Pressure and flow in the systemic arterial system

Impact of tapering of arterial vessels on blood pressure, pulse wave velocity, and wave intensity analysis using one-dimensional computational model

The angle of arterial tapering increases with ageing, and the geometrical changes of the aorta may cause an increase in central arterial pressure and stiffness. The impact of tapering has been primarily studied using frequency-domain transmission line theories. In this work, we revisit the problem of tapering and investigate its effect on blood pressure and pulse wave velocity (PWV) using a time-domain analysis with a 1D computational model. First, tapering is modelled as a stepwise reduction in diameter and compared with results from a continuously tapered segment. Next, we studied wave reflections in a combination of stepwise diameter reduction of straight vessels and bifurcations, then repeated the experiments with decreasing the length to physiological values. As the model's segments became shorter in length, wave reflections and re-reflections resulted in waves overlapping in time. We extended our work by examining the effect of increasing the tapering angle on blood pressure and wave intensity in physiological models: a model of the thoracic aorta and a model of upper thoracic and descending aorta connected to the iliac bifurcation. Vessels tapering inherently changed the ratio between the inlet and outlet cross-sectional areas, increasing the vessel resistance and reducing the compliance compared with non-tapered vessels. These variables influence peak and pulse pressure. In addition, it is well established that pulse wave velocity increases in an ageing arterial tree. This work provides confirmation that tapering induces reflections and offers an additional explanation to the observation of increased peak pressure and decreased diastolic pressure distally in the arterial tree.

A 1D computer model of the arterial circulation in horses: An important resource for studying global interactions between heart and vessels under normal and pathological conditions

Arterial rupture in horses has been observed during exercise, after phenylephrine administration or during parturition (uterine artery). In human pathophysiological research, the use of computer models for studying arterial hemodynamics and understanding normal and abnormal characteristics of arterial pressure and flow waveforms is very common. The objective of this research was to develop a computer model of the equine arterial circulation, in order to study local intra-arterial pressures and flow dynamics in horses. Morphologically, large differences exist between human and equine aortic arch and arterial branching patterns. Development of the present model was based on post-mortem obtained anatomical data of the arterial tree (arterial lengths, diameters and branching angles); in vivo collected ultrasonographic flow profiles from the common carotid artery, external iliac artery, median artery and aorta; and invasively collected pressure curves from carotid artery and aorta. These data were used as input for a previously validated (in humans) 1D arterial network model. Data on terminal resistance and arterial compliance parameters were tuned to equine physiology. Given the large arterial diameters, Womersley theory was used to compute friction coefficients, and the input into the arterial system was provided via a scaled time-varying elastance model of the left heart. Outcomes showed plausible predictions of pressure and flow waveforms throughout the considered arterial tree. Simulated flow waveform morphology was in line with measured flow profiles. Consideration of gravity further improved model based predicted waveforms. Derived flow waveform patterns could be explained using wave power analysis. The model offers possibilities as a research tool to predict changes in flow profiles and local pressures as a result of strenuous exercise or altered arterial wall properties related to age, breed or gender.

Derivation of aortic distensibility and pulse wave velocity by image registration with a physics-based regularisation term

Analysis of the cardiovascular system represents a classical problem in which the solid and fluid phases interact intimately, and so is a rich field of application for state-of-the-art fluid-solid interaction (FSI) analyses. In this paper, we focus on the human aorta. Solution of the full FSI problem requires knowledge of the material properties of the wall and information on vessel support. We show that variation of distensibility along the aorta can be obtained from four-dimensional image data using image registration. If pressure data at one point in the vessel are available, these can be converted to absolute values. Alternatively, values of pulse wave velocity along the vessel can be obtained. The quality of the extracted data is improved by the incorporation into the registration of a regularisation term based on the one-dimensional wave equation. The method has been validated using simulated data. For idealised vessels, the accuracy with which the distensibility and wave velocity can be extracted is high (1%-2%). The method is applied to six clinical datasets from patients with mild coarctation, for which it is shown that wave velocity along the aorta is relatively constant.

Comparative Study of Viscoelastic Arterial Wall Models in Nonlinear One-Dimensional Finite Element Simulations of Blood Flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.

Validation of a one-dimensional model of the systemic arterial tree

A distributed model of the human arterial tree including all main systemic arteries coupled to a heart model is developed. The one-dimensional (1-D) form of the momentum and continuity equations is solved numerically to obtain pressures and flows throughout the systemic arterial tree. Intimal shear is modeled using the Witzig-Womersley theory. A nonlinear viscoelastic constitutive law for the arterial wall is considered. The left ventricle is modeled using the varying elastance model. Distal vessels are terminated with three-element windkessels. Coronaries are modeled assuming a systolic flow impediment proportional to ventricular varying elastance. Arterial dimensions were taken from previous 1-D models and were extended to include a detailed description of cerebral vasculature. Elastic properties were taken from the literature. To validate model predictions, noninvasive measurements of pressure and flow were performed in young volunteers. Flow in large arteries was measured with MRI, cerebral flow with ultrasound Doppler, and pressure with tonometry. The resulting 1-D model is the most complete, because it encompasses all major segments of the arterial tree, accounts for ventricular-vascular interaction, and includes an improved description of shear stress and wall viscoelasticity. Model predictions at different arterial locations compared well with measured flow and pressure waves at the same anatomical points, reflecting the agreement in the general characteristics of the "generic 1-D model" and the "average subject" of our volunteer population. The study constitutes a first validation of the complete 1-D model using human pressure and flow data and supports the applicability of the 1-D model in the human circulation.

Validity of the local nonlinear arterial flow theory: influence of the upstream and downstream conditions

The "local flow" theory provides a simple way to take into account the nonlinear convective terms associated with blood flow in large arteries. The assumption that blood velocity profiles vary slowly with the longitudinal coordinate allows a simple nonlinear resolution via a mathematical approximation. Although validated in vitro by its authors, this theory still needs to be verified in accordance with the range of variation of the hemodynamical parameters. This constitutes the aim of this work where we assess the validity of two models issued from this theory: the "direct model" using the pressure-gradient as an input for the calculations and the "indirect model" using the centre-line velocity. The assessment of these models is made by comparing their solutions to those of the linear theory using numerical simulations. Our main conclusion is that the indirect model has a wide range of validity while the direct one fails in the presence of a strong reflected wave.

Viscoelasticity modulates resonance in the terminal aortic circulation

We used an inertance-viscoelastic windkessel model (IVW) to interpret aortic impedance patterns as seen in the terminal aortic circulation of the dog, and to explain evident oscillatory phenomena in flow measurements. This IVW model consists of an inertance, L, connected in series with a viscoelastic windkessel (VW) where the peripheral resistance, Rp, is connected in parallel with a Voigt cell (a resistor, Rd, in series with a capacitor, C) to account for viscoelasticity. Pressure and flow measurements were taken from the terminal aorta, just downstream of the origin of renal arteries, in three anaesthetised open-chest dogs, under a variety of haemodynamic conditions induced by administering a vasoconstrictor agent (methoxamine) and a vasodilator (sodium nitroprusside). Mean pressure ranged from 40 to 140 mm Hg. The resistance Rp was calculated as the ratio of mean pressure to mean flow. Parameters L, C and Rd were estimated by fitting measured to model predicted flow waves. We found that prominent oscillations observed in flow waves, from midsystole to diastole, are related to resonance that occurs at a frequency, f(o), where reactance of inertance of blood motion matches the reactance of arterial compliance. Estimates of f(o) increased from 2.4 to 10 Hz with increasing pressure and showed a correlation with values of static elastic moduli plotted against mean pressure of dogs' peripheral arteries previously reported by others. Viscous losses, Rd, of arterial wall motion limited the amplitude of resonance peak. We conclude that viscoelasticity, rather than pure elasticity, is a key issue to interpret terminal aortic impedance as it relates to resonance.

Structured tree outflow condition for blood flow in larger systemic arteries

A central problem in modeling blood flow and pressure in the larger systemic arteries is determining a physiologically based boundary condition so that the arterial tree can be truncated after a few generations. We have used a structured tree attached to the terminal branches of the truncated arterial tree in which the root impedance is estimated using a semianalytical approach based on a linearization of the viscous axisymmetric Navier-Stokes equations. This provides a dynamic boundary condition that maintains the phase lag between blood flow and pressure as well as the high-frequency oscillations present in the impedance spectra. Furthermore, it accommodates the wave propagation effects for the entire systemic arterial tree. The result is a model that is physiologically adequate as well as computationally feasible. For validation, we have compared the structured tree model with a pure resistance and a windkessel model as well as with measured data.

Assessment of distributed arterial network models

The aim of this study is to evaluate the relative importance of elastic non-linearities, viscoelasticity and resistance vessel modelling on arterial pressure and flow wave contours computed with distributed arterial network models. The computational results of a non-linear (time-domain) and a linear (frequency-domain) mode were compared using the same geometrical configuration and identical upstream and downstream boundary conditions and mechanical properties. pressures were computed at the ascending aorta, brachial and femoral artery. In spite of the identical problem definition, computational differences were found in input impedance modulus (max. 15-20%), systolic pressure (max. 5%) and pulse pressure (max. 10%). For the brachial artery, the ratio of pulse pressure to aortic pulse pressure was practically identical for both models (3%), whereas for the femoral artery higher values are found for the linear model (+10%). The aortic/brachial pressure transfer function indicates that pressure harmonic amplification is somewhat higher in the linear model for frequencies lower than 6 Hz while the opposite is true for higher frequencies. These computational disparities were attributed to conceptual model differences, such as the treatment of geometric tapering, rather than to elastic or convective non-linearities. Compared to the effect of viscoelasticity, the discrepancy between the linear and non-linear model is of the same importance. At peripheral locations, the correct representation of terminal impedance outweight the computational differences between the linear and non-linear models.

Identification and physiological relevance of an exponentially tapered tube model of canine descending aortic circulation

The aim of this study was to evaluate the effect of incorporating aortic tapering in a tube model of descending aortic circulation. We described the descending aorta and its peripheral load by an exponentially tapered transmission tube terminating in a first-order, low-pass filter load. Under the assumption of adaptation between the transmission tube and the terminal load, the input impedance of this model was characterized by five free parameters, the characteristic impedance, Zce(0), at the tube entrance; the product, qde, between the tapering factor q and the tube length, de, the product ce(0)de, between the compliance, ce(0), at the tube entrance and the tube length; the time constant, tau ne, of the load and the peripheral resistance, Rp. We estimated these parameters making use of experimental pressure and flow measurements taken from the high descending aorta of three anaesthetized dogs. We contrasted the behaviour of this model with that of a competing model constituted by a uniform transmission tube also terminating in a first-order low-pass filter load. We compared the data fits and, with the aid of an extra measurement of pressure in the abdominal aorta, we tested the congruence between the estimates of the transmission tubes' parameters and the physical and geometrical properties of descending thoracic aorta. The tapered tube model showed a slightly better ability in fitting to experimental flow and reproducing input impedance data. However, the estimates of the transmission tube parameters failed to assess the physical properties of descending aorta. By contrast, the estimates of tube parameters provided by the uniform model allowed location of the junction between the tube and its terminal load in the abdominal aorta at level of major branches. These estimates were well correlated with the real system's properties. In conclusion, the complexity added to the uniform tube model by accounting for exponential aortic tapering gave rise only to a better curve fitting, but did not show any identifiable benefits regarding physiological interpretation of the physical properties of the descending aorta.

Wave propagation in coupled left ventricle-arterial system. Implications for aortic pressure

The objective of this study was to examine the effects of wave propagation properties (global reflection coefficient gamma IG; pulse wave velocity, c(ph); and characteristic impedance zeta(o) on the mechanical performance of the coupled left ventricle-arterial system. Specifically, we sought to quantify effects on aortic pressure (P(ao)) and flow Q(ao) while keeping constant other determinants of P(ao) and Q(ao) (left ventricular end-diastolic volume, V(ed), and contractility, heart rate, and peripheral resistance, R(s)). Isolated rabbit hearts were subjected to real-time, computer-controlled physiological loading. The arterial circulation was modeled with a lossless tube terminating in a complex load. The loading system allowed for precise and independent control of all arterial properties as evidenced by accurate reproduction of desired input impedances and computed left ventricular volume changes. While propagation phenomena affected P(ao) and Q(ao) morphologies as expected, their effects on absolute P(ao) values were often contrary to the current understanding. Diastolic (Pd) and mean (Pm) P(ao) and stroke volume decrease monotonically with increases in gamma G, c(ph), or zeta(o) over wide ranges. In contrast, these increase had variable effects on peak systolic P(ao) (Ps): decreasing with gamma G, biphasic with c(ph), and increasing with zeta(o). There was an interaction between gamma G and c(ph) such that gamma G effects on P(m) and P(d) were augmented a higher C(ph) and vice versa. Despite large changes in system parameters, effects on Pm and Ps were modest ( < 10% and < 5%, respectively); effects on Pd were always two to four times greater. Similar results were obtained when the single-tube model of the arterial system was replaced by an asymmetrical T-tube configuration. Our data do not support the prevailing hypothesis that P(s) (and therefore ventricular load) can be selectively and significantly altered by manipulating gamma G, c(ph), and/or zeta o.

Impedance and wave reflection in arterial system: simulation with geometrically tapered T-tubes

The aortic input impedance is simulated by an asymmetric T-tube model loaded with complex loads. A geometric tapering is incorporated to represent the vasculature, assuming a triangular distribution of the wave transmission paths. Parametric analyses using physiological data demonstrate that the predicted impedance and reflection coefficient spectrum (RCS) closely mimic the experimental data. The simulation also reveals several significant features. As diameter tapering can minimise the presence and influence of wave reflections, the impedance modulus stays relatively constant with two distinct minima. The frequency of first minimum of impedance modulus is evidence of the tube elasticity and load compliance in the lower extremity, and the frequency of second minimum is evidence of those in the upper extremity. The high-frequency portion of the impedance modulus is affected by the tube elasticity, but not by the load compliance. The impedance spectrum at higher frequencies shows no notable fluctuations corresponding to a decrease in blood or wall viscosity. Furthermore, the low-frequency range in RCS is dominated by the longer lower body tube, and the high-frequency range by the shorter upper body tube. This geometrically tapered T-tube is considered a more natural model for the description of the systemic arterial system.

Exponentially tapered t-tube model of systemic arterial system in dogs

This study determines the role of an asymmetric T-tube model as a representation of arterial mechanical properties. The model consists of two non-uniform tubes connected in parallel. The non-uniform properties of each tube include geometric and elastic tapering and each tube terminates in a complex load. Pulsatile pressure and flow velocity of the ascending aorta were measured in 10 closed-chest, anaesthetized dogs. An exponentially tapered transmission line is used to describe the non-uniform properties of the vasculature. The phase constant is a function of position along the path length due to geometric and elastic tapers. This non-uniform T-tube model makes it possible to fit the measured pressure waveform in the ascending aorta. Model parameters could be estimated and used to interpret the physical properties of the arterial system. The mathematical and experimental model impedance spectra are similar. There is a close correspondence between the impedance parameters derived from the non-uniform T-tube model and values computed from measurements on dogs. The results suggest that inclusion of tube tapering improves the mathematical model so that it closely represents the experimentally derived arterial impedance in closed-chest dogs. We conclude that the non-uniform properties of wave-transmission paths may play an important role in governing the behaviour of an asymmetric T-tube for the description of the arterial system.

Preoperative assessment and prediction of postoperative results in an artificial arterial network using computer simulation

A computer model has been developed that can be used to describe the human arterial system mathematically. It simulates the complex relationship of morphology and hydraulics in the vessel network. After entering patient data into a standard vessel model, the mean flow velocity, the flow direction, and the blood pressure at each specified point of the flow network can be calculated. The vessel picture can be altered and modified with the help of a graphic editor. Localized or diffuse stenoses, bypasses with simple or multiple anastomoses, end-to-end anastomoses, end-to-side anastomoses, etc., can be studied in terms of the hydraulic effects on the local situation or on the entire vessel system. Experimental results of ultrasonic mean flow data in vessel systems of leg and cerebral arteries of patients are compared with calculated values. The predicted and measured flow velocities show a mean difference of about 10% indicating that such a computer model may be successfully used in the optimal planning of bypass operations.

Computer simulation of arterial flow with applications to arterial and aortic stenoses

A computer model for simulating pressure and flow propagation in the human arterial system is developed. The model is based on the one-dimensional flow equations and includes nonlinearities arising from geometry and material properties. Fifty-five arterial segments, representing the various major arteries, are combined to form the model of the arterial system. Particular attention is paid to the development of peripheral pressure and flow pulses under normal flow conditions and under conditions of arterial and aortic stenoses. Results show that the presence of severe arterial stenoses significantly affects the nature of the distal pressure and flow pulses. Aortic stenoses also have a profound effect on central and peripheral pressure pulse formation. Comparison with the published experimental data suggests that the model is capable of simulating arterial flow under normal flow conditions as well as conditions of stenotic obstructions in a satisfactory manner.

The genesis of the pulse contours of the distal leg arteries in man

In order to clarify the genesis of the human pressure and flow pulse contours of the distal leg arteries, in particular the posterior tibial artery, pulse recordings were performed with transcutaneous techniques under normal conditions and in the state of strong vasodilatation (reactive hyperaemia) in the distal parts of the lower legs. From the experimental results it is concluded that the contour of the incident pressure wave arriving in the leg arteries is very similar to the pressure pulse contour of the abdominal aorta, while the resulting contour in the leg arteries is determined by this incident wave and superimposed reflected waves. The latter arise from positive reflection in the periphery of the lower legs. The travel in retrograde direction, are reflected negatively in proximal regions, particularly in the abdominal aorta, and appear again, with opposite sign, in the leg arteries. In addition, retrograde waves reflected positively at the aortic valve and then traveling in antegrade direction also influence the pulse contours. By considering this wave travel, the genesis of the characteristic contours of the pressure and flow pulses of the lower leg arteries is explained in a satisfactory way. This is demonstrated by a simplified graphical pulse construction as well as by the calculation of pulse contours on the basis of a theoretical tube model of the arterial system with the aid of a digital computer. The results of these calculations are discussed with respect to the findings of previous investigators who used analog and digital models of the arterial system.