6.2 Wave trapping

There is reasonably good evidence that most arterial bifurcations are well-matched for forward waves. A study of 444 human bifurcations from different sites reported that the area ratio between the daughters and the parent vessel was 1.18 ± 0.04 [Papageorgiou GL, Jones NB, Redding VJ and Hudson N (1990) Cardiovasc. Res. 6 478-484]. This is amazingly close to the theoretical value for well-matched symmetrical bifurcations α = 1.15.

In the previous section, we saw that bifurcations that are well-matched for forward wavefronts are necessarily poorly-matched for backward wavefronts approaching in one of the daughter vessels. This asymmetry in the forward and backward reflection coefficients gives rise to a phenomenon that can best be described as wave trapping

Wave trapping can probably best be described by a simple example. Consider a symmetrical tree of four generations of bifurcations that are well-matched for forward waves. Assume that the backward waves are reflected at each bifurcation with R = - ½. Further assume that a forward compression wavefront is generated at the root of this tree with a pressure change of 1. Finally, we assume that one of the terminal vessels is completely occluded with R = 1 so that wavefronts reaching this vessel are totally reflected.

{alternative 'slide show' presentation of this wave trapping example}

The results are shown in the following x-t plot which shows the position of the wavefronts x as a function of time t. The thickness of the line gives an indication of the magnitude of the wavefront. Compression wavefronts are denoted by solid lines and expansion wavefronts by dashed lines.

The first thing we notice is the growing complexity of the wavefronts in this very simple example. Because of the importance of this process in the arterial system (which has many more than four generation), it is probably worth going through the figure in some detail.

The wave starts at the root of the tree with size 1. It propagates through the the first generation vessel with a wave speed indicated by the slope of the wave path (the characteristic direction) in the x-t plane. At the first bifurcation there is no reflection of this forward wavefront and it passes into the second generation vessel unaltered, but travelling with the local wave speed which is different in the different generations. The wavefront carries on unchanged through the third and fourth generation vessels until it reaches the occlusion at the end of the fourth generation vessel. Here it is totally reflected with no change in its magnitude.

When the backward moving reflected wavefront encounters the bifurcation between the fourth and third generation it is reflected as an expansion wavefront with magnitude ½ and transmitted as a compression wave with magnitude ½. There are now two waves in the system.

The backward transmitted in the third generation vessel wave carries on until it encounters the bifurcation between the third and second generations where it is again reflected with R = - ½. This generates a transmitted compression wavefront in the second generation vessel with magnitude ½² and a forward, re-reflected expansion wavefront in the third generation vessel also with magnitude ½².

There is a repetition of this re-reflection when the transmitted wave in the second generation vessel encounters the bifurcation between the second and first generation vessels. This produces a backward transmitted compression wavefront in the first generation vessel with magnitude ½³ and a re-reflected expansion wavefront with magnitude ½³in the second generation vessel.

The first wavefront returning to the root of the tree due to the occlusion in the fourth generation vessel is therefore a compression wavefront with magnitude ½³.

Returning to the first re-reflected wavefront travelling forward in the fourth generation vessel, it very quickly encounters the occlusion and is re-re-reflected as an expansion wavefront. This wave behaves exactly like the first reflected wave except that it is an expansion wavefront with magnitude ½ instead of 1. This wavefront will eventually return to the root as an expansion wavefront with magnitude ½⁴.

The reflected wavefront generated in the fourth generation vessel will be a compression wavefront with magnitude ½². It in turn will be reflected at the closed end and generate a whole new set of waves as it travels back through the arterial tree. It will return to the root as a compression wavefront with magnitude ½⁵.

This process carries on indefinitely generating reflected waves arriving to the root at regular intervals (twice the time of transit of a wave in the fourth generation vessel with magnitudes that decrease as ½ⁿ where n is the number of the reflection. In the figure we have ignored waves of magnitude ½⁷ which are two orders of magnitude smaller than the initial wavefront and probably would not be detected by available measuring techniques.

This pattern of ever decreasing wavefronts due to the repetitive reflection of wavefronts in the fourth generation vessel is regular and it is relatively easy to predict what wavefronts will reach the root. However, it gets more complex when we consider the re-reflected wavefront generated at the bifurcation between the third and second and the second and first generation vessels. Unlike the wavefronts in the fourth generation vessel, they are not immediately re-reflected but propagate back to the periphery with being reflected since they are travelling in the forward direction. When these forward re-reflected wavefronts encounter the closed end they too generate new backward reflected wavefronts that generate a whole new set of wavefronts that eventually reach the root, again at regular intervals with alternating types and ever decreasing magnitude.

In the idealistic example shown, the tree is symmetrical and so some of the waves coincide, sometimes cancelling and other times reinforcing each other. This is unlikely in real arterial trees which are not symmetrical.

The process, even for the 'simple' example, is very complex, but the qualitative results are clear. If an arterial tree is well-matched for forward wavefronts, reflections generated in the periphery of the tree produce a complex network of reflected and re-reflected wavefront, very few of which return to the root of the tree and those that do return are of ever-decreasing magnitude. The effect of the reflection is much stronger closer to the site of reflection where there are many more waves of decreasing, but larger than those returning to the root. In effect, the energy of the reflected wavefronts are 'trapped' in the periphery where they arise with very little of the energy returning to the root of the tree.

This mechanism can explain many of the features of arterial wave mechanics. It can explain a number of previously puzzling experimental results as we will see in the next section.

 

	We see for symmetrical bifurcations, γ = 1, that R = 0 for an area ratio α ~ 1.15. For α less than this the bifurcation acts like a partially closed tube and R is positive. For α greater than this value the bifurcation acts more like an open tube and R is negative.
Reflection coefficient as a function of area ratio for different symmetry ratios

For a symmetrical bifurcation that is well-matched in the forward direction, the area ratio for a wave travelling backwards in one of the daughter vessels is approximately α = 2.7. The reflection coefficient for this backward wave is approximately R = -0/5 which means that approximately half of the energy of the backward wave will be reflected back in the forward direction and that this wave will be of the opposite type as the incident wave (i.e. a compression wavefront will be reflected as an expansion wavefront and an expansion wavefront will be reflected as a compression wavefront.

This is reasonable physically because the backward wave approaching the bifurcation in one of the daughter vessels (now the parent vessel) will see a bifurcation consisting of its twin vessel and the parent vessel with a net area much larger than its own. The bifurcation will therefore act more like an open-end tube and generate a negative reflection coefficient.