7.5 Wave reflections

So far we have discussed propagation of pressure and flow waves in an infinitely long, straight, cylindrical, elastic vessel filled with an incompressible inviscid fluid. However, for real blood vessels we have:

short, curved, tapered, bifurcating vessels
the vessel walls exhibit nonlinear viscoelastic properties
the blood is viscous

It turns out that the effect of nonlinear viscoelasticity is not so severe, as the Womersley number often is sufficiently large. However, the "infinitely long" assumption must be removed. When pressure and flow reach an end they must conform to the end conditions. Consequently, the waves will be modified and reflections will occur. The concept of wave reflection is easily understood for a single pulse which travels to the end of the vessel and is reflected. However, for a train of pulses or continuous oscillations, the reflected waves will interfere with the original pulse. In this case, the only evidence of reflections is spatial variations in amplitude of the waves. Reflections occur wherever there is a change in the characteristic impedance (mismatch in impedance).

The reflection factor: Let $ p_f $ denote the oscillatory pressure associated with the forward propagating wave and $ p_b $ associated with the reflected, backward propagating wave. Further, let the flow wave be split into forward and backward components in the same manner. These components superimpose to form the actual values: $$ \begin{equation} p= p_f + p_b \tag{7.110} \end{equation} $$ The reflection factor $ \Gamma $ is then defined as: $$ \begin{equation} \Gamma \equiv \frac{p_b}{p_f} = -\frac{Q_b}{Q_f} \tag{7.111} \end{equation} $$ The reflection factor may also be expressed in terms of the input impedance $ Z_{in} = p/Q $ and the characteristic impedance $ Z_c = p_f/Q_f = -p_b/Q_b $: $$ \begin{align} Q &= \frac{p}{Z_{in}} = \frac{p_f}{Z_{c}} - \frac{p_b}{Z_{c}} \tag{7.112}\\ \frac{Z_{in}}{Z_c} &= \frac{p_f + p_b}{p_f - p_b} = \frac{1+\Gamma}{1-\Gamma} \tag{7.113} \end{align} $$ And thus: $$ \begin{equation} \Gamma = \frac{Z_{in}-Z_c}{Z_{in}+Z_c} \tag{7.114} \end{equation} $$

7.5.0.1 The quarter wavelength formula

Figure 73: An elastic vessel terminated with $ Z_T $

To illustrate the effect of reflections on the input impedance, let us consider a frictionless vessel with a total reflective impedance at the end, i.e., $ \Gamma = 1 $ (see Figure 73). Let the forward waves at the inlet of the vessel take the form: $$ \begin{equation} p_f = p_0 \; e^{j \omega t}, \quad Q_f = \frac{p_0}{Z_c} \; e^{j \omega t} \tag{7.115} \end{equation} $$ These waves travel the length $ L $ with pulse wave velocity $ c $, are totally reflected, and must then travel the same distance back. Thus, the expressions for the reflected waves are: $$ \begin{equation} p_b = p_0 \; e^{j \omega (t-2L/c)}, \quad Q_b = -\frac{p_0}{Z_c} \; e^{j \omega (t-2L/c)} \tag{7.116} \end{equation} $$ And the input impedance at the inlet of the vessel is: $$ \begin{equation} \tag{7.117} Z_{in} = \frac{p_f + p_b}{Q_f + Q_b} = Z_c \; \frac{e^{j \omega t} + e^{j \omega (t-2L/c)}}{e^{j \omega t} - e^{j \omega (t-2L/c)}} \end{equation} $$ From (7.117) we see that $ Z_{in} = 0 $ whenever: $$ \begin{equation} \frac{2 \omega L}{c} = \pi \tag{7.118} \end{equation} $$ As $ \omega = 2 \pi f $ and $ \lambda = c/f $ the quarter wave length formula is obtained: $$ \begin{equation} \tag{7.119} L = \frac{\lambda}{4} \end{equation} $$ Thus, from the first minimum of $ Z_{in} $ an indication of "the effective length" to the major reflection site of the arterial system may be obtained.

For example if the first minimum of $ |Z_{in}| $ is found at 3.8 Hz, an estimate of $ L \approx 0.33 $ m may be obtained by assuming a typical pulse wave velocity of $ c \approx 5 $ m/s.