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Beyond the aorta: partial transmission of reflected waves from aortic coarctation into supra-aortic branches modulates cerebral hemodynamics and left ventricular load

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Abstract

Wave reflection from the site of aortic coarctation produces a reflected backward compression wave (BCW) that raises left ventricular (LV) afterload. However, not all reflected wave power will propagate back to the LV. This study investigated the hypothesis that the BCW is partially transmitted into supra-aortic vessels as a forward wave and explored the consequences of this phenomenon for cerebral and LV haemodynamic load. In eight sheep, high fidelity pressure and flow were measured in the aortic trunk (AoT) and brachiocephalic trunk (BCT, the single supra-aortic vessel present in sheep) at baseline and during two levels of proximal descending aortic constriction. Wave power analysis showed that aortic constriction produced not only a BCW in the AoT, but also a second forward compression wave (\(\mathrm{FCW}_{2})\) in the BCT that augmented pressure and flow after the initial forward compression wave (\(\mathrm{FCW}_{1})\). Mathematical analysis and a one-dimensional model of the human systemic arteries and aortic coarctation suggested that the relative transmission of waves into supra-aortic vessels versus the aorta was determined by the relative admittances of these vessels. Reducing supra-aortic admittance (1) increased pressure and flow pulsatility in cerebral arteries, (2) produced carotid and middle cerebral arterial flow waveforms with an older adult phenotype, (3) promoted transmission of reflected wave power towards the LV and (4) substantially increased mid- to late-systolic myocardial stress, which may promote LV hypertrophy. These findings suggest that wave transmission into supra-aortic branches has an important impact on both cerebral hemodynamics and LV load in aortic coarctation.

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Acknowledgments

We thank Magdy Sourial, Sarah White, Amy Tilley and Aaron Mocciaro for their assistance with experimental studies.

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Correspondence to Jonathan P. Mynard.

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Funding

JPM was funded by a CJ Martin Early Career Fellowship from the National Health and Medical Research Council of Australia. This work was supported by the Victorian Government’s Operational Infrastructure Support Program.

Appendix

Appendix

When a pressure/flow wave propagating along an arterial segment encounters a branch point, the degree of wave reflection and wave transmission into the respective branches is determined by the relative characteristic admittances (Y) of those branches,

$$\begin{aligned} Y=\frac{1}{Z_{\mathrm{c}} }=\frac{A}{\rho c} \end{aligned}$$
(9)

where A and c are vessel cross-sectional area and wave speed, respectively. Consider the initial pressure- and flow-increasing forward compression wave (\(\mathrm{FCW}_{1}\)) that emerges from the ventricle in early systole. After propagating along the aortic trunk (AoT), it encounters several supra-aortic branches in the human or one brachiocephalic trunk (BCT) in the sheep (for simplicity we here consider the latter) along with the distal aortic arch or ‘aortic isthmus’ (AI). The ratio of the pressure effects of a reflected BCW to that of the incident \(\mathrm{FCW}_{1}\) defines a local pressure reflection coefficient that is related to the branch admittances via

$$\begin{aligned} R^{P}=\frac{\Delta P_{\mathrm{AoT}} \left( {\mathrm{BCW}} \right) }{\Delta P_{\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }=\frac{Y_{\mathrm{AoT}} -Y_{\mathrm{BCT}} -Y_{\mathrm{AI}} }{Y_{\mathrm{AoT}} +Y_{\mathrm{BCT}} +Y_{\mathrm{AI}} } \end{aligned}$$
(10)

Here, \(R^{P}= 0\) signifies a well-matched junction (no reflection), \(R^{P}= 1\) signifies complete reflection of the incident wave, while negative values of \(R^{P}\) up to -1 would indicate a local decrease in admittance and result in a backward expansion wave rather than a BCW. A flow-based reflection coefficient can also be defined (\(R^{Q}= -R^{P}\)).

The pressure transmission coefficients from the AoT to the AI or BCT are the same and are given by

$$\begin{aligned} T^{P}= & {} \frac{\Delta P_{\mathrm{AI}} \left( {\mathrm{FCW}_{1} } \right) }{\Delta P_{\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }=\frac{\Delta P_{\mathrm{BCT}} \left( {\mathrm{FCW}_{1} } \right) }{\Delta P_{\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }\nonumber \\= & {} 1+R^{P}=1+\frac{Y_{\mathrm{AoT}} -Y_{\mathrm{BCT}} -Y_{\mathrm{AI}} }{Y_{\mathrm{AoT}} +Y_{\mathrm{BCT}} +Y_{\mathrm{AI}} } \end{aligned}$$
(11)

Corresponding flow transmission coefficients differ for the respective branches and are given by

$$\begin{aligned}&T_{\mathrm{AoT,AI}}^{Q}=\frac{\Delta Q_{\mathrm{AI}} \left( {\mathrm{FCW}_{1} } \right) }{\Delta Q_{\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }=\frac{2Y_{\mathrm{AI}} }{Y_{\mathrm{AoT}} +Y_{\mathrm{BCT}} +Y_{\mathrm{AI}} } \nonumber \\ \end{aligned}$$
(12)
$$\begin{aligned}&T_{\mathrm{AoT,BCT}}^{Q} =\frac{\Delta Q_{\mathrm{BCT}} \left( {\mathrm{FCW}_{1} } \right) }{\Delta Q_{\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }=\frac{2Y_{\mathrm{BCT}} }{Y_{\mathrm{AoT}} +Y_{\mathrm{BCT}} +Y_{\mathrm{AI}} }\nonumber \\ \end{aligned}$$
(13)

Wave power reflection and transmission coefficients can also be derived from the respective branch admittances (Mynard 2011). The ratio of BCW power to the initial \(\mathrm{FCW}_{1}\) power in the AoT is given by

$$\begin{aligned} \frac{wp_{\mathrm{AoT}} \left( {\mathrm{BCW}} \right) }{wp_{\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }= & {} -\left( {R^{P}} \right) ^{2}\nonumber \\= & {} -\frac{\left( {Y_{\mathrm{AoT}} -Y_{\mathrm{BCT}} -Y_{\mathrm{AI}} } \right) ^{2}}{\left( {Y_{\mathrm{AoT}} +Y_{\mathrm{BCT}} +Y_{\mathrm{AI}} } \right) ^{2}} \end{aligned}$$
(14)

Noting that this expression is always negative, and to differentiate wave reflection arising from an admittance increase versus decrease, the wave power reflection coefficient (\(R_{wi} \)) for a general ‘reflected wave’ (RW) is defined as

$$\begin{aligned} R^{wp}=\mathrm{sign}\left( {Y_{\mathrm{AoT}} -Y_{\mathrm{BCT}} -Y_{\mathrm{AI}} } \right) \frac{wp_{\mathrm{RW}} }{wp_{\mathrm{FCW}} } \end{aligned}$$
(15)

such that \(R^{wp}> 0\) if the FCW is reflected as a (pressure-increasing) backward compression wave (BCW), \(R^{wp}> 0\) if the FCW is reflected as a (pressure-decreasing) backward expansion wave and \(R^{wp}= 0\) if the junction is well-matched. Note that a BCW is more likely to arise if the BCT and/or AI are smaller or stiffer.

Transmission of \(\mathrm{FCW}_{1}\) wave power from AoT to AI is governed by

$$\begin{aligned} T_{\mathrm{{AoT,AI}}}^{wp}= & {} \frac{wp_ {\mathrm{AI}} \left( {\mathrm{FCW}_{1} } \right) }{wp_ {\mathrm{AoT}} \left( {\mathrm{FCW}_{1} } \right) }=\frac{Y_ {\mathrm{AI}} }{Y_ {\mathrm{AoT}} }\left( {T^{P}} \right) ^{2}\nonumber \\= & {} 4\frac{Y_{\mathrm{AoT}} Y_{{ \mathrm AI}} }{\left( {Y_{\mathrm{AoT}} +Y_{\mathrm{AI}} +Y_{\mathrm{BCT}} } \right) ^{2}} \end{aligned}$$
(16)

Importantly, the transmission of \(\mathrm{FCW}_{1}\) power from AoT to AI is dependent on \(Y_{BCT}\), with greater \(T_{\mathrm{AoT,AI}}^{wp} \) at lower \(Y_{BCT} \), such as would occur with a stiffer or smaller BCT. Similarly,

$$\begin{aligned} T_{\mathrm{AoT,BCT}}^{wp}= & {} \frac{wp_\mathrm{{BCT}} \left( {\mathrm{FCW}_{1} } \right) }{wp_\mathrm{{AoT}} \left( {\mathrm{FCW}_{1} } \right) }=\frac{Y_\mathrm{{BCT}} }{Y_\mathrm{{AoT}} }\left( {T^{P}} \right) ^{2}\nonumber \\= & {} 4\frac{Y_\mathrm{{AoT}} Y_\mathrm{{BCT}} }{\left( {Y_\mathrm{{AoT}} +Y_\mathrm{{AI}} +Y_\mathrm{{BCT}} } \right) ^{2}} \end{aligned}$$
(17)

which shows that a stiffer or smaller AI will favour \(\mathrm{FCW}_{1}\) power transmission into the BCT.

Consider now a BCW travelling from the AI towards the AoT, as may arise due to reflection of \(\mathrm{FCW}_{1}\) at the site of coarctation. This wave will be partially transmitted into the AoT as a BCW (that subsequently increases LV load), but also partially transmitted into the BCT as a second forward compression wave (\(FCW_2)\). The magnitude of this \(\mathrm{FCW}_{2}\) can be predicted via

$$\begin{aligned} T_{\mathrm{AI,BCT}}^{wp} =\frac{wp_{\mathrm{BCT}} \left( {\mathrm{FCW}_{2} } \right) }{wp_{\mathrm{AI}} \left( {\mathrm{BCW}} \right) }=4\frac{Y_{\mathrm{AI}} Y_{\mathrm{BCT}} }{\left( {Y_{\mathrm{AoT}} +Y_{\mathrm{AI}} +Y_{\mathrm{BCT}} } \right) ^{2}}\nonumber \\ \end{aligned}$$
(18)

Similarly, transmission of the BCW into the AoT is governed by

$$\begin{aligned} T_{\mathrm{AI,AoT}}^{wp} =\frac{wp_{\mathrm{AoT}} \left( {\mathrm{BCW}} \right) }{wp_{\mathrm{AI}} \left( {\mathrm{BCW}} \right) }=4\frac{Y_{\mathrm{AI}} Y_{\mathrm{AoT}} }{\left( {Y_{\mathrm{AoT}} +Y_{\mathrm{AI}} +Y_{\mathrm{BCT}} } \right) ^{2}}\nonumber \\ \end{aligned}$$
(19)

From Equations (18) and (19), it can be seen that, all else being equal, a stiffer or smaller AoT leads to greater \(\mathrm{FCW}_{2}\) power in the BCT, while a stiffer or smaller BCT leads to greater transmission of BCW power into the AoT.

The passage of waves will alter instantaneous hydraulic pressure power (\(\Pi ={ PQ}\)). It was previously shown (Mynard and Smolich 2016) that incremental changes in the forward and backward components of hydraulic pressure power (\(d\Pi _{\pm } \)) are related to (non-time corrected) wave power (\(d\pi _{\pm } =dP_{\pm } dQ_{\pm } \)) as follows:

$$\begin{aligned} d\Pi _{\pm } \simeq 2\frac{P_{\pm } }{dP_{\pm } }d\pi _{\pm } \simeq 2\frac{Q_{\pm } }{dQ_{\pm } }d\pi _{\pm } \end{aligned}$$
(20)

If we define wave power and hydraulic power reflection coefficients as

$$\begin{aligned} {R^{wp}=\frac{d\pi _{-}}{d\pi _{+} }}, {R^{\Pi }=\frac{d\Pi _{-} }{d\Pi _{+} }} \end{aligned}$$
(21)

it follows from Equation (20) that these two coefficients are related via

$$\begin{aligned} R^{\Pi }\simeq R^{wp}\frac{P_{-} }{P_{+} }\frac{dP_{+} }{dP_{-} }\simeq R^{wp}\frac{Q_{-} }{Q_{+} }\frac{dQ_{+} }{dQ_{-} } \end{aligned}$$
(22)

Since \(d\pi _{\pm } =dP_{\pm } dQ_{\pm } \), it can further be shown that

$$\begin{aligned} R^{\Pi }\simeq R^{Q}\frac{P_{-} }{P_{+} }\simeq R^{P}\frac{Q_{-} }{Q_{+} } \end{aligned}$$
(23)

Thus, (1) wave power and hydraulic power reflection coefficients are related but not equal; and (2) \(R^{\Pi }\) is determined by an interaction between propagating waves (\(dP_{\pm }\) and \(dQ_{\pm } \)) and wave potential (absolute \(P_{\pm } \) and \(Q_{\pm } \)). Similar analysis could be performed for hydraulic power transmission coefficients, but will not be presented here.

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Mynard, J.P., Kowalski, R., Cheung, M.M.H. et al. Beyond the aorta: partial transmission of reflected waves from aortic coarctation into supra-aortic branches modulates cerebral hemodynamics and left ventricular load. Biomech Model Mechanobiol 16, 635–650 (2017). https://doi.org/10.1007/s10237-016-0842-x

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