= Velocity-time integral
{title2=VTI}
The \b[velocity-time integral] is the area under a velocity-versus-time curve over a specified interval:
$$
\mathrm{VTI}=\int_{t_0}^{t_1}v(t)\,dt.
$$
It has units of <length>. In <Doppler echocardiography>, a systolic flow-velocity trace can be integrated to estimate the distance associated with that measured waveform. The interval, sampling site and method of tracing the spectrum are part of the measurement.
The exact volume transported through an opening is
$$
V=\int_{t_0}^{t_1}\int_{A(t)}\mathbf v\cdot\mathbf n\,dA\,dt.
$$
For constant area $A$ and a representative area-mean normal velocity $\bar v(t)$, this reduces to $V=A\int\bar v(t)\,dt$. Approximating the <left ventricular outflow tract> as circular then gives
$$
\mathrm{SV}\simeq\frac{\pi D^2}{4}\,\mathrm{VTI},\qquad
\mathrm{CO}=\mathrm{SV}\times\mathrm{HR},
$$
where $D$ is the measured diameter, SV is <stroke volume>, HR is <heart rate> and CO is <cardiac output>. The primary Doppler study relates integrated velocity and anatomical area to <cardiac output> and compares the result with independent methods.<Determination of cardiac output by Doppler echocardiography by Halfdan Ihlen and others>[Theory and Abstract: velocity integration, aortic size and method comparison]{ref}
For stipulated $D=2.0\,\mathrm{cm}$ and VTI $=20\,\mathrm{cm}$, the estimated <stroke volume> is $20\pi\approx62.8\,\mathrm{mL}$. At 75 beats per minute, <cardiac output> is approximately $4.71\,\mathrm{L/min}$. A triangular waveform with peak $100\,\mathrm{cm/s}$ and duration $0.40\,\mathrm{s}$ has that same $20\,\mathrm{cm}$ integral. A trapezoidal waveform with the same peak and duration, but 0.10-second rise and fall, has integral $30\,\mathrm{cm}$ instead.
\Image[/velocity-time-integral-same-peak.png]
{title=Equal peak flow velocities with different velocity-time integrals}
The calculation depends quadratically on diameter: a 5% diameter overestimate produces a 10.25% area overestimate at fixed VTI. <Flow-profile mismatch in Doppler stroke-volume estimation> is a different error: a peak spectral envelope need not equal the area-mean velocity required by the volume integral. A VTI is also not automatically the path length travelled by one identifiable blood cell, since a fixed sampling region can observe different cells over time.
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