Dynamic Effects of Aortic Arch Stiffening on Pulsatile Energy Transmission to Cerebral Vasculature as A Determinant of Brain-Heart Coupling

Aortic stiffness increases with age and is a robust predictor of brain pathology including Alzheimer’s and other dementias. Aging causes disproportionate stiffening of the aorta compared with the carotid arteries, reducing protective impedance mismatches at their interface and affecting transmission of destructive pulsatile energy to the cerebral circulation. Recent clinical studies have measured regional stiffness within the aortic arch using pulse wave velocity (PWV) and have found a stronger association with cerebrovascular events than global stiffness measures. However, effects of aortic arch PWV on the transmission of harmful excessive pulsatile energy to the brain is not well-understood. In this study, we use an energy-based analysis of hemodynamic waves to quantify the effect of aortic arch stiffening on transmitted pulsatility to cerebral vasculature, employing a computational approach using a one-dimensional model of the human vascular network. Results show there exists an optimum wave condition—occurring near normal human heart rates—that minimizes pulsatile energy transmission to the brain. This indicates the important role of aortic arch biomechanics on heart-brain coupling. Our results also suggest that energy-based indices of pulsatility combining pressure and flow data are more sensitive to increased stiffness than using flow or pressure pulsatility indices in isolation.

Theoretical indicators of an optimum wave condition. As an introduction to the energy-based analysis used in this paper, it is helpful to consider the theoretical relation for the transmitted power through waves. Following the analysis of Alastruey et al. 23 and Passerini 24 , the governing equations for pressure and flow propagation that are employed later in this work can be linearized about a reference state in the space-time domain. Under the assumption of periodicity for the propagated pressure and flow waves inside the vasculature, the solutions for pressure p(x, t) and flow q(x, t) at a time t and at a position x can be sought as harmonic waves of the form where p 0 and q 0 are the (possibly complex) amplitudes of the pressure and flow at = x t ( , ) (0, 0), ω is the frequency of the oscillation, and k is the wavenumber. It can be shown 23 that for a wavenumber k, the corresponding dispersion relation for temporal frequency ω is given by and the subsequent phase velocity c w = ω/k (of particular interest in this work) is given by w C k C k 2 2 2 1 3 3 where = − i 1 and where the constants C 1 , C 2 , and C 3 are combinations of physical parameters that ultimately account for wall compliance, flow inertia and the resistance to the flow. www.nature.com/scientificreports www.nature.com/scientificreports/ Assuming that pressure amplitude p 0 is real-valued, a mass conversation argument yields a resulting complex amplitude for flow as a function of phase velocity 23  where R and I denote the real and imaginary parts of a complex number, respectively. The complex-valued amplitude of the flow given by Eq. (1b) indicates a (frequency-dependent) phase-shift between the pressure p and the flow q that is expressed in Eq. (4b). The corresponding instantaneous transmitted power, derived by multiplying the propagated pressure and flow at a point x, is hence given by px t q x t p q c e p q c e t k x c ( ( , )) ( ( , )) 1 2 ( ( ) ) 1 2 ( ) sin(2 2 ( ) ( )), (5) denotes the complex amplitude of the flow rate and ϕ c ( ) w is a (frequency-dependent) phase difference.
Even for the linearized system of governing equations considered in the above analysis, one can note the complex (nonlinear) dependencies on the wavenumber for oscillation frequency ω (Eq. (2)) and for wave speed c w (Eq. (3)). In particular, these dependencies are prominent in the corresponding transmitted power given by Eq. (5); they may be reasonably assumed to be further nonlinear when incorporating more elastic/viscoelastic effects as well as wave reflections that result from interfaces in vasculature (see Methods). This very simplified analysis suggests that extrema may be found in pulsatility power curves as a function of both frequency (i.e., HR) and wave speed (i.e., PWV). This seems to be the case even in the absence of wave reflections (i.e., branching of the vasculature) and the fully-coupled nonlinear formulation considered later in this paper. Thus, the primary motivation of this work is to investigate these dynamic effects through a computational approach that treats a more physiologically relevant and physically accurate viscoelastic hemodynamics model for understanding the behavior of these physical variables on energy transmission to the brain.
All in all, wave dynamics in a compliant tube depend on three parameters: 1) fundamental frequency of the propagating waves, 2) wave speed as a function of material properties, and 3) reflection sites 25 . In what follows, we have focused primarily on investigating the effect of the first two parameters on the transmitted pulsatile power to the brain. However, the relative change of aortic arch stiffness with respect to the branches will affect the wave reflections as well, and hence the third is implicitly studied through the physiological relevance of the physical and mathematical models.

Methods
Physical model. A validated 1D model of the vascular network based on space-time variables has been employed in this study 26,27 . 1D arterial models have been shown to be a powerful tool for studying hemodynamics and wave dynamics in both large systemic arteries 28 as well as the entire adult circulation 29 . The physical model used in this study consists of 55 larger systemic arteries, where each artery is modeled as a visco-elastic tube characterized by its diameter, length, Young's modulus, viscosity and wall thickness. To consider the effect of viscoelasticity 30 , the Voigt-type model (a combination of a linear spring and a linear viscous dashpot connected in parallel 31 ) has been employed. The arterial wall is assumed to be thin, incompressible, homogenous and isotropic. In this study, our focus was to investigate the effect of proximal aorta stiffness on pulsatility transmission into the brain. For this purpose, the first three portions of the aorta (1-ascending aorta, 2-proximal aortic arch feeding the brachiocephalic and left common carotid arteries, and 3-distal aortic arch feeding the left vertebral and left subclavian arteries) were altered while all the other segments of the aorta and systemic vasculature were kept constant ( Fig. 1(a)).
At the inlet, we impose a physiological flow wave at the aortic root as shown in Fig. 1(b). The blood is assumed to be an incompressible Newtonian fluid with density of ρ = 1050 kg/m 3 and viscosity of µ = 4 mPa•s. Different levels of aortic arch rigidity are considered by employing multiplicative factors of a minimum rigidity level E x ( ) 1 that corresponds to the baseline PWV of c 1 initially prescribed to the model. The baseline properties for aortic segments that were altered in this study are presented in Table 1. In this table, c in and c out refer to the wave speed at the inlet and outlet of the segment, respectively. Mathematical and computational model. Conservation of mass and momentum applied to a 1D impermeable and deformable tubular control volume of an incompressible Newtonian fluid, flowing with a constant axisymmetric velocity profile, yields the system of equations where x is the axial coordinate along the vessel, t is the time, A(x, t) is the cross-sectional area of the lumen, U(x,t) is the average axial velocity, p(x, t) is the average internal pressure over the cross-section, and f is the friction force www.nature.com/scientificreports www.nature.com/scientificreports/ per unit length. For mathematical simplification it is assumed that the Coriolis coefficient (velocity shape factor) is unity, resulting in a flat velocity profile for the 1D model and hence a corresponding friction force per unit length of µπ = − f U 22 32 . In order to close the system of Eq. (6) for the three unknowns A(x, t), U(x, t) and p(x, t), a constitutive relation between the sectional pressure p and area A can be implemented by a Voigt-type viscoelastic tube law. This relationship accounts for the FSI of the problem and can be derived as 33 ext 0 0 0 where p ext is the constant external pressure and A 0 is the constant cross-sectional area at equilibrium state (p, U) = (p ext , 0). The spatially-varying functions G x ( ) and λ x ( ) are related to the elastic and visco-elastic properties of the arterial wall, respectively, and can be given in terms of material properties as is the wall thickness, and ϑ is the Poisson's ratio of the wall (taken to be ϑ = 1/2 assuming the wall is incompressible). Note that following the definition of the local PWV in terms of area and pressure, the wave speed can be written in terms of the elasticity factor G x ( ) as The system of partial differential equations (PDEs) in Eqs. (6 and 7) can be represented in matrix form as  www.nature.com/scientificreports www.nature.com/scientificreports/ . In order to solve this system of hyperbolic PDEs, a discontinuous Galerkin scheme can be employed for simplicity and fast convergence without causing excessive dispersion or diffusion errors 34 U where ϕ x ( ) is an arbitrary function on the domain Ω and is the standard Ω L ( ) 2 inner product. The discrete form of the conservative representation in Eq. (12) can be given by 35 where, following a traditional Galerkin approach, the superscript δ indicates that the variable is approximated in the finite space of piecewise polynomial vector functions (the trial space) and F u is the approximation of the flux at an interface. In the trial space, the expansion basis is chosen to be Legendre polynomials due to their orthogonality with respect to the Ω L ( ) e 2 inner product. Hence, the approximated solution on each elemental region δ U e can be expanded to order M as is the elemental mapping. Substituting Eq. (14) into Eq. (13) and letting ϕ = δ δ U e e yields a system of M+1 ordinary differential equations (ODEs) for each ˆ( ) yields the coefficients required to reconstruct the physical solution given by Eq. (14). To calculate the fluxes at each interface between elements, F u is decomposed into the elastic term F e u and the viscous term F v u . The elastic term is determined by solving the Riemann problem and the viscous term can be treated by the average of the lower and upper limits in an elemental region. In order to numerically resolve ˆt U ( ) e j at a discrete time t = + t n 1 , a second-order Adams-Bashforth time integration scheme is applied to Eq. (15) for each j = 0, …, M and e = 1, …, N el . This yields the iterative sequence with a time-step ∆t 35 . A physiological flow wave ( Fig. 1(b)) has been applied as an inlet flow to the aortic root and scaled to give a cardiac output of 5.2 l/min for any given HR. At arterial segment junctions and bifurcations ( Fig. 1(a)), the boundary conditions are prescribed by enforcing conservation of mass and continuity of the total pressure ρ + . p U 5 2 . By decomposing the governing system of equations (Eq. (6a,b)) into the characteristic variables, the system can be interpreted in terms of forward and backward traveling waves. At any bifurcations and junctions, we have six unknowns: (A U , www.nature.com/scientificreports www.nature.com/scientificreports/ only reach the bifurcation by the forward traveling wave, while within the daughter vessels the information can only reach the bifurcation by the backward traveling wave. Therefore, the first three equations can be obtained by imposing that the characteristic variables in each vessel (parent and daughters) remain constant ( = dW dt / 0, where W is the characteristic variable) 35 . The other three independent equations can be obtained by requiring the conservation of mass and continuity of the momentum balance. The latter condition leads to continuity of the total pressure at the boundary. It has been shown that energy losses at arterial segment junctions only change the mean pressure and flow by less than 0.5% 36 , hence they are disregarded in the model. Finally, at terminal boundaries, three-element RCR Windkessel models are employed as 0D lumped parameters that act as the outflow boundary condition on pressure p(t) and flow q(t) = AU at each peripheral branch. This is given by the ODE for inflow resistance R 1 (that matches the characteristic impedance of the terminal vessel), peripheral compliance C and outflow resistance R 2 -all of which are chosen within the average physiological range 37 . A validated code called "Nektar" was used to solve the discretized equation 16,33,[38][39][40][41] . This code has been developed for solving the nonlinear 1D equations of blood flow in a given network of compliant vessels subjected to boundary and initial conditions. Importantly, the code has been validated against in vitro 33,36 and in vivo 28,39,42 experiments. A Linux operating system has been used to compile the code on a standalone workstation equipped with an Intel Core i7 CPU (6 cores and 3201 MHz) with 32GB memory. Each simulation is run at a time-step of ∆ = µ t 10 s. At least 10 cardiac cycles are simulated in order to ensure that a periodic steady state is reached. The results are then processed and analyzed using MATLAB (The MathWorks, Inc., MA, USA). Hemodynamic analysis. The total power P total transmitted to the brain over a cardiac cycle of length T is calculated as the average of the product of the pressure p(t) and the flow q(t) in each brain segment. The steady power P s is computed as the product of mean pressure p mean and mean flow q mean in each segment. The pulsatile transmitted power P pulse is the difference between the total power and the steady power. Each of these power quantities are respectively given by Common clinical parameters such as flow pulsatility index FPI and pressure pulsatility index PPI 20 are defined as where q q and min m ax (resp. p p and min m ax ) are the minimum and maximum flow (resp. pressure) transmitted to the brain during a cardiac cycle.

Results
Simulations are run for five different levels of aortic arch PWV, starting from the baseline PWV of a healthy individual (c 1 ) and moving towards different levels by multiplicative factors of c 1 given by c 2 = 1.25c 1 , c 3 = 1.5c 1 , c 4 = 2c 1 , c 5 = 3c 1 (see values in Table 1). Each case has been run for eight HRs (30, 47, 63, 75, 94, 126, 150, and 189 beats per minute (bpm)). In all simulations, the Cardiac Output (CO), the peripheral resistance (PR), the terminal compliance, the shape of the inflow wave and all the outflow boundary conditions are kept constant.
Physiological accuracy of the model. A sample of flow and pressure in the left common carotid artery is shown in Fig. 2. The expected fiducial features of pressure and flow waveforms, including the pressure inflection point, the pressure dicrotic notch, and the peaks of the flow (Q 1 and Q 2 ), can be seen in Fig. 2(a,b). Figure 3 demonstrates the comparison between our simulated data with clinical data (reproduced from data by Hashimoto et al. 20  Effect of heart rate on transmitted pulsatility to the brain. Figure 4(a) gives P pulse computed by Eq.
(20) as a function of HR for different levels of the aortic arch PWV. As mentioned previously, the CO is kept constant for all cases. Results are computed from the left common carotid artery hemodynamic waveforms (the only cerebral branch that is directly connected to the aortic arch). As HR increases, the value of P pulse decreases until the HR reaches an optimum point where P pulse is minimized. P pulse increases with HR beyond this optimum point ( Fig. 4(a)). Note that this phenomenon is present for all different multiplicative factors of aortic arch PWV. Figure 4(b) demonstrates the pulsatile power as a function of the aortic arch PWV at different HRs. As expected, pulsatile power in the carotid artery increases at all HRs when the aortic arch PWV increases. Figure 5 depicts a 3D interpolation mapping of the pulsatile power with respect to the aortic arch stiffness (as measured by PWV) and the HR. There is an optimum wave condition region in which pulsatile power transmission is minimized (red arrow in Fig. 5). This optimal region occurs at the baseline aortic arch PWV (for a normal adult) around a value of HR = 75 bpm.
Flow and pressure pulsatility indices versus pulsatile power percentage. Figure 6 compares pulsatility indices and PPP at different aortic arch PWVs for three cases of the HR: (1) a normal heart rate, (2) the highest simulated heart rate (HR = 189 bpm), and (3) the lowest simulated heart rate (HR = 30 bpm). This range of HRs covers a large domain of different wave conditions.

Discussion
In this study, we have employed a physiologically accurate computational model of the systemic vasculature to investigate the effect of aortic arch stiffening on the transmission of excessive wave pulsatility to the cerebral circulation. Our results suggest that: (1) there exists an optimum wave condition in the aorta that minimizes the harmful pulsatile energy (power) transmission to the brain, (2) at different wave conditions (i.e. different HR and aortic arch PWV), this optimum wave condition occurs around a value near the normal human HR (75 bpm), and (3) an index based on pulsatile power (i.e., a percentage of it) is a more sensitive measure for excessive pulsatility transmission to the brain compared to conventional measures such as pressure and flow pulsatility indices alone.
Transmission of arterial pulsatility to the brain has long been known to promote vascular events such as ischemic and hemorrhagic stroke. More recently, hypertension has been shown to promote dementia, accounting for up to 30% of cases 43 . The hemodynamic mechanisms underlying these associations have not been well characterized. Aortic stiffening is the primary cause of systolic hypertension with aging 44 . In our prior work we have shown that the aortic arch stiffening that occurs with aging 45 is a much more powerful predictor of insult to the microvasculature in the brain than blood pressure or the presence of hypertension treatment 14 . Our results affirm that aortic stiffening does indeed increase transmission of harmful pulsatility to the brain. More importantly we also saw that this increase was several folds more severe when other parameters such as HR also become suboptimal. Results from simulations have been compared to published clinical data in order to verify the clinical relevancy of the computational model (Fig. 3). As it has been demonstrated, the calculated carotid flow augmentation index and pressure augmentation index from simulation data are well within the range of clinical data, and follow similar trends. This confirms the physiological accuracy of our study for purposes of investigating pulsatility transmission to the cerebral vasculature. This may help us identify with much greater accuracy those persons at risk of cerebrovascular events and accelerated brain aging due to harmful effects of excessive arterial pulsatility.
We studied the effect of aortic arch stiffening on the pulsatile energy transmission to the brain across a physiological range of HRs while keeping CO(=5.2 L/min) and other vascular parameters constant. Our results show that there is an optimum HR at which the transmitted pulsatile energy to the brain is minimized (Fig. 4(a)). The pulsatile energy decreases with increasing HR until it reaches this minimum value. Beyond the value, waves transmitted to the brain start acting destructively and, as a result, the pulsatile power starts elevating as HR increases.  Table 1) and HR of 75 bpm.

Scientific RepoRtS |
(2020) 10:8784 | https://doi.org/10.1038/s41598-020-65616-7 www.nature.com/scientificreports www.nature.com/scientificreports/ The existence of an optimum wave reflection has been shown in different contexts related to ventricular workload in animals 21,46 , in-vitro experimental data 21 , and computational data 22 . Results are consistent with previous studies that have suggested the idea that aortic wave optimization is one of the design characteristics found in the mammalian cardiovascular system 21 . However, the connection of the wave optimization in a heart-brain coupling framework has been thus far unknown. In this work, we have demonstrated (Fig. 5) the presence of an optimum wave condition -found near the normal human heart beat-for the transmitted energy to the brain across different aortic arch rigidities and HRs.
Unfortunately, as people age and become frail, the resting HR increases 47 even when it is impaired from appropriately increasing in response to physical exertion 48 . Effects of harmful pulsatile energy transmission to the brain that result from aortic stiffening with aging is likely compounded by a harmful interaction with increased HR among frail elderly at rest and by a decreased HR during exertion. This finding has not been reported before from prior in-vivo experiments that involve evaluating arterial stiffness and blood flow for a person at rest. In vivo-work fails to capture the impact on cerebral hemodynamics of the changes in HR that occur throughout the day due to physical exertion.
Pressure and flow pulsatility indices (PPI and FPI) are the conventional dimensionless parameters for monitoring hemodynamic pulsatility transmission to the brain. Figure 6 displays results based on the PPI and FPI. As expected, these indices capture the effect of the aortic arch stiffness on pulsatility, and they additionally support the presence of an optimum wave condition when passing from a low HR to a high HR. In other words, increasing or decreasing the HR (the green and blue curves in Fig. 6) will render the transmitted wave suboptimal. Since aortic aging affects the transmitted flow and pressure waves to the brain simultaneously, there is a need to employ  a parameter which considers the combined effects of pressure and flow pulsatilities. Hence, we have utilized an energy-based index defined as the ratio of the pulsatile power over the total power transmitted to the brain per cardiac cycle. We have compared the performance of this index with PPI and FPI. The results show that the increase in PPP is much more significant than the other two indices. This suggests that the PPP is more sensitive to changes in wave dynamics and can provide better insight into the effect of aortic arch stiffness and its subsequent behavior on excessive pulsatile power transmission to the brain. Additionally, on the effect of the HR, it has been found that flow pulsatility is more consistent with PPP. Both the FPI as well as the PPP have their highest values at higher HRs (blue curves in Fig. 6), while the PPI has its highest value at lower HRs (green curve in Fig. 6). This work shows that pressure and flow pulsatility may not be considered as interchangeable measures of cerebral pulsatility. A combined consideration of pressure and flow are needed to properly understand the power transmitted to the brain, and future clinical studies should include both assessments.
A limitation of this study lies in the 1D vasculature model formulation. The model used here may not necessarily reveal all aspects of flow distribution in the arterial network, especially those of the cerebral arteries and the morphology of carotid arteries. There are numerous mechanisms that may impact transmission of pulsatility to the brain which are still under investigation. Schubert et al. 49 have recently demonstrated that the contorted shape of the distal internal carotid artery may attenuate the flow pulsatility and hence may provide a protective effect on downstream cerebral vasculature. In another study, Seong et al. 50 investigated the dilation of carotid sinus to reduce the blood flow pulsatility to protect the brain. Further three-dimensional modeling and phase contrast MR at the skull based should be done to investigate these phenomena. In addition, a detailed exploration of how pulsatility is conveyed in the brain and the effect of the circle of Willis on the transmitted pulsatility to the brain are beyond the scope of the current work. In this work, we have focused on the interaction between the aorta and the cerebral circulation; the model has demonstrated capability of capturing the main features of flow and pressure wave pulsatility in larger arteries (Fig. 2). Therefore, it is a good starting point for investigating the effects of arch stiffness on transmitted pulsatility to the brain. Future works involves studying the widespread  www.nature.com/scientificreports www.nature.com/scientificreports/ variability in the circle of Willis and its potential effects on the transmission of arterial pulsatility into different cerebrovascular beds.
Additionally, employing a linear model to describe the dynamics of vessel walls may introduce errors relating to the wall stress relaxation. However, it has been shown previously that, under normal physiological conditions, this error is not very significant 33 . A further limitation of the current model is the approximation of the dynamics of the heart as a flow source imposed as a flow wave at the inlet. Although in general the heart is neither a flow nor pressure source, the behavior of the normal heart is closer to a flow source, and this interpretation has been employed in literature to validate the 1D model and has been shown to be a reasonable approximation 21,22 .

Conclusion
We have demonstrated that at different aortic arch rigidities, there is an optimum wave condition that minimizes the pulsatile energy transmitted to the brain. This optimum condition occurs near the normal HR and remains constant across a wide range of aortic arch stiffnesses. Based on an energy-based analysis of the waves at the carotid artery, pulsatile power percentage was used as an index to consider the combined effects of pressure and flow changes as the aortic arch become stiffer. This non-dimensional parameter was compared across different wave conditions with pulsatility indices that are based on pressure and flow. Results demonstrate that pulsatile power percentage can capture the transmitted pulsatility to the brain more clearly than the other pulsatility indices due to a higher sensitivity to different wave conditions. Previous work has discussed pathological wavesdefined as abnormalities in aortic and coronary wave dynamics-as a potential trigger towards cardiac death in the presence of the cardiovascular disease 51 . The pathology in the wave dynamics and the detection of wave condition signatures for different aortic arch rigidities may provide further insight into the underlying wave dynamics of the arterial system, particularly for the brain-heart coupling portion of the vasculature. Understanding the physics can potentially be a first step towards contributing in the development of new therapeutic strategies for neurodegenerative diseases like Alzheimer's dementia.