Modeling of intraorgan arterial vasculature. II. Propagation of pressure waves

The dependence of the wave conductance in self-similar dichotomous models of intraorgan arterial vasculatures on the model parameters was studied. It was found that, with different sets of parameters, it is possible to simulate the suction effect induced by negative reflections of waves from arterial branchings and to model the resonance properties of arterial beds. It was shown that the choice of an adequate model for a given intraorgan arterial vasculature should be based on agreement between the biophysical characteristics of the model and the bed that characterize the propagation and reflection of pulse waves.     

Wave Transmission through an Assembly of Randomly Branching Elastic Tubes

Calculations are presented of the transmission of oscillations through an assembly of randomly branching elastic tubes, as a model of not only the major arteries, but also a peripheral vascular bed. It appears that the viscosity of the arterial wall must be the major source of attenuation in the larger arteries, while the viscosity of the blood plays a significant role only in the smaller vessels. In all situations, variations of cross-sectional area have a considerable effect on wave transmission, causing a general decrease in amplitude and an accentuation of reflection from the terminations. The effects of variation in cross-sectional area are sufficiently great to indicate that they should be included in future models of the arterial system. Finally, it is argued that because of the presence of random branching and elastic nonuniformity, the determination of the reflection coefficient for a system such as the arterial tree may be quite misleading.     

Modeling of intraorgan arterial vasculatures. I. A stationary blood flow at low Reynolds numbers

Based on the statistical relationships between the lengths and diameters of vessels in the arterial beds, obtained by measurements on plastic casts, a method for the generation of models of intraorgan arterial vasculatures was proposed. The dependence of the total hydraulic conductivity of the system on the model parameters was computed. The problem of selecting the adequate models by a comparison of the corresponding biophysical characteristics of vasculatures is discussed.     

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.     

Investigation of the structure of human coronary vasculature

Some results of a morphometric study of the parameters of coronary arteries are presented. The parameters that characterize the structure of the arterial vasculature as an optimal branching system have been calculated. Statistically reliable correlations between the diameter of the bigger of two daughter vessels in a bifurcation with the diameter of the parent vessel as well as between the diameter of the smaller daughter vessel and the asymmetry coefficient have been obtained. Differences in the structural parameters of the two types of coronary arteries that provide blood delivery and distribution have been revealed. The relationships between the lengths and diameters of the arteries of different subsystems have been obtained. It is shown that asymmetrical branching is characteristic of the coronary vasculature, and self-similar asymmetric tree-like systems may be used for its modeling.     

Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels.

Studies of the origin of pulmonary blood flow heterogeneity have highlighted the significant role of vessel branching structure on flow distribution. To enable more detailed investigation of structure-function relationships in the pulmonary circulation, an anatomically based finite element model of the arterial and venous networks has been developed to more accurately reflect the geometry found in vivo. Geometric models of the arterial and venous tree structures are created using a combination of multidetector row X-ray computed tomography imaging to define around 2,500 vessels from each tree, a volume-filling branching algorithm to generate the remaining accompanying conducting vessels, and an empirically based algorithm to generate the supernumerary vessel geometry. The explicit generation of supernumerary vessels is a unique feature of the computational model. Analysis of branching properties and geometric parameters demonstrates close correlation between the model geometry and anatomical measures of human pulmonary blood vessels. A total of 12 Strahler orders for the arterial system and 10 Strahler orders for the venous system are generated, down to the equivalent level of the terminal bronchioles in the bronchial tree. A simple Poiseuille flow solution, assuming rigid vessels, is obtained within the arterial geometry of the left lung, demonstrating a large amount of heterogeneity in the flow distribution, especially with inclusion of supernumerary vessels. This model has been constructed to accurately represent available morphometric data derived from the complex asymmetric branching structure of the human pulmonary vasculature in a form that will be suitable for application in functional simulations.     

Modeling of intraorgan arterial vasculature. I. Steady flow at low Reynolds numbers

Based on the statistical relationships between lengths and diameters of vessels in arterial beds obtained from measurements on plastic casts, a method is proposed for building models of intraorgan arterial vascualtures. The dependences of full hydraulic conductance on the model parameter values have been calculated. Discussed is the choice of adequate models based upon collation of the biophysical characteristics of the vasculatures.     

Pulsatile blood flow in the entire coronary arterial tree: theory and experiment

The pulsatility of coronary circulation can be accurately simulated on the basis of the measured branching pattern, vascular geometry, and material properties of the coronary vasculature. A Womersley-type mathematical model is developed to analyze pulsatile blood flow in diastole in the absence of vessel tone in the entire coronary arterial tree on the basis of previously measured morphometric data. The model incorporates a constitutive equation of pressure and cross-section area relation based on our previous experimental data. The formulation enables the prediction of the impedance, the pressure distribution, and the pulsatile flow distribution throughout the entire coronary arterial tree. The model is validated by experimental measurements in six diastolic arrested, vasodilated porcine hearts. The agreement between theory and experiment is excellent. Furthermore, the present pulse wave results at low frequency agree very well with previously published steady-state model. Finally, the phase angle of flow is seen to decrease along the trunk of the major coronary artery and primary branches toward the capillary vessels. This study represents the first, most extensive validated analysis of Womersley-type pulse wave transmission in the entire coronary arterial tree down to the first segment of capillaries. The present model will serve to quantitatively test various hypotheses in the coronary circulation under pulsatile flow conditions.     

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.     

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.     

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.     

Assessment of Model Based (Input) Impedance,Pulse Wave Velocity,and Wave Reflection in the Asklepios Cohort

Objectives

Arterial stiffness and wave reflection parameters assessed from both invasive and non-invasive pressure and flow readings are used as surrogates for ventricular and vascular load. They have been reported to predict adverse cardiovascular events, but clinical assessment is laborious and may limit widespread use. This study aims to investigate measures of arterial stiffness and central hemodynamics provided by arterial tonometry alone and in combination with aortic root flows derived by echocardiography against surrogates derived by a mathematical pressure and flow model in a healthy middle-aged cohort.

Methods

Measurements of carotid artery tonometry and echocardiography were performed on 2226 ASKLEPIOS study participants and parameters of systemic hemodynamics, arterial stiffness and wave reflection based on pressure and flow were measured. In a second step, the analysis was repeated but echocardiography derived flows were substituted by flows provided by a novel mathematical model. This was followed by a quantitative method comparison.

Results

All investigated parameters showed a significant association between the methods. Overall agreement was acceptable for all parameters (mean differences: -0.0102 (0.033 SD) mmHg*s/ml for characteristic impedance, 0.36 (4.21 SD) mmHg for forward pressure amplitude, 2.26 (3.51 SD) mmHg for backward pressure amplitude and 0.717 (1.25 SD) m/s for pulse wave velocity).

Conclusion

The results indicate that the use of model-based surrogates in a healthy middle aged cohort is feasible and deserves further attention.     

Constructive and destructive addition of forward and reflected arterial pulse waves

Although the physics of arterial pulse wave propagation and reflection is well understood, there is considerable debate as to the effect of reflection on vascular input impedance (Z(in)), pulsatile pressure, and stroke work (SW). This may be related to how reflection is studied. Conventionally, reflection is experimentally abolished (thus radically changing unrelated parameters), or a specific model is assumed from which reflection can be removed (yielding model-dependent results). The present work proposes a simple, model-independent method to evaluate the effect of reflection directly from measured pulsatile pressure (P) and flow (Q). Because characteristic impedance (Z(0)) is Z(in) in the absence of reflection, the P with reflection theoretically removed can be calculated from Q x Z(0). Applying this insight to an illustrative case indicates that reflection has the least effect on P and SW at normal pressure but a greater effect with vasodilation and vasoconstriction. Z(in), P, and SW are increased or decreased depending on the relative amount of constructive and destructive addition of forward and reflected arterial pulse waves.     

Dominance of geometric overelastic factors in pulse transmission through arterial branching

The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.     

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is approximately 2.2, which deviates from Murray's prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg.ml(-1).s(-1).g(-1) for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.     

Studies on the structure of human coronary vasculature

Based on morphometric data, we calculate the structural parameters of the coronary vasculature as an optimal branching bed. We show (i) significant correlations between the diameters of the larger daughter and the parent vessel and between the diameter of the smaller daughter vessel and the asymmetry coefficient; (ii) differences in the structural parameters for two types of artery that deliver and distribute blood in the cardiac muscle; and (iii) the length-diameter relationships for different arteries. The coronary vasculature is characterized by asymmetrical branching and thus should be modeled with self-similar asymmetrical tree-like systems.     

Computational assessment of model-based wave separation using a database of virtual subjects

The quantification of arterial wave reflection is an important area of interest in arterial pulse wave analysis. It can be achieved by wave separation analysis (WSA) if both the aortic pressure waveform and the aortic flow waveform are known. For better applicability, several mathematical models have been established to estimate aortic flow solely based on pressure waveforms. The aim of this study is to investigate and verify the model-based wave separation of the ARCSolver method on virtual pulse wave measurements.The study is based on an open access virtual database generated via simulations. Seven cardiac and arterial parameters were varied within physiological healthy ranges, leading to a total of 3325 virtual healthy subjects. For assessing the model-based ARCSolver method computationally, this method was used to perform WSA based on the aortic root pressure waveforms of the virtual patients. As a reference, the values of WSA using both the pressure and flow waveforms provided by the virtual database were taken.The investigated parameters showed a good overall agreement between the model-based method and the reference. Mean differences and standard deviations were −0.05 ± 0.02 AU for characteristic impedance, −3.93 ± 1.79 mmHg for forward pressure amplitude, 1.37 ± 1.56 mmHg for backward pressure amplitude and 12.42 ± 4.88% for reflection magnitude.The results indicate that the mathematical blood flow model of the ARCSolver method is a feasible surrogate for a measured flow waveform and provides a reasonable way to assess arterial wave reflection non-invasively in healthy subjects.     

Model-based cardiovascular disease diagnosis: a preliminary in-silico study

In this study, we developed and examined the feasibility of a model-based system identification approach to cardiovascular disease diagnosis. The basic premise of the approach is that it may be possible to diagnose cardiovascular disease from disease-induced alterations in the arterial mechanical properties manifested in the proximal and distal arterial blood pressure waveforms. It first individualizes the lumped-parameter model of wave propagation and reflection in the artery using the measurement of proximal and distal arterial blood pressure waveforms. Then, it employs a diagnosis logic, in the form of disease-specific patterns in model parameters, referred as \(\alpha , \beta \) and pulse transit time. The longitudinal change in these parameters is used to diagnose the presence of peripheral artery disease and arterial stiffening. We illustrated the feasibility of the proposed approach by testing it in a full-scale in-silico arterial tree simulation. The results showed that the approach exhibited superior sensitivity to ankle-brachial index and convenience to carotid-femoral pulse wave velocity: The model parameters \(\alpha \) and \(\beta \) responded with up to 100 and 40 % changes to peripheral artery disease with up to 50 % arterial blockage whereas the change in ankle-brachial index was \({<}5\,\%\); the same parameters responded with up to 300 and 40 % changes to up to 100 % arterial stiffening while pulse transit time changed by up to 24 %. Together with the development of more convenient techniques for the measurement of arterial blood pressure waveforms, the proposed approach may evolve into a viable alternative to the state-of-the-art techniques for cardiovascular disease diagnosis.     

A model of growing vascular structures

Increasing attention is being paid to the configuration and development of vascular structures and their possible correlations with physiological events. The study of angiogenesis in normal and pathological states as well as in the embryo and adult has provided new insights into the mechanism of vessel growth and organization of the vasculature. Various mathematical branching models have been developed. These constructions are mainly geometrical and only involve a branching phenomenon. We propose the use of a deterministic non-linear model based on physiological laws and hydrodynamics. Growth, branching and anastomosis, the three actual main events occurring in vascular growth, are included in this model. Space growth, including cells and vessels, is defined by a decreasing transformation. Space density and the length of new sprouts are controlled by a set of parameters. The conditions on these parameters are well established, which allows the production of realistic patterns.     

A Bio-Inspired Approach for the Reduction of Left Ventricular Workload

Previous studies have demonstrated the existence of optimization criteria in the design and development of mammalians cardiovascular systems. Similarities in mammalian arterial wave reflection suggest there are certain design criteria for the optimization of arterial wave dynamics. Inspired by these natural optimization criteria, we investigated the feasibility of optimizing the aortic waves by modifying wave reflection sites. A hydraulic model that has physical and dynamical properties similar to a human aorta and left ventricle was used for a series of in-vitro experiments. The results indicate that placing an artificial reflection site (a ring) at a specific location along the aorta may create a constructive wave dynamic that could reduce LV pulsatile workload. This simple bio-inspired approach may have important implications for the future of treatment strategies for diseased aorta.