Pressure and flow in the systemic arterial system

The dynamic characteristics of the proximal arterial system are studied by solving the nonlinear momentum and mass conservation equations for pressure and flow. The equations are solved for a model systemic arterial system that includes the aorta, common iliacs, and the internal and external iliac arteries. The model includes geometric and elastic taper of the aorta, nonlinearly elastic arteries, side flows, and a complex distal impedance. The model pressure wave shape, inlet and outlet impedance, wave travel, and apparent wave velocity compare favorably with the values measured on humans. Calculations indicate that: (i) reflections are the major factor determining the shape and distal amplification of the pressure wave in the arterial tree; (ii) although important in attenuating the proximal transmission of reflecting waves, geometric taper is not the major cause of the distal pressure wave amplification; (iii) the dicrotic wave is a result of peripheral reflection and is not due to the sudden change in flow at the end of systole; (iv) the elastic taper and nonlinearity of the wall elasticity are of minor significance in determining the flow and pressure profiles; and (v) in spite of numerous nonlinearities, the system behaves in a somewhat linear fashion for the lower frequency components.     

Influence of geometric taper on the derivation of the true propagation coefficient using a three point method

We studied the effect of geometric taper on the derivation of the true propagation coefficient from three pressures determined 10 cm apart ('three-point method'). For this purpose the true propagation coefficients of a uniform latex tube (length 50 cm, outer diameter 12.73 mm, Womersley phase velocity 6.23-6.42 ms-1 (1-10 Hz), Womersley damping coefficient 0.05-0.14 m-1 (1-10 Hz) and of a tapered latex tube (length 50 cm, outer diameter varying from 15.88 to 9.45 mm, in the middle section with same properties as the uniform tube) were determined. The real part of the true propagation coefficient (the damping coefficient) was compared with apparent damping, and with the damping coefficient calculated using Womersley's theory. The imaginary part of the true propagation coefficient (the phase coefficient) was expressed in terms of phase velocity. True phase velocity was compared with measurements of apparent phase velocity, foot-to-foot velocity, and calculations of phase velocity parameters Womersley's theory and the Moens-Korteweg equation. The results show that in the uniform tube the three-point propagation coefficient is in agreement with all other estimates. Taper causes an error in the three-point propagation coefficient. At some frequencies the damping is reversed to amplification (values up to -2 m-1) and the phase velocity may be both overestimated or underestimated (up to 50%). The overestimation of true damping as reported in the literature cannot be explained from vessel taper.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Effect of taper geometries and launch angle on evanescent wave penetration depth in optical fibers

A large penetration depth of an evanescent wave is the key to success for developing an ultra high-resolution fiber-based evanescent wave biosensor. Tapering the fiber and launching light at an angle has the potential of increasing the penetration depth of evanescent wave manifolds. The effects of tapering, launch angle and taper length of the fiber have been explored in detail using a ray-tracing model to calculate the highest possible penetration depth of the evanescent field. Evanescent wave penetration depths of the order of the size of living cells have been achieved by optimizing the parameters relating geometry of tapered fibers.     

Arterial pulse attenuation prediction using the decaying rate of a pressure wave in a viscoelastic material model

The present study examines the possibility of attenuating blood pulses by means of introducing prosthetic viscoelastic materials able to absorb energy and damp such pulses. Vascular prostheses made of polymeric materials modify the mechanical properties of blood vessels. The effect of these materials on the blood pulse propagation remains to be fully understood. Several materials for medical applications, such as medical polydimethylsiloxane or polytetrafluoroethylene, show viscoelastic behavior, modifying the original vessel stiffness and affecting the propagation of blood pulses. This study focuses on the propagation of pressure waves along a pipe with viscoelastic materials using the Maxwell and the Zener models. An expression of exponential decay has been obtained for the Maxwell material model and also for low viscous coefficient values in the Zener model. For relatively high values of the viscous term in the Zener model, the steepest part of the pulse can be damped quickly, leaving a smooth, slowly decaying wave. These mathematical models are critical to tailor those materials used in cardiovascular implants to the mechanical environment they are confronted with to repair or improve blood vessel function.     

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

     

Wave Propagation through a Viscous Fluid Contained in a Tethered, Initially Stressed, Orthotropic Elastic Tube

To give a realistic representation of the pulse propagation in arteries a theoretical analysis of the wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube is considered. The tube is assumed to be orthotropic and its longitudinal motion is constrained by a uniformly distributed additional mass, a dashpot and a spring. The fluid is assumed to be Newtonian. The analysis is restricted to propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. Elimination of arbitrary constants from the general solutions of the equations of motion of the fluid and the wall gives a frequency equation to determine the velocity of propagation. Two roots of this equation give the velocity of propagation of two distinct outgoing waves. One of the waves propagates slower than the other. The propagation properties of s lower waves are very slightly affected by the degree of anisotropy of the wall. The velocity of propagation of faster waves decreases as the ratio of the longitudinal modulus of elasticity to the circumferential modulus decreases; transmission of these waves is very little affected. The influence of the tethering on the propagation velocity of slower waves is negligibly small; transmission of these waves is seriously affected. In tethered tubes faster waves are completely attenuated.     

Role of tapering in aortic wave reflection: hydraulic and mathematical model study

Pressure and flow have been measured simultaneously at six locations along the aorta of an anatomically correct 1:1 scale hydraulic elastic tube model of the arterial tree. Our results suggest a discrete reflection point at the level of the renal arteries based on (i) the quarter-wavelength formula and (ii) the comparison of foot-to-foot (c(ff)) and apparent phase velocity (c(app)). However, separation of the pressure wave into an incident and reflected wave at all six locations indicates continuous reflection: a reflected wave is generated at each location as the forward wave passes by. We did a further analysis using a mathematical transmission line model with a simple tapering geometry (length 50 cm, 31 and 11 mm proximal and distal diameter, respectively) for a low (0.32 ml/mmHg), normal (1.6 ml mmHg) and high (8 ml/mmHg) value of total arterial compliance. Using the quarter-wavelength formula, a discrete reflection point is found at x = 33 cm, the level of the renal arteries, independent of the value of total compliance. However, local analysis comparing c(ff) and c(app) does not reveal a marked reflection site, and the analysis of incident and reflected waves merely suggests a continuous reflection. We therefore conclude that the measured in vivo aortic wave reflection indices are the result of at least two interacting phenomena: a continuous wave reflection due to tapering, and local reflections arising from branches at the level of the diaphragm. The continuous reflection is hidden in the input impedance pattern. Using the quarter-wavelength formula or the classical wave separation theory, it appears as a reflection coming from a single discrete site, confusingly also located at the level of the diaphragm. Therefore, the quarter-wavelength formula and the linear wave separation theory should be used with caution to identify wave reflection zones in the presence of tapering, i.e., in most mammalian arteries.     

Wave Propagation in a Viscous Fluid Contained in an Orthotropic Elastic Tube

The problem of pressure wave propagation through a viscous fluid contained in an orthotropic elastic tube is considered in connection with arterial blood flow. Solutions to the fluid flow and elasticity equations are obtained for the presence of a reflected wave. Numerical results are presented for both isotropic and orthotropic elastic tubes. In particular, the pressure pulse, flow rate, axial fluid velocity, and wall displacements are plotted vs. time at various stations along the ascending aorta of man. The results indicate an increase in the peak value of the pressure pulse and a decrease in the flow rate as the pulse propagates away from the heart. Finally, the velocity of wave propagation depends mainly on the tangential modulus of elasticity of the arterial wall, and anisotropy of the wall accounts in part for the reduction of longitudinal movements and an increase in the hydraulic resistance.     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

Determination of viscoelastic moduli of arteries

A model for the arterial wall, motivated by wave propagation findings, is developed. The wall is taken to be a viscoelastic, orthotropic, prestressed shell which is materially characterized at any prestress level by ten incremental moduli. By using the momentum equations and the wave propagation characteristics for three axisymmetric modes, the ten moduli are found in terms of the three wave speeds, the three attenuation coefficients and the prestresses.     

A multiscale approach for modelling wave propagation in an arterial segment

A mathematical model of blood flow through an arterial vessel is presented and the wave propagation in it is studied numerically. Based on the assumption of long wavelength and small amplitude of the pressure waves, a quasi-one-dimensional (1D) differential model is adopted. It describes the non-linear fluid-wall interaction and includes wall deformation in both radial and axial directions. The 1D model is coupled with a six compartment lumped parameter model, which accounts for the global circulatory features and provides boundary conditions. The differential equations are first linearized to investigate the nature of the propagation phenomena. The full non-linear equations are then approximated with a numerical finite difference method on a staggered grid. Some numerical simulations show the characteristics of the wave propagation. The dependence of the flow, of the wall deformation and of the wave velocity on the elasticity parameter has been highlighted. The importance of the axial deformation is evidenced by its variation in correspondence of the pressure peaks. The wave disturbances consequent to a local stiffening of the vessel and to a compliance jump due to prosthetic implantations are finally studied.     

Fluid waves in renal tubules.

Autoregulation of renal blood flow is ineffective when arterial pressure perturbations occur at frequencies above 0.05 Hz. To determine whether wave propagation velocity to the macula densa is rate limiting, we estimated compliances of the proximal tubule and the loop of Henle, and used these values in a model of pressure and flow as functions of time and distance in the nephron. Compliances were estimated from measurements of pressures and flows in early proximal, late proximal, and early distal tubules in rats under normal and Ringer-loaded conditions. A model of steady pressure and flow in a compliant, reabsorbing tubule was fitted to these results. The transient model was a set of nonlinear, hyperbolic partial differential equations with split, nonlinear boundary conditions, and was solved with finite difference methods. The loop of Henle compliance was larger than the proximal tubule compliance, and impulses in glomerular filtration rate were attenuated in magnitude and delayed in time in the loop of Henle. Simulated step forcings revealed a similar pattern. Periodic variations of GFR were attenuated at frequencies greater than 0.05 Hz, and there was a delay of 5 s between variations in GFR and macula densa flow rate. The high compliance of the loop slows wave propagation to the macular densa and reduces the amplitude of high frequency waves originating in the glomerulus, but other parts of the signal chain also contribute to the slow response of macula densa feedback.     

Studies on the relations between pressure and flow in the circulation

Simple theoretical models are proposed for the study of the interdependence between cardiac contraction, arterial pressure, and capillary drainage. The relation between pressure and flow is derived for a model of branching distensible tubes taking into account the finite pulse wave velocity. Equations are derived both for the case where the pulse wave is non-distorted and for the case where the wave is damped and distorted to a limited extent. Following the model of J. W. Remington and W. F. Hamilton (1947), the former case is applied to the larger arteries. Expressions are developed for the stroke volume, cardiac ejection, and systolic arterial storage in both the steady and non-steady states. Expressions for the percentage discrepancy involved in the computation of these quantities from a single tube model as contrasted with a multi-branched model are derived. For typical cases these discrepancies are small and thus credence is lent to the further use of the simpler single tube model which requires fewer independent parameters. It is also shown that the formulae for stroke volume and arterial storage are only slightly sensitive to changes in pulse wave velocities, and that for some purposes it would seem permissible to assume an infinite velocity. The problem of capillary drainage is discussed, and the consequences of equations developed for the case of a distorted wave are shown to compare favorably with published experimental data. An approximate boundary condition for capillary drainage is derived. Finally, A. V. Hill's velocity load equation for muscle is used to obtain a first approximation for the velocity of cardiac contraction in terms of the initial arterial pressure, the heart radius, and the parameters of the heart musculature. It is shown how methods developed for stroke volume determination from the pressure contour may be used to estimate the heart and “air chamber” parameters. Use of these parameters and those obtained by other independent measurements permits the principle variables to be determined numerically.     

Spectra from stimulated Raman scattering of short relativistically strong laser pulses in an underdense plasma

A set of equations describing large-angle stimulated Raman scattering (SRS) of a short, relativistically strong laser pulse propagating in an underdense plasma is derived and investigated numerically. It is shown that the SRS spectrum depends strongly on the pulse shape. If a pulse with a sharp leading edge excites a strongly nonlinear wake wave, the scattering occurs in relativistic electron flows and is accompanied by the Doppler frequency shift. When the electron flow is directed oppositely to the pulse propagation direction, the frequency upshift is maximum for the direct-backward SRS and decreases with decreasing scattering angle.     

Attenuation of flow disturbances in tapered arterial grafts

Flow disturbances in tapered arterial grafts of angles of taper between 0.5 and 1.0 deg were measured in vitro using a pulsed ultrasound Doppler velocimeter. The increase in transition Reynolds numbers with angle of taper and axial distance was determined for steady flow. The instantaneous centerline velocities were measured distal to a 50 percent area stenosis (as a model of a proximal anastomosis), in steady and pulsatile flow, from which the disturbance intensities were calculated. A significant reduction in post-stenotic disturbance intensity was recorded in the tapered grafts, relative to a conventional cylindrical graft. In pulsatile flow with a large backflow component, however, there was an increase in disturbance intensity due to diverging flow during flow reversal. This was observed only in the 1.0 deg tapered graft. These findings indicate that taper is an important consideration in the design of vascular prostheses.     

Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.     

Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube

This paper studies the peristaltic transport of a viscoelastic fluid (with the fractional second-grade model) through an inclined cylindrical tube. The wall of the tube is modelled as a sinusoidal wave. The flow analysis is presented under the assumptions of long wave length and low Reynolds number. Caputo's definition of fractional derivative is used to formulate the fractional differentiation. Analytical solutions are developed for the normalized momentum equations. Expressions are also derived for the pressure, frictional force, and the relationship between the flow rate and pressure gradient. Mathematica numerical computations are then performed. The results are plotted and analysed for different values of fractional parameter, material constant, inclination angle, Reynolds number, Froude number and peristaltic wave amplitude. It is found that fractional parameter and Froude number resist the flow pattern while material constant, Reynolds number, inclination of angle and amplitude aid the peristaltic flow. Furthermore, frictional force and pressure demonstrate the opposite behaviour under the influence of the relevant parameters emerging in the equations of motion. The study has applications in uretral biophysics, and also potential use in peristaltic pumping of petroleum viscoelastic bio-surfactants in chemical engineering and astronautical applications involving conveyance of non-Newtonian fluids (e.g. lubricants) against gravity and in conduits with deformable walls.