A constrained mixture model for developing mouse aorta

Mechanical stresses influence the structure and function of adult and developing blood vessels. When these stresses are perturbed, the vessel wall remodels to return the stresses to homeostatic levels. Constrained mixture models have been used to predict remodeling of adult vessels in response to step changes in blood pressure, axial length and blood flow, but have not yet been applied to developing vessels. Models of developing blood vessels are complicated by continuous and simultaneous changes in the mechanical forces. Understanding developmental growth and remodeling is important for treating human diseases and designing tissue-engineered blood vessels. This study presents a constrained mixture model for postnatal development of mouse aorta with multiple step increases in pressure, length and flow. The baseline model assumes that smooth muscle cells (SMCs) in the vessel wall immediately constrict or dilate the inner radius after a perturbation to maintain the shear stress and then remodel the wall thickness to maintain the circumferential stress. The elastin, collagen and SMCs have homeostatic stretch ratios and passive material constants that do not change with developmental age. The baseline model does not predict previously published experimental data. To approximate the experimental data, it must be assumed that the SMCs dilate a constant amount, regardless of the step change in mechanical forces. It must also be assumed that the homeostatic stretch ratios and passive material constants change with age. With these alterations, the model approximates experimental data on the mechanical properties and dimensions of aorta from 3- to 30-day-old mice.     

Numerical simulation of sinusoidal flow in a straight elastic tube: effects of phase angles

Numerical simulations of flow in straight elastic (moving wall) tubes subjected to a sinusoidal pressure gradient were performed for conditions prevailing in large and medium sized arteries. The effects of varying the phase angle between the pressure gradient and the tube radius, the amplitude of wall motion, and the unsteadiness parameter (alpha) on flow rate and wall shear stress were investigated. Mean and peak flow rates and shear stresses were found to be strongly affected by the phase angle between the pressure gradient and the tube radius with greater sensitivity at higher diameter variation and higher alpha. In large artery simulations (alpha = 12), means flow rate was found to be 60% higher and peak flow rate to be 73% higher than corresponding rigid tube values for certain phase angles, while a threefold increase in mean wall shear stress and sevenfold increase in peak wall shear stress were observed in a sensitive phase angle range. Significant reversal in the wall shear stress direction occurred in the sensitive phase angle range even when there was negligible flow rate reversal. All effects were greatly diminished in simulations of medium sized vessels (alpha = 4). Some experimental evidence to support the predictions of a strong effect of phase angle on wall shear stress in large vessels is presented. Finally, physiological implications of the present work are discussed from a basis of aortic input impedance data, and a physical explanation for the extreme sensitivity of the flow field to small amplitude wall motion at high alpha is given.     

Smooth muscle relaxation and local hydraulic impedance properties of the aorta.

Smooth muscle relaxation is expected to yield beneficial effects on hydraulic impedance properties of large vessels. We investigated the effects of intravenous diltiazem infusion on aortic wall stiffness and local hydraulic impedance properties. In seven anesthetized, closed-chest dogs, instantaneous cross-sectional area and pressure of the descending thoracic aorta were measured using transesophageal echocardiography combined with acoustic quantification and a micromanometer, respectively. Data were acquired during a vena caval balloon inflation, both at the control condition and with diltiazem infusion. At the operating point, diltiazem reduced blood pressure in all dogs but did not alter aortic dimensions or wall stiffness. Over the observed pressure range, aortic area-pressure relationships were linear. Whereas diltiazem affected the slope of this relationship variably (no change in 3 dogs, increase in 1 dog, decrease in 3 dogs), the zero-pressure area intercept was significantly increased in every case such that higher area was observed at any given pressure. When comparisons were made at a common level of wall stress, wall stiffness was either increased or unchanged during diltiazem infusion. In contrast, diltiazem decreased wall stiffness in every case when comparisons were made at a common level of aortic midwall radius. Aortic characteristic impedance and pulse wave velocity, components of left ventricular hydraulic load that are determined by aortic elastic and geometric properties, were affected variably. A comparison of wall stiffness at matched wall stress appears inappropriate for assessing changes in smooth muscle tone. Because of the competing effects of changes in vessel diameter and wall stiffness, smooth muscle relaxation is not necessarily accompanied by the expected beneficial changes in local aortic hydraulic impedance. These results can be reconciled by recognizing that components other than vascular smooth muscle (e.g., elastin, collagen) contribute to aortic wall stiffness.     

Role of elastic fibers in cooling-induced relaxation

The objective of this work was to confirm the main role of elastic fibers in differing responses of certain vessels during cooling from 37 to 8 degrees C. Previous results have shown that the nature of the vessel (conduit vessel vs muscular vessel) determines the different behavior (dilatation vs contraction) of isolated vessel segments when temperature decreases from 37 to 8 degrees C. In this work, it has been demonstrated that vessels with a great amount of elastic fibers show a dilatation when cooling. On the other hand, muscular vessels with fewer elastic fibers, such as the renal artery, undergo a contraction. The output of calcium from intracellular stores causes contraction of the renal artery during cooling. In this vessel, vasodilatation occurs only when mechanisms of smooth muscle contraction are inactive, as is the case with vessels that have undergone a cold storage period of 48 h. The results presented in this work confirm that there are two main effects, which directly depend on the vessel origin. In conduit arteries, the decrease of temperature induces a vascular relaxation, dependent on the elastic component of the vessel wall. In muscular vessels, the predominant effect is cooling-induced contraction due to an increase of intracellular calcium. This cooling-induced contraction needs the vessel to be in optimal conditions with an active metabolism of the muscular cells. These results are a crucial issue in the sense of explaining several biomedical mechanisms where hypothermia is implicated. The type of vessel implicated in procedures, such as isolated organ perfusion, extracorporeal circulation, and bypass surgery, must be taken into account.     

Nonlinear elastic analysis of blood vessels

Based on the theory of Green and Adkins [9], a strain energy function is proposed to describe the nonlinear mechanical behavior of arteries. The arterial tissue is assumed to be a nonlinear elastic, incompressible material with local triclinicity and transverse isotropy. Although the arterial tissue shows viscous phenomena, experimental results have indicated that viscosity is only a second-order effect as compared to the nonlinear elasticity of the tissue. The advantage of the formulation presented herein is that it is relatively simple and results in an accurate stress-strain relation that can be used readily for the study of wave propagations in the blood vessels. For nonlinear finite strain elasticity of the order two, ten elastic constants are needed to describe the material nonlinearity of the arterial tissue. Based on the orthogonal, transverse isotropies and the incompressibility conditions, ten constraint equations may be established and the elastic constants can be uniquely determined by correlating with the experimental results. An example of calculating these elastic constants is made by using the experimental data of Patel, et al. [14-17] for the intercoastal arteries in living dogs. The predicted mechanical behavior of canine arteries is quite satisfactory as compared to the experimental data except when the longitudinal and the circumferential stretches exceed 1.60. However, such a strain magnitude may not be expected in in-vivo arteries because of the constraints of peripheral connecting tissues. Otherwise, the strain energy function including higher order strain terms should be used.(ABSTRACT TRUNCATED AT 250 WORDS)     

Biomechanical properties of decellularized porcine common carotid arteries

To analyze the effects of decellularization on the biomechanical properties of porcine common carotid arteries, decellularization was performed by a detergent-enzymatic procedure that preserves extracellular matrix scaffold. Internal diameter, external diameter, and wall thickness were measured by optical microscopy on neighboring histological sections before and after decellularization. Rupture tests were conducted. Inner diameter and wall thickness were measured by echo tracking during pressure inflation from 10 to 145 mmHg. Distensibility and incremental elastic modulus were computed. At 10 mmHg, mean diameter of decellularized arteries was 5.38 mm, substantially higher than controls (4.1 mm), whereas decellularized and control arteries reached the same internal diameter (6.7 mm) at 145 mmHg. Wall thickness decreased 16% for decellularized and 32% for normal arteries after pressure was increased from 10 to 145 mmHg. Decellularized arteries withstood pressure >2,200 mmHg before rupture. At 145 mmHg, decellularization reduced compliance by 66% and increased incremental elastic modulus by 54%. Removal of cellular elements from media led to changes in arterial dimensions. Collagen fibers engaged more rapidly during inflation, yielding a stiffer vessel. Distensibility was therefore significantly lower (by a factor of 3) in decellularized than in normal vessels: reduced in the physiological range of pressures. In conclusion, decellularization yields vessels that can withstand high inflation pressures with, however, markedly different geometrical and biomechanical properties. This may mean that the potential use of a decellularized artery as a scaffold for the creation of xenografts may be compromised because of geometrical and compliance mismatch.     

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.     

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.     

Lymphatic vessels in the developing diaphragm of the rat.

Diaphragms of fetal, neonatal and young albino rats have been observed both under light and electron microscopes to examine the presence and distribution of lymphatic vessels and their morphological features. In fetal diaphragms of between 18 and 22 days of gestation, no normal lymphatic vessels can be seen; only after birth, specifically in neonatal and 2-day-old rats, small lymphatic vessels appear; they are in close proximity to the blood vessels in the inner areas of the muscle. As the rats get older, lymphatic vessels are also observed in the subserosa where an abundant connective tissue is present. The fine structure of diaphragmatic lymphatic vessels is different at different ages. In neonatal rats of up to 2 days, the endothelial wall is very thin and often holed. The relationships between contiguous endothelial cells are characterized by simple end-to-end or overlapping structures. The basement membrane is virtually absent. Within the first week of life, the endothelial wall becomes more complex; along the wall, complex interdigitations between two contiguous endothelial cells often touch. A discontinuous basement membrane and collagen and elastic fibers surround the vessels. In the older rats (from 14 to 25 to 140 days), next to the complex interdigitations which characterize the junction between two contiguous endothelial cells, cellular flaps interdigitate forming a channel which opens out either to the exterior or the interior of the vessel. Dense bundles of elastic and collagen fibers are closely apposed to the endothelial wall.     

Possibilities and limitations of histomorphometric studies of arterial vessels of the microcirculation

The choice of a procedure for measurements of blood vessels (b.v.) should depend on the type of b.v. investigated: For vessels defined anatomically and in respect to their function of organ supply the method of SUWA et al. (1961) is the most adequate one, provided that b.v. dilatations developing during lifetime and persisting after death as a result of special methods of fixation or shock-freezing or being generated by postmortal perfusion can be excluded. Changes in wall thickness of defined b.v. sectioned transversally can also be determined by measurement of the wall area and/or the maximal chord length. At undefined vessels procedures which measure the wall-to-lumen ratio are useless, because lumen changes are not quantifiable. Thus lumen changes can mask of mimick changes in wall thickness. The especially interesting b.v. either with a pathologically changed or without an elastic membrane are not measurable. The unsable method to detect changes in wall thickness at undefined b.v. is founded on the determination of the vessel wall area and/or the number of all the b.v. classified according to their diameter of a certain organ region. But the investigation of the smallest precapillary vessels presupposes their visualization by a special modification of the ATPase reaction.     

Viscoelasticity reduces the dynamic stresses and strains in the vessel wall: implications for vessel fatigue

The mechanical behavior of blood vessels is known to be viscoelastic rather than elastic. The functional role of viscoelasticity, however, has remained largely unclear. The hypothesis of this study is that viscoelasticity reduces the stresses and strains in the vessel wall, which may have a significant impact on the fatigue life of the blood vessel wall. To verify the hypothesis, the pulsatile stress in rabbit thoracic artery at physiological loading condition was investigated with a quasi-linear viscoelastic model, where the normalized stress relaxation function is assumed to be isotropic, while the stress-strain relationship is anisotropic and nonlinear. The artery was subjected to the same boundary condition, and the mechanical equilibrium equation was solved for both the viscoelastic and an elastic (which has a constant relaxation function) model. Numerical results show that, compared with purely elastic response, the viscoelastic property of arteries reduces the magnitudes and temporal variations of circumferential stress and strain. The radial wall movement is also reduced due to viscoelasticity. These findings imply that viscoelasticity may be beneficial for the fatigue life of blood vessels, which undergo millions of cyclic mechanical loadings each year of life.     

Ventricular and arterial wall stresses based on large deformation analyses

Assuming a spherical geometry for the left ventricle and a cylindrical geometry for arteries, wall stresses and elastic stiffnesses are evaluated on the basis of a large elastic deformation theory. On the basis of canine pressure-volume data, the numerical results indicate marked gradients of stress in the endocardial layers even for thin-walled vessels, a result not predicted by the classical theory of elasticity. These high gradients of stress are due to the fact that the elastic stiffness of the wall material increases with the stress which reaches maximum levels in the endocardial layers. The high stresses may be responsible for ischemia of the left ventricle and be a triggering mechanism for atherosclerosis.     

Examination of elastic non-uniformity in the arterial system using a hydraulic model

Pressure wave propagation has been examined in a model artery with spatially varying compliance. Although results were affected by viscous losses, appropriate allowance for such losses produced agreement between experimental findings and predictions of linear wave transmission theory. Particularly, the ability of non-uniformity of the tube wall to generate amplification of the pressure wave was confirmed. However, extrapolation to the physiological situation suggests that reflections from discrete sites in peripheral beds have a greater effect on pressure wave propagation than does elastic non-uniformity of major vessels. A theoretical analysis has demonstrated that the effects of elastic non-uniformity can be interpreted as the integrated effects of infinitesimal reflections from each progressive increment in wall stiffness.     

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.     

A Rapid Digestive Technique to Expose Networks of Vascular Elastic Fibers for Sem Observation

The NaOH sonication digestion technique permits rapid isolation and exposure of intact networks of elastic fibers in vascular tissue for 3-dimensional observation with the SEM. The configuration of the network of elastic fibers within the vascular wall of large elastic arteries (aorta) is generally agreed to be a flexible framework through which smooth muscle cells and collagenous fibers are interwoven. However, the configuration of elastic fiber networks in muscular arteries, medium sized veins and smaller vessels remains unknown. When the lengthy standard biochemical elastin purification techniques were applied to vessels containing lesser amounts of elastic tissue and finer elastic fibers, the vessels were completely digested. In contrast, the digestion and sonication technique isolated and exposed intact networks of delicate elastic fibers in blood vessels which do not contain large amounts of elastic tissue. Unfixed vessels were cut into short segments, placed in 0.5 N NaOH and sonicated for 20-40 min. The specimens were rinsed in deionized distilled H₂O, then autoclaved for 30 min. The tissue was rinsed a second time, fixed and processed routinely for SEM. Elastic stains and enzymatic digestion with chromatographically purified elastase and collagenase confirmed that the digestion and sonication technique produced clean, isolated networks of elastic fibers. Knowledge of the configuration of the networks of elastic fibers in different vessels enhances understanding of distensibility characteristics of individual vessels and serves as a baseline for studying alterations in the elastic framework which occur during aging and disease processes such as atherosclerosis, arterial hypertension and aneurysms.     

Effects of elastic property of the wall on flow characteristics through arterial stenoses

Hemodynamic characteristics of blood flow through arterial stenoses are numerically investigated in this work. The blood is assumed as a Newtonian fluid and the pulsatile nature of flow is modeled by using measured values of the flowrate and pressure for the canine femoral artery. An isotropic elastic and incompressible material is assumed for the wall at each axial section, but a non-uniform distribution of the shear modulus in axial direction is used to model the high stiffness of the wall at the stenosis location. Full Navier equations for a thick wall are used as the governing equations for the wall displacements. A continuous grid extending over the flow field and the wall is considered and governing equations are transformed for use in the computational domain. Discretized forms of the transformed wall and flow equations, which are coupled through the boundary conditions at their interface, are obtained by control volume method and simultaneously solved using the well-known SIMPLER algorithm. To study the effects of wall deformability, solutions are obtained for both rigid and elastic walls. The results indicate that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Displacement and stress distributions in the wall are presented.     

Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.     

A rapid digestive technique to expose networks of vascular elastic fibers for SEM observation

The NaOH sonication digestion technique permits rapid isolation and exposure of intact networks of elastic fibers in vascular tissue for 3-dimensional observation with the SEM. The configuration of the network of elastic fibers within the vascular wall of large elastic arteries (aorta) is generally agreed to be a flexible framework through which smooth muscle cells and collagenous fibers are interwoven. However, the configuration of elastic fiber networks in muscular arteries, medium sized veins and smaller vessels remains unknown. When the lengthy standard biochemical elastin purification techniques were applied to vessels containing lesser amounts of elastic tissue and finer elastic fibers, the vessels were completely digested. In contrast, the digestion and sonication technique isolated and exposed intact networks of delicate elastic fibers in blood vessels which do not contain large amounts of elastic tissue. Unfixed vessels were cut into short segments, placed in 0.5 N NaOH and sonicated for 20-40 min. The specimens were rinsed in deionized distilled H2O, then autoclaved for 30 min. The tissue was rinsed a second time, fixed and processed routinely for SEM. Elastic stains and enzymatic digestion with chromatographically purified elastase and collagenase confirmed that the digestion and sonication technique produced clean, isolated networks of elastic fibers. Knowledge of the configuration of the networks of elastic fibers in different vessels enhances understanding of distensibility characteristics of individual vessels and serves as a baseline for studying alterations in the elastic framework which occur during aging and disease processes such as atherosclerosis, arterial hypertension and aneurysms.     

Nonlinear Anisotropic Elastic Properties of the Canine Aorta

A nonlinear theory of large elastic deformations of the aortic tissue has been developed. The wall tissue has been considered to be incompressible and curvilinearly orthotropic. The strain energy density function for the tissue is expressed as a polynomial in the circumferential and longitudinal Green-St. Venant strains. Limiting application to states of strains wherein the geometric axes are the principal axes and truncating the energy expression to include terms with highest degrees 2, 3, and 4, three expressions with 3, 7, and 12 constitutive constants are obtained. Results of application of these expressions to data from three series of in vitro and in vivo experiments involving 31 dogs have been presented. Whereas all the three expressions are found to be applicable to various degrees, the third-degree expression for the strain energy density function with seven constitutive constants is particularly recommended for general use.     

The Mechanical Buckling of Curved Arteries^*

Though tortuosity and kinking are often observed in various arteries and arterioles, little is known about the underlying mechanisms. This paper presents a biomechanical analysis of bent buckling in long arterial segments with a small initial curvature using a thick-walled elastic cylindrical arterial model. The critical buckling pressure was established as a function of wall stiffness, wall dimensions, and the axial tension (or axial stretch ratio). The effects of both wall dimensions and axial stretch ratio on the critical pressure, as well as the thin-walled approximation were discussed. The buckling equation sheds light on the biomechanical mechanism of artery tortuosity and provides guidance for the development of new techniques to treat and prevent artery tortuosity and kinking.