Blood Vessel Adaptation with Fluctuations in Capillary Flow Distribution

Throughout the life of animals and human beings, blood vessel systems are continuously adapting their structures – the diameter of vessel lumina, the thickness of vessel walls, and the number of micro-vessels – to meet the changing metabolic demand of the tissue. The competition between an ever decreasing tendency of luminal diameters and an increasing stimulus from the wall shear stress plays a key role in the adaptation of luminal diameters. However, it has been shown in previous studies that the adaptation dynamics based only on these two effects is unstable. In this work, we propose a minimal adaptation model of vessel luminal diameters, in which we take into account the effects of metabolic flow regulation in addition to wall shear stresses and the decreasing tendency of luminal diameters. In particular, we study the role, in the adaptation process, of fluctuations in capillary flow distribution which is an important means of metabolic flow regulation. The fluctuation in the flow of a capillary group is idealized as a switch between two states, i.e., an open-state and a close-state. Using this model, we show that the adaptation of blood vessel system driven by wall shear stress can be efficiently stabilized when the open time ratio responds sensitively to capillary flows. As micro-vessel rarefaction is observed in our simulations with a uniformly decreased open time ratio of capillary flows, our results point to a possible origin of micro-vessel rarefaction, which is believed to induce hypertension.     

Contributions to the mathematical biophysics of the cardiovascular system

The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

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.     

Vascular wall energetics in arterioles during nitric oxide-dependent and -independent vasodilation.

The objective of this study was to evaluate whether the nitric oxide (NO) released from vascular endothelial cells would decrease vessel wall oxygen consumption by decreasing the energy expenditure of mechanical work by vascular smooth muscle. The oxygen consumption rate of arteriolar walls in rat cremaster muscle was determined in vivo during NO-dependent and -independent vasodilation on the basis of the intra- and perivascular oxygen tension (Po2) measured by phosphorescence quenching laser microscopy. NO-dependent vasodilation was induced by increased NO production due to increased blood flow, whereas NO-independent vasodilation was induced by topical administration of papaverine. The energy efficiency of vessel walls was evaluated by the variable ratio of circumferential wall stress (amount of mechanical work) to vessel wall oxygen consumption rate (energy cost) in the arteriole between normal and vasodilated conditions. NO-dependent and -independent dilation increased arteriolar diameters by 13 and 17%, respectively, relative to the values under normal condition. Vessel wall oxygen consumption decreased significantly during both NO-dependent and -independent vasodilation compared with that under normal condition. However, vessel wall oxygen consumption during NO-independent vasodilation was significantly lower than that during NO-dependent vasodilation. On the other hand, there was no significant difference between the energy efficiency of vessel walls during NO-dependent and -independent vasodilation, suggesting the decrease in vessel wall oxygen consumption produced by NO to be related to reduced mechanical work of vascular smooth muscle.     

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.     

"Sausage-string" appearance of arteries and arterioles can be caused by an instability of the blood vessel wall

Vascular damage induced by acute hypertension is preceded by a peculiar pattern where blood vessels show alternating regions of constrictions and dilations ("sausages on a string"). The pattern occurs in the smaller blood vessels, and it plays a central role in causing the vascular damage. A related vascular pattern has been observed in larger vessels from several organs during angiography. In the larger vessels the occurrence of the pattern does not appear to be related to acute hypertension. A unifying feature between the phenomenon in large and small vessels seems to be an increase in vascular wall tension. Despite much research, the mechanisms underlying the sausage pattern have remained unknown. Here we present an anisotropic model of the vessel wall and show that the sausage pattern can arise because of an instability of the vessel wall. The model reproduces many of the key features observed experimentally. Most importantly, it suggests that the "sausaging" phenomenon is neither caused by a mechanical failure of the vessel wall due to a high blood pressure nor is it due to standing pressure waves caused by the beating of the heart. Rather, it is the expression of a general instability phenomenon. Experimental data suggest that the structural changes induced by the instability may cause secondary damage to the wall of small arteries and arterioles in the form of endothelial hyperpermeability followed by local fibrinoid necrosis of the vascular wall.     

The effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes

An analytical solution for pulsatile flow of a generalized Maxwell fluid in straight rigid tubes, with and without axial vessel motion, has been used to calculate the effect of blood viscoelasticity on velocity profiles and shear stress in flows representative of those in the large arteries. Measured bulk flow rate Q waveforms were used as starting points in the calculations for the aorta and femoral arteries, from which axial pressure gradient ▿P waves were derived that would reproduce the starting Q waves for viscoelastic flow. The ▿P waves were then used to calculate velocity profiles for both viscoelastic and purely viscous flow. For the coronary artery, published ▿P and axial vessel acceleration waveforms were used in a similar procedure to determine the separate and combined influences of viscoelasticity and vessel motion.Differences in local velocities, comparing viscous flow to viscoelastic flow, were in all cases less than about 2% of the peak local velocity. Differences in peak wall shear stress were less than about 3%.In the coronary artery, wall shear stress differences between viscous and viscoelastic flow were small, regardless of whether axial vessel motion was included. The shape of the wall shear stress waveform and its difference, however, changed dramatically between the stationary and moving vessel cases. The peaks in wall shear stress difference corresponded with large temporal gradients in the combined driving force for the flow.     

不同纬度对鹅掌楸次生木质部结构的影响

对生长在北纬23至32度的鹅掌楸〖WTBX〗（Liriodendron chinense〖WTBZ〗 Sarg.）的次生木质部进行了比较解剖学研究。观察的特征有：导管分子的长度、直径、壁厚、梯状穿孔板横闩的数目;导管密度、纤维长度、直径、壁厚;射线的高度、宽度,单列射线的比例,射线的密度等。结果表明,随着纬度的增加,导管分子变短变细,壁变薄,梯状穿孔板的横闩数目减少;纤维也变短变细,壁变薄;木射线变矮变窄,单列射线的比例和射线密度增加。     

不同纬度对鹅掌楸次生木质部结构的影响

对生长在北纬23至32度的鹅掌楸（Liriodendron chinense Sarg.）的次生木质部进行了比较解剖学研究。观察的特征有：导管分子的长度、直径、壁厚、梯状穿孔板横闩的数目;导管密度、纤维长度、直径、壁厚;射线的高度、宽度,单列射线的比例,射线的密度等。结果表明,随着纬度的增加,管分子变短变细,壁变薄,梯状穿孔板的横闩数目减少;纤维也变短变线,壁变薄;木射线变敌变窄,单列射线的比例和射线密度增加。     

Readdressing the issue of thermally significant blood vessels using a countercurrent vessel network

A physiologically realistic arterio-venous countercurrent vessel network model consisting of ten branching vessel generations, where the diameter of each generation of vessels is smaller than the previous ones, has been created and used to determine the thermal significance of different vessel generations by investigating their ability to exchange thermal energy with the tissue. The temperature distribution in the 3D network (8178 vessels; diameters from 10 to 1000 microm) is obtained by solving the conduction equation in the tissue and the convective energy equation with a specified Nusselt number in the vessels. The sensitivity of the exchange of energy between the vessels and the tissue to changes in the network parameters is studied for two cases; a high temperature thermal therapy case when tissue is heated by a uniformly distributed source term and the network cools the tissue, and a hypothermia related case, when tissue is cooled from the surface and the blood heats the tissue. Results show that first, the relative roles of vessels of different diameters are strongly determined by the inlet temperatures to those vessels (e.g., as affected by changing mass flow rates), and the surrounding tissue temperature, but not by their diameter. Second, changes in the following do not significantly affect the heat transfer rates between tissue and vessels; (a) the ratio of arterial to venous vessel diameter, (b) the diameter reduction coefficient (the ratio of diameters of successive vessel generations), and (c) the Nusselt number. Third, both arteries and veins play significant roles in the exchange of energy between tissue and vessels, with arteries playing a more significant role. These results suggest that the determination of which diameter vessels are thermally important should be performed on a case-by-case, problem dependent basis. And, that in the development of site-specific vessel network models, reasonable predictions of the relative roles of different vessel diameters can be obtained by using any physiologically realistic values of Nusselt number and the diameter reduction coefficient.     

Mathematical analysis of several parameters of the microcirculatory bed of muscles and muscular organs]

The authors propose a complex mathematical method for analysis of the microcirculatory link of the vascular bed of muscles and muscular organs. On the basis of the data of the amount and diameters of the vessels the method permits calculating the following parameters according to the proposed pattern of the table-report and formulas: 1. the square surface of the cross-section of the vascular bed; 2. the square surface of the exchange or contact with the tissue surface of the vascular bed; 3. the density of the vascular network in percentage; 4. the blood volume in the vascular bed in volumetric units and average data; 5. the blood volume in one vessel on the average; 6. the volume of the tissue fed by one vessel; 7. the volume of blood per a unit of the surface of the vascular wall.     

The influence of the blood vessel diameter on the full scattering profile from cylindrical tissues: experimental evidence for the shielding effect

Optical methods for detecting physiological state based on light–tissue interaction are noninvasive, inexpensive, simplistic, and thus very useful. The blood vessels in human tissue are the main cause of light absorbing and scattering. Therefore, the effect of blood vessels on light–tissue interactions is essential for optically detecting physiological tissue state, such as oxygen saturation, blood perfusion and blood pressure. We have previously suggested a new theoretical and experimental method for measuring the full scattering profile, which is the angular distribution of light intensity, of cylindrical tissues. In this work we will present experimental measurements of the full scattering profile of heterogenic cylindrical phantoms that include blood vessels. We show, for the first time that the vessel diameter influences the full scattering profile, and found higher reflection intensity for larger vessel diameters accordance to the shielding effect. For an increase of 60% in the vessel diameter the light intensity in the full scattering profile above 90° is between 9% to 40% higher, depending on the angle. By these results we claim that during respiration, when the blood‐vessel diameter changes, it is essential to consider the blood‐vessel diameter distribution in order to determine the optical path in tissues.

A CT scan of the measured silicon‐based phantoms. The phantoms contain the same blood volume in different blood‐vessel diameters.     

The Dissipation and Dispersion of Small Waves in Arteries and Veins with Viscoelastic Wall Properties

Theoretical and experimental evidence suggests that the dissipation of high frequency pressure waves in blood vessels is caused primarily by the viscoelastic behavior of the vessel wall. In this theoretical analysis the vessels are considered as fluid-filled circular cylindrical shells whose walls have isotropic and homogeneous viscoelastic properties and are subjected to an initial axial stretch and a transmural pressure. If the wall material is incompressible and behaves as a Voigt solid in shear, the results predict a decrease in wave amplitude per wavelength which is essentially independent of frequency over a wide range. This finding is in qualitative agreement with recent experiments on anesthetized dogs. A parametric study also shows a great sensitivity of the dissipation to changes in transmural pressure and axial stretch. Axisymmetric waves are only mildly dispersive, while all nonaxisymmetric waves are highly dispersive and exhibit much stronger damping per wavelength at low frequencies than do axisymmetric waves.     

Numerical simulation of two-phase non-Newtonian blood flow with fluid-structure interaction in aortic dissection

The behavior of blood cells and vessel compliance significantly influence hemodynamic parameters, which are closely related to the development of aortic dissection. Here the two-phase non-Newtonian model and the fluid-structure interaction (FSI) method are coupled to simulate blood flow in a patient-specific dissected aorta. Moreover, three-element Windkessel model is applied to reproduce physiological pressure waves. Important hemodynamic indicators, such as the spatial distribution of red blood cells (RBCs) and vessel wall displacement, which greatly influence the hemodynamic characteristics are analyzed. Results show that the proximal false lumen near the entry tear appears to be a vortex zone with a relatively lower volume fraction of RBCs, a low time-averaged wall shear stress (TAWSS) and a high oscillatory shear index (OSI), providing a suitable physical environment for the formation of atherosclerosis. The highest TAWSS is located in the narrow area of the distal true lumen which might cause further dilation. TAWSS distributions in the FSI model and the rigid wall model show similar trend, while there is a significant difference for the OSI distributions. We suggest that an integrated model is essential to simulate blood flow in a more realistic physiological environment with the ultimate aim of guiding clinical treatment.     

Assessment of energy requirement for the retinal arterial network in normal and hypertensive subjects

The retinal arterial network structure can be altered by systemic diseases such as hypertension and diabetes. In order to compare the energy requirement for maintaining retinal blood flow and vessel wall metabolism between normal and hypertensive subjects, 3D hypothetical models of a representative retinal arterial bifurcation were constructed based on topological features derived from retinal images. Computational analysis of blood flow was performed, which accounted for the non-Newtonian rheological properties of blood and peripheral vessel resistance. The results suggested that the rate of energy required to maintain the blood flow and wall metabolism is much lower for normal subjects than for hypertensives, with the latter requiring 49.2% more energy for an entire retinal arteriolar tree. Among the several morphological factors, length-to-diameter ratio was found to have the most significant influence on the overall energy requirement.     

Theoretical comparison of wall-derived and erythrocyte-derived mechanisms for metabolic flow regulation in heterogeneous microvascular networks

The objective of this study is to compare the effectiveness of metabolic signals derived from erythrocytes and derived from the vessel wall for regulating blood flow in heterogeneous microvascular networks. A theoretical model is used to simulate blood flow, mass transport, and vascular responses. The model accounts for myogenic, shear-dependent, and metabolic flow regulation. Metabolic signals are assumed to be propagated upstream along vessel walls via a conducted response. Arteriolar tone is assumed to depend on the conducted metabolic signal as well as local wall shear stress and wall tension, and arteriolar diameters are calculated based on vascular smooth muscle mechanics. The model shows that under certain conditions metabolic regulation based on wall-derived signals can be more effective in matching perfusion to local oxygen demand relative to regulation based on erythrocyte-derived signals, resulting in higher extraction and lower oxygen deficit. The lower effectiveness of the erythrocyte-derived signal is shown to result in part from the unequal partition of hematocrit at diverging bifurcations, such that low-flow vessels tend to receive a reduced hematocrit and thereby experience a reduced erythrocyte-derived metabolic signal. The model simulations predict that metabolic signals independent of erythrocytes may play an important role in local metabolic regulation of vascular tone and flow distribution in heterogeneous microvessel networks.     

As human pial arteries (internal diameter 200-1000 microm) get smaller, their wall thickness and capacity to develop tension relative to their diameter increase.

Pial arteries play a key role in the regulation of human cerebral blood flow. However, many of the features and mechanisms that regulate the tone and diameters of these vessels cannot be studied in situ. One approach is to study in vitro segments of arteries obtained during neurosurgical procedures. The ratios of arterial media thickness to lumen diameter and of the capacity to develop wall force to lumen diameter have important functional consequences and are known to change in disease. Experiments were carried out on pial arteries from normotensive humans to determine the way in which these parameters vary with vessel size. Vessel dimensions--media thickness and lumen diameter were derived from fixed sections using quantitative morphometry. Wall force was measured using a resistance artery myograph. The ratio of media thickness to lumen diameter and of maximum tension developed to lumen diameter both increased as vessel diameter decreased. These ratios do not change over the age range of 15-75 years. These findings show that although in vivo intralumenal pressure falls as human pial arteries become smaller, their media thickness and capacity to develop tone increase.     

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.