The mathematical model used in the present work is based on previous work conducted in our laboratory (Reymond et al., 2009, 2011). This model of the cardiovascular system includes a detailed description of the circulation in the main systemic arteries as well as a model for cardiac contractility. Blood circulation in the arterial network is described by the one-dimensional (1-D) form of the Navier–Stokes equations coupled with a constitutive law for the arterial wall elasticity. At the terminal sites, the arteries are connected with three-element Windkessel models that represent the periphery. This arterial model has been thoroughly validated against in vivo human data and has been demonstrated to accurately predict pressure and flow curves throughout the arterial network (Reymond et al., 2009, 2011).

At the proximal boundary, the arterial tree is connected to a 0-D model of the heart’s LV, which is represented by a time-varying elastance function (Suga and Sagawa, 1974; Sagawa et al., 1977). A pressure source feeds the cardiac model, assumed to have a constant value (filling pressure – Pfill). The systolic function of the LV is dictated by a linear end-systolic pressure-volume relation (ESPVR) equal to E_es⋅(V_LV−V_d), where E_es is the end-systolic elastance and V_d is the dead volume (Sagawa et al., 1977). The diastolic relaxation is described by an exponential end-diastolic pressure-volume relation (EDPVR) equal to P₀⋅exp(β⋅V_LV), where P_0 is the dead pressure and β a diastolic stiffness parameter (Burkhoff et al., 2005). At any given moment, the pressure-volume relation is described by the combination of the ESPVR and EDPVR, weighted according to a time-varying activation function, ∈ (t):

The simulation results for the three hemodynamic states were processed using pulse wave analysis. Concretely, we calculated the following hemodynamic parameters: (1) aortic systolic blood pressure (aSBP), (2) aortic diastolic blood pressure (aDBP), (3) MAP, (4) aortic pulse pressure (aPP), 5) augmentation pressure (AP) defined according to the characteristic inflection point or “shoulder” on the aortic pressure waveform as proposed by Murgo et al. (1980), (6) augmentation index (AIx) calculated as the ratio of the AP over the pulse pressure (Murgo et al., 1980). Additionally, the pressure waveforms were classified to either Type A or Type C based on the timing of the inflection, where the Type A phenotype has the peak systolic pressure occurring after the shoulder and cAIx10%, and the Type C pressure waveform has the peak systolic pressure preceding the inflection point and cAIx0 (Murgo et al., 1980).

The aortic pressure and flow waves were also used in a subsequent frequency-based wave separation analysis. The input impedance was calculated as the ratio of the Fourier transformed pressure over flow signals. The characteristic impedance (Zc) was isolated after averaging the input impedance modulus in the frequency range from 3 to 9 harmonics (Westerhof et al., 1972). The total pressure wave was then separated into its forward and backward components as proposed by Westerhof et al. (1972):

The backward and forward wave amplitudes were computed as well as their ratio, hereby called the reflection coefficient. The forward wave’s relative contribution to the total pulse pressure was also calculated as the ratio of the forward wave amplitude over the pulse pressure, PPf/PP.

The complete analysis as described in Figure 2 was repeated for two additional baseline model configurations. More specifically, the short-term and long-term effects of an 80% decrease in proximal aortic compliance were investigated de novo using an average 30 and 70 year-old model, according to our previously published aging cardiovascular model (Pagoulatou and Stergiopulos, 2017). The arterial parameters of these models were adjusted according to previous literature, i.e., arterial compliance was adjusted based on the expected evolution of central and peripheral pulse wave velocity with age and peripheral resistance was tuned to achieve the expected increase in MAP. Venous return was increased with aging in order to keep the cardiac output constant despite the increased afterload. Systolic and diastolic LV properties were also altered to incorporate the effects of age-induced hypertrophy and diastolic stiffening. More details on the derivation of the aging models can be found in the original publication (Pagoulatou and Stergiopulos, 2017).

Table 4 summarizes the simulation results we obtained after using the model configurations for a 30 and 70-year old adult for (i) baseline, (ii) acutely after banding, and (iii) after LV remodeling. Acutely after banding, we note for both cases a significant decrease in the total arterial compliance accompanied by an increase in the Zc, which is however more prominent for the young compliant model (loss of 46% of the total arterial compliance for the young subject versus 30% for the old subject). Aortic banding leads to an acute increase in the AP and AIx, which is subsequently alleviated after LV remodeling for both models, although the effect is less visible for the old model (Table 4). Additionally, both cases display the expected trend for an increase in SBP and PP with banding, which is driven by the significant enhancement of the forward pressure wave. Indeed, the contribution of the forward wave to the total pulse pressure increases throughout the manipulation for both cases, for the young subject from 0.69 at baseline to 0.78 acutely to 0.88 after LV remodeling and for the old subject from 0.74 at baseline to 0.77 acutely to 0.82 after LV remodeling.

In this work, we investigated the acute and long-term impact of proximal aortic compliance decrease on central arterial hemodynamics. Concretely, we used a physiologically relevant mathematical model of the cardiovascular system to simulate compliance decrease due to non-stenotic aortic banding and the subsequent remodeling of the LV as a response to pressure overload. We demonstrated that our mathematical model’s predictions were highly consistent with aortic banding experiments in swine. The major findings of this study can be summarized as follows: (a) reduction of the proximal aortic compliance leads acutely to hypertension, with an increase in the aortic SBP and PP, which is sustained yet slightly alleviated after LV remodeling, (b) the primary mechanism for the increase in PP due to banding is an increase in the forward wave amplitude, which is even more enhanced after LV remodeling, (c) additionally to an increase in its magnitude, the forward pressure wave alters its shape after LV remodeling, adopting a pronounced upstroke and an earlier peak, and (d) LV remodeling is the principal cause of the transformation of the pressure waveform from the Type A to Type C phenotype.

This study supplements previous literature that has explored acute and chronic effects of reduced aortic compliance on hemodynamics by making use of the banding procedure. In the present, aortic banding was achieved via an 80% reduction of the proximal aortic compliance, which yielded a 40% reduction in the total arterial compliance. This is in line with the current thinking that the proximal aorta accounts for approximately half of the total arterial compliance.

Note that in this work, only proximal aortic banding was considered while peripheral compliance was left unaltered. The motivation behind this choice comes from the facts that (a) the compliance of the proximal aorta is a major component of afterload regulating pulsatility, and (b) there are processes, such as physiological aging or aortic reconstruction with prosthetic grafts, which decrease mainly or even exclusively the proximal aortic compliance, thereby impairing cardiovascular function.

As expected, aortic compliance reduction was immediately linked with an increase in systolic blood pressure and pulse pressure. This can be understood as the result of an increase in the aortic Zc and the subsequent amplification of the forward pressure wave (Figure 5). Indeed, in our analysis, the forward wave amplitude was almost doubled after banding and the relative contribution of the forward wave to the total pulse pressure increased by 13%. The amplified forward wave also drove the amplification of the backward wave. However, there was a minor change in the relative ratio between the two wave amplitudes, i.e., the reflection coefficient. We might attribute this minor effect on the value of the reflection coefficient to the fact that the major reflection sites in the arterial circulation are at the level of the abdominal bifurcations and other peripheral sites, the geometry and compliance of which was not altered in our simulations.

Our findings have important implications for our understanding of the development of hypertension, particularly for the paradigm of cardiovascular aging. Concretely, the aging process is associated with progressive stiffening of the arteries (14) and the subsequent gradual increase in the central SBP and PP with advancing age, often referred to as isolated systolic or “old age” hypertension (Yip et al., 2009). Traditionally, it is hypothesized that the stiffer arterial tree produces pronounced reflections (O’Rourke and Nichols, 2004). Hypertension is, therefore, the result of the arrival of these augmented reflections back to the heart. Nevertheless, in this line of thinking, we do not take into consideration the fact that aging induces non-uniform arterial stiffening, whereby the proximal aorta becomes significantly stiffer and peripheral compliance hardly changes (Boutouyrie et al., 1992; Kimoto et al., 2003). In light of the evidence provided in the present study, it is likely that old age hypertension might be primarily the result of an augmentation of the forward wave ejected by the heart due to proximal stiffening rather than solely increased reflections coming back from the periphery. This hypothesis has also been investigated and proven plausible in our previous modeling work (Pagoulatou and Stergiopulos, 2017). Similar findings of the relative contribution of the forward wave to the development of hypertension have also been reported in clinical studies on old adults (Mitchell et al., 2004) as well as children and adolescents (Zocalo et al., 2018).

The heart and the arterial network form a coupled system and adapt to maintain optimal coupling conditions. Increased arterial stiffness and hypertension can trigger LV remodeling under the form of concentric hypertrophy, which is a powerful mechanism to normalize tensile stress exerted on the heart. This mechanism has been thoroughly described in the literature; it is supported by multiple previous observations, including evidence of increased wall thickness under pressure overload, and at later stages LV dilatation (Xie et al., 2013). In the present work, we simulated LV adaptation in response to arterial compliance decrease and the resulting pressure increase, similarly to our previous computational efforts (Pagoulatou and Stergiopulos, 2017). Our algorithm predicted that a compliance decrease by 40% leads to systolic pressure increase by 42%; the LV needs to increase its wall thickness and, consequently, its contractility by 35% to normalize tensile stress. This result is consistent with prior evidence highlighting the interaction between hemodynamic load and myocardial contractile state (Burkhoff et al., 2005).

Our analysis showed that the primary mechanism that drives the increase in SP and PP acutely after banding is the enhancement of the forward wave amplitude. After LV remodeling, there is an even more pronounced increase in the forward pulse pressure while the total pulse pressure decreases slightly. In other words, the forward wave accounts for a larger part of the total pulse pressure once the heart becomes stiffer. This finding is supported by our previous numerical and clinical observations (Pagoulatou et al., 2020), demonstrating that increased cardiac contractility leads to the generation of a more pronounced forward traveling wave, which is both larger in amplitude and steeper, both of which are direct consequences of the increase in LV contractility.

Building on this framework, we can also offer a plausible explanation as to why the pressure wave phenotype is altered after LV remodeling. The blood pressure and flow at each time point result from the interaction between the heart and the arterial system. At baseline, these two systems operate under optimal coupling conditions. When aortic banding is performed, the aortic compliance instantaneously decreases to a great extent. This entails that the coupling conditions between the heart and the arterial tree will change and likely become less favorable. Indeed, acutely after performing aortic banding, the aortic pressure phenotype has the characteristics of those found in patients with isolated systolic hypertension (Murgo et al., 1980) with a pronounced late systolic peak, i.e., the Type A phenotype. Subsequently, the pressure overload will trigger LV remodeling aiming to restore the matching between the two systems. When the heart becomes stiffer and more contractile, it will be able to meet the increased afterload by pumping a steeper, more pronounced forward pressure wave. It is precisely this alteration in the shape of the generated forward wave that drastically alters central hemodynamics, restoring the Type C pressure phenotype. These theoretical findings match very well the results obtained in the swine aortic banding experiments. We also reported similar acute changes in the pressure phenotypes and the forward wave amplitude on aortic valve stenosis patients undergoing transcatheter aortic valve replacement (TAVR) (Pagoulatou et al., 2020).

In light of this evidence, we might need to reconsider our understanding of Type A and Type C pressure phenotypes. Indeed, pressure waveforms characterized as Type A are often linked to old individuals with stiff arteries and increased wave reflections. At the same time, the Type C phenotype is assumed representative of young adults with elastic arteries and small reflections (Murgo et al., 1980). Here, we demonstrate that these phenotypes are, in fact, the result of the coupling between the cardiac and arterial systems and therefore depend on both components. The same comment can be extended for the AP and AIx, which are traditionally considered sole arterial measures dependent on reflections (Vlachopoulos et al., 2011).

The main study conclusions regarding the acute and long-term effects of afterload increase on central hemodynamics were confirmed after repeating the analysis for different baseline models (Table 4). More specifically, after using the parameters for the average 30-year old and 70-year old subject, we were able to verify that hypertension induced after aortic banding is achieved by an increase in the forward wave amplitude, which becomes the major contributor to total PP after LV remodeling. Interestingly, this mechanism seems to attenuate with advancing age (Table 4). This may be attributed to the fact that the proximal aorta of a young person is highly compliant and therefore aortic banding is expected to increase disproportionally the aortic impedance as compared to an older subject. Additionally, for all hemodynamic cases, we found that the shape of the pressure wave is similarly affected by the afterload increase and the subsequent adaptation of the LV, i.e., the AIx is increased acutely after banding and thereafter restored due to LV remodeling. Interestingly, the later observation is less prominent for the old subject, whose pressure waveform belongs to the characteristic Type A phenotype already at baseline.

When interpreting our results, the reader should consider that the data presented above reflect computational simulations. Nevertheless, this limitation is mitigated by the following facts: (a) the state-of-the-art model used here has been thoroughly validated and found capable of accurately representing the hemodynamics of healthy young as well as old individuals (Reymond et al., 2011; Pagoulatou and Stergiopulos, 2017), (b) the simulations were set up in order to closely imitate plausible hemodynamic states, based on the reported literature and widely accepted remodeling mechanisms, (c) the simulation results were in perfect qualitative agreement with previous experimental data performed in swine (Ioannou et al., 2009).

In our simulations, only proximal aortic stiffening was imposed while peripheral circulation was left unaltered. However, it is known that the peripheral compliance and geometry might adjust to increases in afterload, which could have important implications on the wave reflection profile. Moreover, previous studies have shown a marginally significant increase in the heart rate due to the banding of the aorta (Ioannou et al., 2009). This constitutes an acute mechanism of adaptation to the increased afterload, and its effect wears off in the long-term (Ioannou et al., 2009). As we had no available data that would allow us to simulate this compensatory mechanism, we decided to exclude it altogether from our study.

Previous literature has shown that vascular smooth muscle activation during hypertension can acutely improve wall buffering function by attenuating pressure oscillations and diminishing the stress tension on the vascular wall (Grignola et al., 2007). Additional long-term compensatory mechanisms that are related to the nervous system pressure regulation might come into play and alter cardiac contractility, arterial properties, preload and cardiac output (Nobrega et al., 2014). These mechanisms are not included in our computational model, which constitutes an important limitation. However, it should be highlighted that Ioannou et al. (2009) did not observe any significant changes in the arterial compliance, resistance and Zc between the acute and long-term stages in their experiments.

The remodeling process was simulated based on previous literature on the effect of increased afterload on cardiac function. Of note, we chose to keep the dead volume constant during this manipulation, given that data on V_d changes during remodeling were lacking. In vivo, invasive measurements of the LV pressure-volume loop would be needed in order to include such effects in our future work.

In our cardiac model, we assumed that the normalized activation function is not altered during banding and remodeling, although certain studies have shown a significant variation of the normalized elastance with afterload and introduced correction models (Shishido et al., 2000). This feature will be included in our future computational efforts. Finally, other cardiac contractility models have been proposed in the literature, such as the one developed by Lumens et al. (2009) that includes cavity and sarcomere mechanics. The use of such a model might be relevant for the investigation of the LV remodeling process. This possibility will be explored in future works.