The measurement of afterload, vascular input impedance, and power distribution in aorto-femoral bypass

The effects of aorto-femoral bypass grafts on the vascular input impedance, and the ratio of pulsatile to total power were studied in eight dogs. Unilateral ileo-femoral stenosis was simulated and comparisons were made between the input impedance and power distribution in healthy and simulated disease situations. Input impedance magnitude spectra and phase were displayed graphically and it was shown that the presence of the simulated disease increases the ratio of pulsatile to total power as measured in the abdominal aorta from 7.5 to 14.8% (p less than 0.05). This suggests that the presence of the stenosis creates an impedance mismatch thus causing reflected waves to propagate proximally towards the heart. It was concluded that the way in which the heart transfers fluid power into the arterial bed was compromised by the presence of the ileo-femoral partial stenoses. It is further suggested that the system described in the paper makes it possible to quantitatively assess afterload, vascular input impedance and cardiovascular power distribution as a quasi-real time diagnostic procedure.     

Microcirculation impedance analysis in cat lung

An analysis of pulsatile microcirculation in cat lung, with special attention to the pulmonary microvascular impedance, is presented. A theoretical calculation is made on the basis of a complete set of experimental data on the morphology and elasticity of cat's pulmonary capillary sheets. The transfer matrix of the pulmonary microvascular impedance is obtained. The input impedance at the capillary entrance and exit are determined. The input impedance at the pulmonary arterial trunk is compared under various physiological conditions. It is shown that although the impact of pulmonary microcirculation on the relationship between the steady mean flow and pressure in the pulmonary arteries and veins is decisively large, the influence of the alveolar microcirculation on the input impedance at the pulmonary arterial trunk is small.     

Instantaneous blood flow, impedance and elastic properties computed in man from aortic pulse waves

A mathematical model of the pressure-flow relationship in the arterial circulation and its possible use in routine hemodynamics in man are described. The instantaneous blood flow velocity in the ascending aorta can be calculated from two pressure curves simultaneously recorded 5 cm apart. The mechanical aortic input impedance is computed from the recorded pressure and the calculated blood flow velocity curves. Projection of the pulse waves on a time-length plane leads to the determination of the pulse wave velocity and then an estimation of the elastic modulus of the aortic wall.     

Transient behavior of arterial systems in response to flow pulses

The arterial system is characterized geometrically as a system of branched elastic fluid lines whose frequency response is then known in the sense of the Fourier transform. For convenience of visualization the transient response of the individual tube to an input pressure-flow pair is represented in the time domain by kernel functions indicating the hybrid effect of viscosity and momentum on the line impedance and damping characteristics. The system as a whole is then divided into a zone of smaller tubes (below 3 mm) and a zone of larger tubes extending up to the aorta. It is shown that as a system each labyrinth of tubes below the 3 mm size may be replaced by a single impedance transformation which is dominantly resistive-capacitive. In the larger tubes, the transformation of the pulse wave at different stations is considered a point of interest. Therefore hand calculated examples are worked to derive the response of a system involving some of the larger vessels to a pressure or flow pulse of the typical shape seen near the heart. The result suggests that the dicrotic wave seen in the pressure pulse of mammals is due to the hybrid viscosity-momentum nature of the longer fluid lines in relation to the gradation of unmatched terminal impedances with which they are terminated. Damping of the higher frequency components is also accounted for.     

Four-point electrode measurement of impedance in the vicinity of bovine aorta for quasi-static frequencies

Results are presented here of experimental measurements using a four-point electrode technique to measure the complex impedance of bovine aorta submerged in Ringer's solution. Impedance measurements were taken at 250 microm intervals, ranging from 0 (the electrode directly on the surface of the tissue) to 10 mm. Frequencies ranged from 1 kHz to 10 MHz. Throughout this range, the measured impedance changed by an average of 400% when the electrode was 10 mm from the tissue as compared to when the electrode was in direct contact with the tissue. The change in impedance made it possible to determine when the electrode made contact with the arterial wall.     

Influence of the distensibility of large arteries on the longitudinal impedance: application for the development of non-invasive techniques to the diagnosis of arterial diseases

Background

This study shows that the arterial longitudinal impedance constitutes a hemodynamic parameter of interest for performance characterization of large arteries in normal condition as well as in pathological situations. For this purpose, we solved the Navier?CStokes equations for an incompressible flow using the finite element analysis method and the Arbitrary Lagrangian Eulerian (ALE) formulation. The mathematical model assumes a two-dimensional flow and takes into account the nonlinear terms in the equations of fluid motion that express the convective acceleration, as well as the nonlinear deformation of the arterial wall. Several numerical simulations of the blood flow in large vessels have been performed to study the propagation along an arterial vessel of a pressure gradient pulse and a rate flow pulse. These simulations include various deformations of the wall artery leading to parietal displacements ranging from 0 (rigid wall) to 15% (very elastic wall) in order to consider physiological and pathological cases.

Results

The results show significant changes of the rate flow and the pressure gradient wave as a function of aosc, the relative variation in the radius of the artery over a cardiac cycle. These changes are notable beyond a critical value of aosc equal to 0.05. This critical value is also found in the evolution of the longitudinal impedance. So, above a variation of radius of 5%, the convective acceleration, created by the fluid-wall interactions, have an influence on the flow detectable on the longitudinal impedance.

Conclusions

The interpretation of the evolution of the longitudinal impedance shows that it could be a mean to test the performance of large arteries and can contribute to the diagnosis of parietal lesions of large arteries. For a blood vessel with a wall displacement higher than 5% similar to those of large arteries like the aorta, the longitudinal impedance is substantially greater than that obtained in the absence of wall displacement. This study also explains the effects of convective acceleration, on the shape of the decline of the pressure gradient wave and shows that they should not be neglected when the variation in radius is greater than 5%.     

Input impedance for studying hydraulic parameters of the vessel system

Vascular input impedance can be used as an effective tool in estimating hydraulic parameters of arterial bed. These parameters may be interpreted as hydraulic resistance, elastance and inertance of particular sites of the arterial system. There is no significant difference between these parameters and those obtained through a direct measurement.     

Application of 1D blood flow models of the human arterial network to differential pressure predictions

A new application of 1D models of the human arterial network is proposed. We take advantage of the sensitivity of the models predictions for the pressure profiles within the main aorta to key model parameter values. We propose to use the patterns in the predicted differences from a base case as a way to infer to the most probable changes in the parameter values. We demonstrate this application using an impedance model that we have recently developed (Johnson, 2010). The input model parameters are all physiologically related, such as the geometric dimensions of large arteries, various blood properties, vessel elasticity, etc. and can therefore be patient specific. As a base case, nominal values from the literature are used. The necessary information to characterize the smaller arteries, arterioles, and capillaries is taken from a physical scaling model (West, 1999). Model predictions for the effective impedance of the human arterial system closely agree with experimental data available in the literature. The predictions for the pressure wave development along the main arteries are also found in qualitative agreement with previous published results. The model has been further validated against our own measured pressure data in the carotid and radial arteries, obtained from healthy individuals. Upon changes in the value of key model parameters, we show that the differences seen in the pressure profiles correspond to qualitatively different patterns for different parameters. This suggests the possibility of using the model in interpreting multiple pressure data of healthy/diseased individuals.     

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.     

Hydraulic input impedance measurements in physical models of the arterial wall

A white noise method was used to measure the hydraulic input impedance and transmission characteristics in physical models of an arterial system made of single, unbranched latex tubes. The experimentally obtained impedance curves show a rise in modulus and a positive phase at high frequencies in the absence of wave reflections. Using the impedance moduli in the presence of wave reflections, wave velocity and attenuation were calculated. The influence of wall nonlinearity on hydraulic impedance was also examined. It is concluded that, in the model used neither wave reflections nor wall nonlinearity can account for the deviations of the experimental impedance curves from the theoretically predicted ones. Impedance moduli in the presence of reflections may be used to study transmission characteristics (wave velocity and attenuation) of the model.     

A novel servo-control system that imposes desired aortic input impedance on in situ rat heart

To clarify the pathophysiological role of dynamic arterial properties in cardiovascular diseases, we attempted to develop a new control system that imposes desired aortic impedance on in situ rat left ventricle. In 38 anesthetized open-chest rats, ascending aortic pressure and flow waveforms were continuously sampled (1,000 Hz). Desired flow waveforms were calculated from measured aortic pressure waveforms and target impedance. To minimize the difference between measured and desired aortic flow waveforms, the computer generated commands to the servo-pump, connected to a side branch of the aorta. By iterating the process, we could successfully control aortic impedance in such a way as to manipulate compliance and characteristic impedance between 60 and 160% of their respective native values. The error between desired and measured aortic flow waveforms was 70 +/- 34 microl/s (root mean square; 4.4 +/- 1.4% of peak flow), indicating reasonable accuracy in controlling aortic impedance. This system enables us to examine the importance of dynamic arterial properties independently of other hemodynamic and neurohumoral factors in physiological and clinical settings.     

Using hydraulic input impedance in determination of changes in the arteries of different calibre

In anaesthetised cats, the arterial input impedance in combination with seven-element lumped-parameter model was used to estimate the resistance change in arteries of different caliber. The results show that the method gives reasonable estimations of changes in hydraulic resistance of arterial vessels of different caliber. We found that the method of vascular input impedance permits to reveal and assess quantitatively local constrictions and dilations as well as hemodynamically insignificant stenosis of conduit arteries.     

In-vitro investigation of a potential wave pumping effect in human aorta

An impedance pump – also known as Liebau pump – is a simple valveless pump that operates based on the principles of wave propagation and reflection. It has been shown in embryonic zebrafish that a similar mechanism is responsible for the pumping action in the embryonic heart during the early stages before valve formation. Recent studies suggest that the cardiovascular system is designed to take advantage of wave propagation and reflection phenomena in the arterial network. In this study we report the results of an in-vitro study that examines the hypothesis that the adult human aorta acts as a passive pump based on Liebau effect. A hydraulic model with different compliant models of an artificial aorta was used for a series of in-vitro experiments. Our result indicates that wave propagation and reflection can result in a pumping mechanism in a compliant aorta.     

Distribution of stress and strain along the porcine aorta and coronary arterial tree

The existence of a homeostatic state of stresses and strains has been axiomatic in the cardiovascular system. The objective of this study was to determine the distribution of circumferential stress and strain along the aorta and throughout the coronary arterial tree to test this hypothesis. Silicone elastomer was perfused through the porcine aorta and coronary arterial tree to cast the arteries at physiological pressure. The loaded and zero-stress dimensions of the vessels were measured. The aorta (1.8 cm) and its secondary branches were considered down to 1.5 mm diameter. The left anterior descending artery (4.5 mm) and its branches down to 10 microm were also measured. The Cauchy mean circumferential stress and midwall stretch ratio were calculated. Our results show that the stretch ratio and Cauchy stress were lower in the thoracic than in the abdominal aorta and its secondary branches. The opening angle (theta) and midwall stretch ratio (lambda) showed a linear variation with order number (n) as follows: theta = 10.2n + 63.4 (R(2) = 0.989) and lambda = 4.47 x 10(-2)n + 1.1 (R(2) = 0.995). Finally, the stretch ratio and stress varied between 1.2 and 1.6 and between 10 and 150 kPa, respectively, along the aorta and left anterior descending arterial tree. The relative uniformity of strain (50% variation) from the proximal aorta to a 10-microm arteriole implies that the vascular system closely regulates the degree of deformation. This suggests a homeostasis of strain in the cardiovascular system, which has important implications for mechanotransduction and for vascular growth and remodeling.     

Effects of diabetes and gender on mechanical properties of the arterial system in rats: aortic impedance analysis

We determined the effects of diabetes and gender on the physical properties of the vasculature in streptozotocin (STZ)-treated rats based on the aortic input impedance analysis. Rats given STZ 65 mg/kg i.v. were compared with untreated age-matched controls. Pulsatile aortic pressure and flow signals were measured and were then subjected to Fourier transformation for the analysis of aortic input impedance. Wave transit time was determined using the impulse response function of the filtered aortic input impedance spectra. Male but not female diabetic rats exhibited an increase in cardiac output in the absence of any significant changes in arterial blood pressure, resulting in a decline in total peripheral resistance. However, in each gender group, diabetes contributed to an increase in wave reflection factor, from 0.47 +/- 0.04 to 0.84 +/- 0.03 in males and from 0.46 +/- 0.03 to 0.81 +/- 0.03 in females. Diabetic rats had reduced wave transit time, at 18.82 +/- 0.60 vs 21.34 +/- 0.51 msec in males and at 19.63 +/- 0.37 vs 22.74 +/- 0.57 msec in females. Changes in wave transit time and reflection factor indicate that diabetes can modify the timing and magnitude of the wave reflection in the rat arterial system. Meanwhile, diabetes produced a fall in aortic characteristic impedance from 0.023 +/- 0.002 to 0.009 +/- 0.001 mmHg/min/kg/ml in males and from 0.028 +/- 0.002 to 0.014 +/- 0.001 mmHg/min/kg/ml in females. With unaltered aortic pressure, both the diminished aortic characteristic impedance and wave transit time suggest that the muscle inactivation in diabetes may occur in aortas and large arteries and may cause a detriment to the aortic distensibility in rats with either sex. We conclude that only rats with male gender diabetes produce a detriment to the physical properties of the resistance arterioles. In spite of male or female gender, diabetes decreases the aortic distensibility and impairs the wave reflection phenomenon in the rat arterial system.     

A theoretical evaluation of cardiac output as a function of mean arterial pressure in the human cardiovascular system

A correlation function of cardiac output and mean arterial pressure is presented for the human cardiovasular system. The function is developed using an energy transfer balance for a unit volume of blood which flows in the vascular system between the aorta and the vena cava. The energy transfer balance equates the energy utilized in the vascular system to the algebraic sum of the pulse energy, the kinetic energy and the potential energy in the vascular system. Each of these energies is defined in terms of the physiology of the cardiovascular system. Pulse energy is defined in terms of the work done by the heart on the aorta. Kinetic energy is defined in terms of the cardiac output and the potential energy is defined in terms of the diastolic pressure in the aorta. The utilization energy is equivalent to the energy transfer in the work done by the blood on the viscoelastic blood vessels, and to the frictional energy loss due to drag on the blood mass as it flows through the vascular system.The correlation function of cardiac output with mean arterial pressure demonstrates that the cardiac output is a double-valued function of the mean arterial pressure. The function also varies with the ratio of the fourth power of the Shear Modulus of the blood vessels to the third power of Young's Modulus. The function shows that mean arterial pressure minimizes for a cardiac output of approximately 51 per min when one holds the ratio of the elastic moduli constant. Further discussion indicates how clinicians can use the function, developed in this research, to interpret the experimental data obtained from cardiac output studies.     

Computer modeling of the human systemic arterial tree

A model of the human systemic arterial tree has been devised, based on a lumped-parameter-circuit approximate form. This model has been set up and studied on an analog computer. A feature of this simulation is the division of the arterial system into sections whose lengths are inversely proportional (approximately) to their cross-sectional area-or what is termed ‘equal-volume’ modeling.

Great care was exercised in the determination of the model parameters, using expressions for these parameters from a recent paper by Rideout and Dick on fluid flow in distensible tubes, with numerical values based on measurements reported in the medical literature.

The simulated pressure and flow waveforms obtained with the model compare favorably with data recorded from the normal adult human, and exhibit such well-known features as distal delay and peaking of pressure pulses. The aortic input impedance vs. frequency curve checks well against measurements on the human. The model also provides a simple means for determination of cardiac output, cardiac work and cardiac power under various assumed conditions such as variation of heart rate.     

High frequency characteristics of the arterial system

Pressure, flow and diameter were measured in the abdominal aorta of five anesthetized dogs during normal heart beats and heart beats with a superimposed impulse (generated by rapidly injecting a small volume of saline into the system). From Fourier analysis it was found that the impulse enhanced the amplitudes of the higher harmonics so that frequencies up to 80 Hz could be studied. Both the input impedance and apparent phase velocity above 20 Hz were independent of frequency and their average values were designated as characteristic impedance and true phase velocity. Average characteristic impedance for all five animals was 2.0 +/- 0.1 X 10(8) Nsm-5 and average phase velocity was 8.3 +/- 0.6 ms-1. Phase velocities calculated from characteristic impedance (1.76-2.39 X 10(8) Nsm-5) and from the slope of the pressure-diameter relation (0.102-0.25 X 10(-8) Nm-3) were similar to the true phase velocity as defined above (6.79-9.85 ms-1). It may be concluded that the input impedance converges to characteristic impedance and apparent phase velocity converges to phase velocity for high frequencies.     

Measurement of aortic input impedance in mice: effects of age on aortic stiffness

Mice are used with increasing frequency as models of human cardiovascular diseases, but significant gaps exist in our knowledge of vascular function in the aging mouse. We determined aortic input impedance spectra, pulse wave velocity, and augmentation index in adult (8-mo-old) and old (29-mo-old) mice to determine whether arterial stiffening occurred with age in mice as it does in humans. Pressure and blood velocity signals measured simultaneously from the same location in the ascending aorta were used to determine input impedance spectra (0-10 harmonics). The first minimum of the impedance modulus occurred at the second harmonic in adult mice but shifted to the fourth harmonic in old mice. Characteristic impedance (average of 2nd-10th harmonic) was 57% higher in old mice: 471 +/- 62 vs. 299 +/- 10 (SE) dyn.s.cm-3 (P < 0.05). Pulse pressure and augmentation index, determined from the aortic pressure signals, were also higher in old mice: 42 +/- 2.2 vs. 29 +/- 4.9 mmHg (P < 0.05) and 37 +/- 5 vs. 14 +/- 2% (P < 0.005). Aortic pulse wave velocity measured from the timing of upstrokes of the Doppler velocity signals was 45% higher in old mice: 416 +/- 22 vs. 286 +/- 14 cm/s (n = 3, P < 0.01). These results reproduce age-related findings reported in humans and confirm that mice may be used as models of age-related vascular stiffening.     

Pulse transmission and impedance characteristics of a non-uniform circulatory model

In recent years, the technique of non-uniform transmission lines has been utilized in the synthesis of lines feeding antennae from transmitters, thereby maintaining matching over a wide band of frequencies and with varying load. For this reason, an electrical model based on that technique stems from the resemblance between the two systems: in both the feeding medium is non-uniform, and it is necessary to maintain a good response over a wide range of frequencies. Our non-uniform transmission line model of the arterial system, introduced in an earlier publication, supplies comprehensive answers to many questions dealing with the research into this system. Whereas in our earlier model the ohmic resistance R was considered small and not considered in the calculations, it is included in the present study. We have calculated the variation of the input impedance with the frequency and distance from the source, the effect of occluded main branches, augmentation of the pressure wave, the relationship between body size and heart rate and the matching of impedances at large bifurcations. We found that our calculated results agree very well with the quantitative results measured by other investigators in the field.