Dynamic Analysis of Aortic Flows: Transient Response of Flow-Compliance and Flow-Resistance When Subjected to Sinusoidal Variations of Input Pressure

The ascending aorta plays a pivotal role in maintaining efficient cardiovascular function by serving as a compliant reservoir that buffers the pulsatile nature of blood flow. During each cardiac cycle, the aorta expands and recoils, regulating blood pressure and ensuring steady perfusion of distal tissues [1]. The property that characterizes this dynamic behavior, known as “flow capacitance,” is essentially the innate ability of the aorta to accommodate changes in blood volume that are concomitant with severe fluctuations in the internal pressure profile. Accurate quantification of aortic flow capacitance is of significant clinical and engineering interest, as it provides insights into conditions such as hypertension, atherosclerosis, arrhythmia, and aneurysms. Such insights can also inform the development and evaluation of vascular grafts and stents, especially for personalized medicine and formulating novel therapeutic strategies [2–4].

Early attempts to model the mechanical behavior of the aorta and other large vessels often employed linear elasticity assumptions, treating the vascular wall as a simple, homogeneous, and linearly elastic material [5]. Although these models offered initial approximations and insights, their inherent simplifications fail to capture the complex, nonlinear, and anisotropic nature of aortic tissue. In reality, the aortic wall is composed of multiple layers with collagen, elastin, and smooth muscle cells arranged in intricate microstructures. These constituents endow the aorta with a markedly nonlinear stress–strain response, which becomes particularly important at physiological and supra-physiological pressure ranges [6–8]. Consequently, linear elasticity models often underestimate vessel compliance, resulting in inaccurate predictions of hemodynamic parameters and limited applicability in modern clinical and research settings.

To address these limitations, researchers have increasingly adopted hyperelastic constitutive models that can accommodate large deformations and nonlinear elasticity. Among these, the Mooney–Rivlin model has garnered significant attention for its relative simplicity and ability to approximate the mechanical response of biological soft tissues, including the ascending aorta [9]. Mooney–Rivlin and other hyperelastic models have been successful in improving the fidelity of capacitance predictions, while also elucidating the mechanical heterogeneity of the blood vessels. This can also help with guiding more accurate simulations of fluid–structure interactions (FSI) [10]. When compared against experimental data, hyperelastic models have repeatedly demonstrated superior congruence, underscoring their importance in biomedical engineering workflows such as the design of vascular implants and the computational modeling of patient-specific hemodynamics [11].

In addition, the use of advanced constitutive models allows researchers to explore the vessel's response under various pressure inputs and boundary conditions. Considering the ascending aorta as part of a Windkessel-like system that incorporates both resistance and compliance, sinusoidal input pressures analysis can reveal how complex loading histories influence vessel deformation and blood flow distribution [12]. By accurately capturing the nonlinear mechanical behavior, hyperelastic models open avenues for more realistic numerical simulations, enhancing our understanding of pathophysiological states and improving the predictive capabilities of computational hemodynamics tools [13]. Ultimately, these improvements translate into tangible benefits for clinical diagnostics, treatment planning, and the design of biomimetic prosthetics that better replicate the nuanced mechanical environment of the human aorta.

The aim of this study is to compare the predictions for flow capacitance obtained from a traditional linear elasticity model with that of a hyperelastic Mooney–Rivlin model. This comparative study was implemented using the ascending aorta as a representative large vessel. By utilizing this strategy, the improved accuracy of hyperelastic formulations was demonstrated along with their importance in capturing the true mechanical response under realistic physiological conditions. Furthermore, these analyses were extended to account for the nonlinear compliance of the aorta and to demonstrate how such nonlinear behavior influences the outlet pressure profiles under varying input conditions, thus providing a comprehensive perspective that can guide more informed clinical decisions and engineering designs.

In this study, the mechanical deformations in hyperelastic cylindrical conduits were related to the fluid flow parameters by utilizing numerical models that are analogous to those used in electrical circuit network analyses. By treating the vessel as a deformable “capacitor” and fluid flowrate as the equivalent of current that is subject to “resistance,” key expressions were derived for obtaining the equivalent flow capacitance, fluid flow resistance, stretch ratio, and pressure drops (i.e., flow potential). This approach provides a robust framework for parametric analysis and computational simulations, for better cognition of the nature (and behavior) of the flow transients in compliant vessels, such as the ascending aorta. It may be noted that the aim of this study is to estimate the values of flow capacitance arising from structural deformations in the flow conduits caused by the net pressure drop. Hence, to keep the analysis procedure simple and tractable the fluid flow portion of this study has been limited to Newtonian fluids. While it is well-established that blood exhibits non-Newtonian behavior under certain flow conditions, particularly in smaller vessels or at low shear rates, the assumption of Newtonian fluid dynamics remains widely used and justified in the study of blood flow in large arteries. Studies comparing Newtonian and non-Newtonian fluid models, such as one employing the Carreau model, have shown that the difference in systolic and diastolic flow velocities between the two models is relatively small (2% and 9%, respectively) [14]. Similarly, while non-Newtonian models may result in higher wall shear stress (WSS) predictions during systole (30% higher), the average diastolic WSS predictions differ by only 20% in favor of the Newtonian model. These findings suggest that the Newtonian assumption provides a reasonable approximation for large arteries, where shear rates are typically high and non-Newtonian effects are less pronounced [14]. Future studies can adapt the techniques developed in this study for non-Newtonian fluids (in addition to the Newtonian fluid models that were utilized in this study).

To describe the vessel's ability to accommodate volume changes against pressure variations, two different models for capacitance calculations were formulated, as follows:

1. Hyperelastic Mooney–Rivlin model:

Here v represents the Poisson's ratio, E is the Young's modulus, and t is the wall thickness. This model is formulated using a linear relationship between stress and strain, which may not accurately represent the mechanical behavior of biological tissues under large deformations (such as in the aorta) but serves as a good benchmark.

The mechanical deformation response of the vessel geometry that is subjected to fluctuating pressure and fluid flow transients can be modeled to be analogous to an “RC circuit,” where the capacitance (C) and resistance (R) are the key parameters that govern the transient response of the vessel for the imposed values of the pressure transients. This analogy simplifies complex and computationally laborious FSI models and facilitates the use of electrical engineering analysis tools for systems analysis (e.g., analogous to the lumped-capacitance models deployed in transient heat transfer problems). By incorporating these mechanical and electrical analogies, we can effectively simulate and study the dynamic behavior of compliant vessels under various loading conditions.

Typically, flow resistance arises from frictional effects along the pipe walls. This resistance is influenced by variables such as the length of the pipe, the viscosity of the fluid, and the pipe's radius, all of which are encapsulated in Poiseuille's equation for laminar flows (as shown in Eq. (2)). Correspondingly, the flow C represents the mechanical compliance of the conduit (or vessel). Hence, C represents the ability of the compliant walls of the vessel (or the flow conduits) to cyclically store and discharge hydraulic energy amid pressure variations or flow oscillations. Analogous to how a capacitor accumulates charge (or discharges electrical charge) under changing electrical voltage, the compliant walls of the pipe store fluid mass and hydraulic energy in response to pressure shifts (i.e., transients in the flow potential values). The formulation for flow capacitance accounts for the initial radius of the pipe, wall elasticity, and the stretch ratio.

In this analogy, the R and C components of the equivalent RC circuit are configured in a series–parallel arrangement, effectively mimicking the fluid dynamics through a single hyperelastic pipe. Within such a configuration, the differential flowrate (Q) is regulated as the flexible element temporarily retains a portion of the fluid, as shown in Fig. 1.

In conclusion, integrating these capacitance models within our methodology provides a comprehensive and versatile toolset for studying the effects of nonlinear elastic behavior on fluid flow in hyperelastic conduits. This approach enhances our ability to understand cardiovascular dynamics and can inform the development of medical devices and diagnostics.

In scientific modeling of the ascending aorta, the initial condition can be set as: $Po(0)=0$⁠. This enables a practical approach under specific experimental or theoretical frameworks, i.e., particularly useful in isolating specific vascular responses. This assumption facilitates the calibration of experimental setups and validation of measurement techniques by ensuring that any observed pressure changes are attributable solely to induced conditions, rather than pre-existing stresses. Additionally, in computational models, this initial condition can be used to simplify the boundary conditions, thus enhancing the clarity of the results obtained from these analyses, since these steps are intended for studying the effects of sudden physiological changes, such as surgical interventions or controlled pressure modulations. While this starting condition does not mirror the continuous baseline pressure observed under normal physiological states, it provides a clear, simplified baseline from which the dynamic responses of the aorta to external stresses can be accurately assessed, free from the complexities of pre-existing vascular pressures.

The goal of this study is estimating the flow capacitance properties of the ascending aorta, and is modeled using an equivalent cylindrical conduit. Two different approaches were leveraged for model formulations used in this study: (a) the hyperelastic Mooney–Rivlin model, which considers the complex behaviors of materials under stress; and (b) the traditional linear elasticity model, which simplifies these behaviors into straightforward stress–strain relationships. Our goal was to compare the efficacy of these models for predicting the dynamic behavior (and transient response) of tissue materials, akin to those in the human aorta, when subjected to pressure transients and variations in fluid flowrates. The parameters used for modeling the ascending aorta (along with the references and sources for the values for these parameters) are listed in Table 1. The specified geometric parameters and material properties for both the hyperelastic Mooney–Rivlin model and the traditional linear elasticity model are provided in Table 1. The parameters listed in Table 1 were used for calculating the flow capacitance of the ascending aorta (which is modeled as a cylindrical pipe).

The material and fluid properties used were average values typical for a healthy individual, without differentiation by sex or age (or disease progression). Blood pressure typically ranges from 80 mmHg (diastolic) to 120 mmHg (systolic) under normal conditions. The stretch ratio was estimated using Eq. (1) (which models the expansion of the ascending aorta). This value was then incorporated into the capacitance equation to quantify the compliance of the aorta (which is the ability of the aorta to store or discharge hydraulic energy), and is influenced by the corresponding dynamic mechanical properties. On the other hand, the traditional linear elasticity model was implemented using Young's modulus and Poisson's ratio, thus approximating the material behavior as nearly incompressible (i.e., using linear stress–strain behavior model). This simplified model does not account for the large deformations that are typical of hyperelastic materials like the aortic wall.

Table 2 shows results obtained from the two different strategies that were used to calculate the flow capacitance values of the ascending aorta. The predictions obtained from the hyperelastic Mooney–Rivlin model are consistent with the experimental data (obtained from the literature). This model predicts the flow capacitance value to be 2.1 × 10⁻⁹ m³/Pa, and is similar to the experimental data (2.85 × 10⁻⁹ m³/Pa). It's good at showing how the aorta behaves under different pressures because it takes into account how the materials and the shape of the aorta work together. On the other hand, the traditional linear model yields a much lower value of 1.36 × 10⁻¹⁰ m³/Pa, which is not consistent for predicting the characteristics of biological tissues, such as the aorta. So, using more detailed and nonlinear models, such as the hyperelastic Mooney–Rivlin model, could enable better evaluation of cardiac health conditions and enable more precise therapeutic strategies.

After performing the experimental validation of the the flow capacitance and fluid flow resistance estimates obtained in this study, the transient response of the ascending aorta for two specific input conditions was then estimated in this study, as follows:

• sudden changes that mimic the beat-to-beat shifts in the heart from diastolic to systolic pressures; and

• more gradual, rhythmic changes that represent a normal heartbeat cycle.

The model was then deployed to predict how the flow characteristics within the aorta respond to sudden and sharp changes in blood pressure. Essentially the kind of stress the artery could be subjected to multiple times a day due to a variety of disease conditions.

Diastolic Pressure: Set at 80 mmHg, this represents the quieter phase of the cardiac cycle when the heart relaxes and the chambers fill with blood, a lower-pressure state that's critical for understanding how the aorta behaves when the heart is at rest (Fig. 2).

Systolic Pressure: Set at 120 mmHg, this value captures the peak stress during the contraction of the heart (as shown in Fig. 3), when blood is actively being pumped out. It's a test of the strength and elasticity of the tissue in the aorta in the most stressful condition for normal and regular cardiac cycles. The cardiac response for these two phases provides insights into the ability of the aorta to manage and adapt to these rapid variations in the input conditions, thus shedding light on the structural integrity and functional resilience of the cardiac system.

A more realistic scenario was explored using the model developed in this study by simulating the response of the aorta to sinusoidal pressure changes that range between 80 mmHg and 120 mmHg, with the imposed frequency value of 1 Hz, which closely matches the normal and natural rhythm of heartbeats of healthy subjects, as shown in Fig. 4.

Such predictions obtained in this study provide insights into the dynamic pressure response profiles. The results demonstrate how the outlet pressure of the aorta adjusts to continuous, rhythmic changes. The aorta thus acts as a damper and reduces the acuteness of the pressure peaks to the organs located downstream in the human body. By softening the impact of each heartbeat, the aorta protects smaller vessels further downstream and protects organ systems (such as the kidneys and liver) from experiencing sudden pressure peaks.

In addition to examining outlet pressure responses, the models developed in this study also enable the estimation of the transient profiles of the effective flow capacitance and fluid flow resistance values, as they adjust to the sinusoidal pressure variations, and are as plotted in Fig. 5. Such predictions are crucial as they reveal how effectively the aorta can store and manage varying amounts of blood volume under cyclic pressure conditions that mimic natural heart rhythms. Understanding the changes in capacitance and resistance provides insights into the role of the aorta as well as the associated physical properties for influencing the resultant blood flow dynamics. This provides deeper insights into the role of the aorta in preserving cardiovascular health and its ability to respond to changes in heart function. This knowledge is vital for designing therapeutic strategies that can enhance the resilience and functionality of aorta under various physiological conditions.

The dynamic response of ascending aorta was explored in this study using Mooney–Rivlin model. Flow capacitance and viscous resistance to fluid flow enabled the detailed analyses of the transient response of the ascending aorta to sinusoidal pressure inputs. The results illuminate the critical role of aorta in damping the pulsatile nature of blood flow, thereby protecting organs and blood vessels downstream from the aortic arch. The numerical predictions obtained from the hyperelastic Mooney–Rivlin model enabled a more accurate representation of aortic compliance (compared to that of the traditional linear elasticity models), and were observed to be consistent with the experimental data available in the literature. This demonstrates the significance of deploying hyperelastic models for accurate predictions of the complex biomechanics of tissues, organs, and biosystems.

Blood flow was assumed to be consistent with Newtonian fluid flow models in this study. However, a more accurate representation would involve power-law type fluid flow models that were specifically developed in the literature for capturing the essence of blood flows more accurately and more consistently (i.e., non-Newtonian fluid flow models). Hence, the models developed in this study can be made more sophisticated in future research studies by expanding the scope of these models to include more patient-specific parameters. Such approaches could enhance the predictive power of these simulations, especially for clinical settings (e.g., personalized medicine). Investigating the impact of variable material properties within the aortic wall and including factors such as age, disease state, and genetic predispositions could enable more personalized insights into the cardiovascular health of individual patients. Additionally, integrating these models into real-time monitoring systems could potentially lead to the development and implementation of better and more effective diagnostic tools, which in turn can enhance their efficacy in targeted treatments and therapeutic strategies for tackling cardiovascular diseases. Furthermore, exploring the interactions between the aorta and other vascular components in a more integrated cardiovascular system model could yield deeper understanding of the overall hemodynamics and lead to improvements in surgical planning (as well as the design optimization of medical devices).

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

No data, models, or code were generated or used for this paper.