Wave reflection effects in the central circulation of American alligators (Alligator mississippiensis): what the heart sees

A large central compliance is thought to dominate the hemodynamics of all vertebrates except birds and mammals. Yet large crocodilians may adumbrate the avian and mammalian condition and set the stage for significant wave transmission (reflection) effects, with potentially detrimental impacts on cardiac performance. To investigate whether crocodilians exhibit wave reflection effects, pressures and flows were recorded from the right aorta, carotid artery, and femoral artery of six adult, anesthetized American alligators (Alligator mississippiensis) during control conditions and after experimentally induced vasodilation and constriction. Hallmarks of wave reflection phenomena were observed, including marked differences between the measured profiles for flow and pressure, peaking of the femoral pressure pulse, and a diastolic wave in the right aortic pressure profile. Pulse wave velocity and peripheral input impedance increased with progressive constriction, and thus changes in both the timing and magnitude of reflections accounted for the altered reflection effects. Resolution of pressure and flow waves into incident and reflected components showed substantial reflection effects within the right aorta, with reflection coefficients at the first harmonic approaching 0.3 when constricted. Material properties measured from isolated segments of blood vessels revealed a major reflection site at the periphery and, surprisingly, at the junction of the truncus and right aorta. Thus, while our results clearly show that significant wave reflection phenomena are not restricted to birds and mammals, they also suggest that rather than cope with potential negative impacts of reflections, the crocodilian heart simply avoids them because of a large impedance mismatch at the truncus.     

Effects of stent stiffness on local haemodynamics with particular reference to wave reflections

The placement of a rigid stent within an elastic vessel produces wave reflection sites at the entrance to and exit from the stent. The net haemodynamic effects of these reflections depend critically on the degree of stiffness of the stent and on its length and position within the diseased vessel, variables that have been found to affect the clinical performance of a stent. Here these effects are examined analytically, using a segmented tube model. The results indicate that the presence of the stent within the larger diseased vessel has the effect of producing higher pressure at the vessel entrance than that at exit. This pressure difference, when superimposed on the underlying pressure distribution within the vessel, has the net effect of actually aiding rather than impeding the flow, but the extent of this depends on the length and position of the stent. A short stent placed near the entrance of the diseased vessel may be favoured clinically for producing the least perturbation in the underlying haemodynamics and thus reducing the chance of restenosis, while a long stent placed near the exit may be favoured for producing a positive pressure difference and thus aiding the flow.     

Mechanics of blood supply to the heart: wave reflection effects in a right coronary artery.

Mechanics of blood flow in the coronary circulation have in the past been based largely on models in which the detailed architecture of the coronary network is not included because of lack of data: properties of individual vessels do not appear individually in the model but are represented collectively by the elements of a single electric circuit. Recent data from the human heart make it possible, for the first time, to examine the dynamics of flow in the coronary network based on detailed, measured vascular architecture. In particular, admittance values along the full course of the right coronary artery are computed based on actual lengths and diameters of the many thousands of branches which make up the distribution system of this vessel. The results indicate that effects of wave reflections on this flow are far more significant than those generally suspected to occur in coronary blood flow and that they are actually the reverse of the well known wave reflection effects in the aorta.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Smaller, stiffer coronary bypass can moderate or reverse the adverse effects of wave reflections.

The question of whether the mechanical stiffness of a coronary bypass or that of a diseased coronary artery can have a significant effect on the hemodynamics in these vessels is addressed analytically, with emphasis on the effects of wave reflections. The analysis is based on a model of the vessels involved, and the results show the essential hemodynamic effects in each vessel. It is found that in the absence of a bypass graft, wave reflections resulting from a narrowing and stiffening of a diseased coronary artery have the effect of actually aiding the flow in the diseased vessel. In the presence of a bypass graft, however, the effects of wave reflections are reversed and become adverse to flow in both the bypass graft and the diseased coronary artery. A stiffer bypass moderates these effects and is therefore preferable to a more elastic bypass. The adverse effects also depend critically on the relative diameter of the bypass. Here the results indicate that a bypass of smaller diameter than that of the native coronary artery can moderate and even reverse the adverse effects of wave reflections resulting from the presence of the bypass.     

Calculations of pulsatile flow across bifurcations in distensible tubes

A simple method is given for extending across junctions the numerical methods previously used to study fluid flow in nonuniform, nonbranching, distensible tubes. Calculations with this method suggest that the ratio of downstream to upstream cross-sectional areas and the pressure wave velocity ratio in mammals are optimal for energy transfer across the junction.     

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.     

Contributions to the mathematical biophysics of branched circulatory systems

A derivation is given of the reflection coefficient of pressure waves in a vessel whose end branches into many smaller vessles. This coefficient depends on the number of these smaller vessels and their sizes relative to the size of the main vessel. Estimations are made of the order of magnitude of the coefficient. Assuming the main vessel to be of the order of size of an artery, it is shown that the reflection coefficient has a value close to one for reflections at branchings into vessels of arteriolar size. It is pointed out that the result may support the idea that the standing waves in the arterial system are due to reflections at the site of the arterioles.     

The nanostructure of myoendothelial junctions contributes to signal rectification between endothelial and vascular smooth muscle cells

Micro-anatomical structures in tissues have potential physiological effects. In arteries and arterioles smooth muscle cells and endothelial cells are separated by the internal elastic lamina, but the two cell layers often make contact through micro protrusions called myoendothelial junctions. Cross talk between the two cell layers is important in regulating blood pressure and flow. We have used a spatiotemporal mathematical model to investigate how the myoendothelial junctions affect the information flow between the two cell layers. The geometry of the model mimics the structure of the two cell types and the myoendothelial junction. The model is implemented as a 2D axi-symmetrical model and solved using the finite element method. We have simulated diffusion of Ca(2+) and IP(3) between the two cell types and we show that the micro-anatomical structure of the myoendothelial junction in itself may rectify a signal between the two cell layers. The rectification is caused by the asymmetrical structure of the myoendothelial junction. Because the head of the myoendothelial junction is separated from the cell it is attached to by a narrow neck region, a signal generated in the neighboring cell can easily drive a concentration change in the head of the myoendothelial protrusion. Subsequently the signal can be amplified in the head, and activate the entire cell. In contrast, a signal in the cell from which the myoendothelial junction originates will be attenuated and delayed in the neck region as it travels into the head of the myoendothelial junction and the neighboring cell.     

Gap junction structures: Analysis of the x-ray diffraction data

Models for the spatial distribution of protein, lipid and water in gap junction structures have been constructed from the results of the analysis of X-ray diffraction data described here and the electron microscope and chemical data presented in the preceding paper (Caspar, D. L. D., D. A. Goodenough, L. Makowski, and W.C. Phillips. 1977. 74:605-628). The continuous intensity distribution on the meridian of the X-ray diffraction pattern was measured, and corrected for the effects of the partially ordered stacking and partial orientation of the junctions in the X-ray specimens. The electron density distribution in the direction perpendicular to the plane of the junction was calculated from the meridional intensity data. Determination of the interference function for the stacking of the junctions improved the accuracy of the electron density profile. The pair-correlation function, which provides information about the packing of junctions in the specimen, was calculated from the interference function. The intensities of the hexagonal lattice reflections on the equator of the X-ray pattern were used in coordination with the electron microscope data to calculate to the two-dimensional electron density projection onto the plane of the membrane. Differences in the structure of the connexons as seen in the meridional profile and equatorial projections were shown to be correlated to changes in lattice constant. The parts of the junction structure which are variable have been distinguished from the invariant parts by comparison of the X-ray data from different specimens. The combination of these results with electron microscope and chemical data provides low resolution three- dimensional representations of the structures of gap junctions.     

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.     

The cost of departure from optimal radii in microvascular networks

In the Murray optimality model of branching vasculatures, the radii of vessels are related to blood viscosity, vascular metabolic rate, and blood flow rate, in such a way as to minimize the total work (hydraulic and metabolic) of the system. The model predicts that flow is proportional to the cube of a vessel radius, and that at junctions the cube of the radius of the parent vessel equals the sum of the cubes of the daughter radii. In comparing real vasculatures to the Murray model, we have previously had no expressions for evaluating the apparent energy cost for departures from the optimal junction exponent of 3. Such expressions are derived here. They show that junction exponents, from about 1.5 to large positive values, are within 5% of the energy minimum. With the new equations, observed individual junctions or entire vascular trees can be compared, energy-wise, with the Murray optimum. Junctions in the transverse arteriolar trees of cat sartorius muscle were compared to the Murray optimality model, using these new expressions. The junction exponents for these small pre-capillary vessels had a broad range, with a median value greater than the Murray optimum of 3. The exponents were restricted, however, to values requiring, at individual junctions, little increase in energy. The majority of junctions had energy costs less than 1% above the Murray minimum. For entire trees involving many junctions the departures from optimality averaged less than 10%. Thus, while the branching geometry for these microvascular trees deviates significantly from the Murray optimum in the direction of larger daughter to parent ratios, the departures are small in energy terms.     

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.     

Branch junctions and the flow of water through xylem in Douglas-fir and ponderosa pine stems

Water flowing through the xylem from the roots to the leaves of most plants must pass through junctions where branches have developed from the main stem. These junctions have been studied as both flow constrictions and components of a hydraulic segmentation mechanism to protect the main axes of the plant. The hydraulic nature of the branch junction also affects the degree to which branches interact and can respond to changes in flow to other branches. The junctions from shoots of two conifer species were studied, with particular emphasis on the coupling between the downstream branches. Flow was observed qualitatively by forcing stain through the junctions and the resulting patterns showed that flow into a branch was confined to just part of the subtending xylem until a considerable distance below the junction. Junctions were studied quantitatively by measuring flow rates in a branch before and after flow was stopped in an adjacent branch and by measuring the hydraulic resistance of the components of the junction. Following flow stoppage in the adjacent branch, flow into the remaining branch increased, but considerably less than predicted based on a simple resistance analogue for the branch junction that assumes the two branches are fully coupled. The branches downstream from a junction, therefore, appear to be limited in their interconnectedness and hence in their ability to interact.     

Study of hydrodynamics in microcarrier culture spinner vessels: A particle tracking velocimetry approach

Three-dimensional particle tracking velocimetry (3-D PTV), a modern, quantitative, visualization tool, has been applied to the characterization of the flow field in the impeller region of cell culture reactor vessels. The experimental system used here is a 250-mL microcarrier spinner vessel. The studies were conducted at three different agitation rates, 90, 150, and 210 rpm, corresponding to healthy, mildly damaging, and severely damaging shear intensities, respectively. The flow can be classified into three regions: a predominantly tangential (azimuthal) flow generated by the impeller; a trailing vortex region coming off the impeller tip; and a converging flow region close to the center of the vessel. The latter two are the regions of highest velocity gradients. Energy dissipation rates due to mean velocity gradients were also calculated to characterize the impeller stream. Local specific energy dissipation rates > 10,000 erg/(cm(3)sec) . have been measured. It is proposed that the critical regions for microcarrier culture damage due to impeller hydrodynamics are the trailing vortex region and the high energy converging flow region. Graphical representation of the mean velocity flow fields and the distribution of energy dissipation rates in the impeller region are also presented here. The merits of using the dissipation function (measure of specific energy dissipation rate) as a possible scale-up parameter are also discussed. (c) 1996 John Wiley & Sons, Inc.     

A three-dimensional model of intercellular calcium signaling in epithelial cells

We have developed a fully three-dimensional (3D) model of calcium signaling in epithelial cells based on a set of reaction diffusion equations that are solved on a large-scale finite-element code in three dimensions. We have explicitly included the cellular compartments including the cell nucleus, cytoplasm, and gap junctions. The model allows for buffering of free Ca2+, calcium-induced calcium release, and the explicit inclusion of mobile buffers. To make quantitative comparisons to experimental results, we used fluorescence microscopy images of cells to generate an accurate mesh describing cell morphology. We found that Ca2+ wave propagation through the tissue is a function of both initial conditions used to start the wave and various geometrical parameters that affect propagation such as gap junction density and distribution, and the presence of nuclei. The exogenous dyes used in experimental imaging also affect wave propagation.     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.     

Vessel distensibility and flow distribution in vascular trees

 In a class of model vascular trees having distensible blood vessels, we prove that flow partitioning throughout the tree remains constant, independent of the nonzero driving flow (or nonzero inlet to terminal outlet pressure difference). Underlying assumptions are: (1) every vessel in the tree exhibits the same distensibility relationship given by $D/D_0 = f(P)$ where $D$ is the diameter which results from distending pressure $P$ and $D_0$ is the diameter of the individual vessel at zero pressure (each vessel may have its own individual $D_0$). The choice of $f(P)$ includes distensibilities often used in vessel biomechanics modeling, e.g., $f(P) = 1 + \alpha P$ or $f(P) = b + (1-b) \exp(-c P)$, as well as $f(P)$ which exhibit autoregulatory behavior. (2) Every terminal vessel in the tree is subjected to the same terminal outlet pressure. (3) Bernoulli effects are ignored. (4) Flow is nonpulsatile. (5) Blood viscosity within any individual vessel is constant. The results imply that for a vascular tree consistent with assumptions 2–5, the flow distribution calculations based on a rigid geometry, e.g., $D=D_0$, also gives the flow distribution when assuming the common distensibility relationships. Received: 30 October 2001 / Published online: 14 March 2002     

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

The lymphatic system is an open-ended network of vessels that run in parallel to the blood circulation system. These vessels are present in almost all of the tissues of the body to remove excess fluid. Similar to blood vessels, lymphatic vessels are found in branched arrangements. Due to the complexity of experiments on lymphatic networks and the difficulty to control the important functional parameters in these setups, computational modeling becomes an effective and essential means of understanding lymphatic network pumping dynamics. Here we aimed to determine the effect of pumping coordination in branched network structures on the regulation of lymph flow. Lymphatic vessel networks were created by building upon our previous lumped-parameter model of lymphangions in series. In our network model, each vessel is itself divided into multiple lymphangions by lymphatic valves that help maintain forward flow. Vessel junctions are modeled by equating the pressures and balancing mass flows. Our results demonstrated that a 1.5 s rest-period between contractions optimizes the flow rate. A time delay between contractions of lymphangions at the junction of branches provided an advantage over synchronous pumping, but additional time delays within individual vessels only increased the flow rate for adverse pressure differences greater than 10.5 cmH₂O. Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes. We observed that peak flow rates normally occurred under transmural pressures between 2 to 4 cmH₂O (for multiple pressure differences and network sizes). Networks with 10 lymphangions per vessel had the highest pumping capability under a wide range of adverse pressure differences. For favorable pressure differences, pumping was more efficient with fewer lymphangions. These findings are valuable for translating experimental measurements from the single lymphangion level to tissue and organ scales.