The Frank / Bramwell-Hill equation

For an incompressible fluid (blood) in a compressible (elastic) tube (e.g. an artery):^[11]

${\displaystyle PWV={\sqrt {\frac {V\cdot dP}{\rho \cdot dV}}}}$,

where ${\displaystyle V}$ is volume per unit length and ${\displaystyle P}$is pressure. This is the equation derived by Otto Frank,^[12] and John Crighton Bramwell and Archibald Hill.^[13]

Alternative forms of this equation are:

${\displaystyle PWV={\sqrt {\frac {r\cdot dP}{\rho \cdot 2\cdot dr}}}}$, or ${\displaystyle PWV={\frac {1}{\sqrt {\rho \cdot D}}}}$,

where ${\displaystyle r}$ is the radius of the tube and ${\displaystyle D}$ is distensibility.

The Moens–Korteweg equation

The Moens–Korteweg equation:

${\displaystyle \mathrm {PWV} ={\sqrt {\dfrac {E_{\mathrm {inc} }\cdot h}{2\cdot r\cdot \rho }}}}$,

characterises PWV in terms of the incremental elastic modulus ${\displaystyle {E_{\mathrm {inc} }}}$of the vessel wall, the wall thickness ${\displaystyle h}$, and the radius. It was derived independently by Adriaan Isebree Moens and Diederik Korteweg and is equivalent to the Frank / Bramwell Hill equation:^[11]: 64

These equations assume that:

1. there is little or no change in vessel area.
2. there is little or no change in wall thickness.
3. there is little or no change in density (i.e. blood is assumed incompressible).
4. ${\displaystyle \operatorname {d} \!v(\operatorname {d} \!r^{-1})\operatorname {d} \!x\cdot \operatorname {d} \!t}$ is negligible.

Using two simultaneously measured pressure waveforms

PWV, by definition, is the distance traveled (${\displaystyle \Delta x}$) by the pulse wave divided by the time (${\displaystyle \Delta t}$) for the wave to travel that distance:

${\displaystyle \mathrm {PWV} ={\dfrac {\Delta x}{\Delta t}}}$,

in practice this approach is complicated by the existence of reflected waves.^[11] It is widely assumed that reflections are minimal during late diastole and early systole.^[11] With this assumption, PWV can be measured using the `foot' of the pressure waveform as a fiducial marker from invasive or non-invasive measurements; the transit time corresponds to the delay in arrival of the foot between two locations a known distance apart. Locating the foot of the pressure waveform can be problematic.^[15] The advantage of the foot-to-foot PWV measurement is the simplicity of measurement, requiring only two pressure wave forms recorded with invasive catheters, or non-invasively using pulse detection devices applied to the skin at two measurement sites, and a tape measure.^[16]

Using pressure and volume, or pressure and diameter

This is based on the method described by Bramwell & Hill^[17] who proposed modifications to the Moens-Kortweg equation. Quoting directly, these modifications were:

"A small rise ${\displaystyle \delta P}$ in pressure may be shown to cause a small increase, ${\displaystyle \delta y=y^{2}\delta P/(Ec)}$, in the radius ${\displaystyle y}$ of the artery, or a small increase, ${\displaystyle \delta V=2\pi y^{3}\delta P/(Ec)}$, in its own volume ${\displaystyle V}$ per unit length. Hence ${\displaystyle 2y/Ec=\operatorname {d} \!V/(V\operatorname {d} \!P)}$"

where ${\displaystyle c}$ represents the wall thickness (defined as ${\displaystyle h}$above), ${\displaystyle E}$ the elastic modulus, and ${\displaystyle y}$ the vessel radius (defined as ${\displaystyle r}$above). This permits calculation of local PWV in terms of ${\displaystyle {\sqrt {V\cdot dP/(\rho \cdot dV)}}}$, or ${\displaystyle {\sqrt {r\cdot dP/\rho \cdot 2\cdot dr}}}$, as detailed above, and provides an alternative method of measuring PWV, if pressure and arterial dimensions are measured, for example by ultrasound^[18][19] or magnetic resonance imaging (MRI).^[20]

Using pressure-flow velocity, pressure-volumetric flow relationships or characteristic impedance

The Water hammer equation expressed either in terms of pressure and flow velocity,^[21] pressure and volumetric flow, or characteristic impedance^[22] can be used to calculate local PWV:

${\displaystyle \mathrm {PWV} =P/\left(v\cdot \rho \right)=P/Q\cdot A/\rho =Z_{\mathrm {c} }\cdot A/\rho }$,

where ${\displaystyle v}$ is velocity, ${\displaystyle Q}$is volumetric flow, ${\displaystyle Z_{\mathrm {c} }}$ is characteristic impedance and ${\displaystyle A}$is the cross-sectional area of the vessel. This approach is only valid when wave reflections are absent or minimal, this is assumed to be the case in early systole.^[23]

Using diameter-flow velocity relationships

A related method to the pressure-flow velocity method uses vessel diameter and flow velocity to determine local PWV.^[24] It is also based on the Water hammer equation:

${\displaystyle dP_{\pm }=\pm \rho \cdot PWV\cdot dv_{\pm }}$,

and since

${\displaystyle dP_{+}+dP_{-}={\frac {2\cdot \rho \cdot PWV^{2}}{S}}\cdot (dS_{+}+dS_{-})}$,

where ${\displaystyle S}$is diameter; then:

${\displaystyle PWV={\frac {S}{2}}\cdot {\frac {(dv_{+}+dv_{-})}{(dS_{+}+dS_{-})}}}$,

or using the incremental hoop strain, ${\displaystyle dS/S=d\ln S}$,

PWV can be expressed in terms of ${\displaystyle v}$and ${\displaystyle S}$

${\displaystyle PWV=\pm {\frac {1}{2}}\cdot {\frac {dv_{\pm }}{d\ln S_{\pm }}}}$,

therefore plotting ${\displaystyle \ln S}$against ${\displaystyle v}$gives a 'lnDU-loop', and the linear portion during early systole, when reflected waves are assumed to be minimal, can be used to calculate PWV.

Clinical methods

Clinically, PWV can be measured in several ways and in different locations. The 'gold standard' for arterial stiffness assessment in clinical practice is cfPWV,^[3][4] and validation guidelines have been proposed.^[25] Other measures such as brachial-ankle PWV and cardio-ankle vascular index (CAVI) are also popular.^[26] For cfPWV, it is recommended that the arrival time of the pulse wave measured simultaneously at both locations, and the distance travelled by the pulse wave calculated as 80% of the direct distance between the common carotid artery in the neck and the femoral artery in the groin.^[3] Numerous devices exist to measure cfPWV;^[27][28] some techniques include:

• use of a transducer to record the time of arrival of the pulse wave at the carotid and femoral arteries.
• use of cuffs placed around the limbs and neck to record the time of arrival of the pulse wave oscillometrically.
• use of Doppler ultrasound or magnetic resonance imaging to record the time of arrival of the pulse wave based on the flow velocity waveform.

Newer devices that employ an arm cuff,^[29] fingertip sensors^[30] or special weighing scales^[31] have been described, but their clinical utility remains to be fully established.

Interpretation

Current guidelines by the European Society of Hypertension state that a measured PWV larger than 10 m/s can be considered an independent marker of end-organ damage.^[8] However, the use of a fixed PWV threshold value is debated, as PWV is dependent on blood pressure.^[14] A high pulse wave velocity (PWV) has also been associated with poor lung function.^[32]