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.     

Validation of a patient-specific one-dimensional model of the systemic arterial tree

The aim of this study is to develop and validate a patient-specific distributed model of the systemic arterial tree. This model is built using geometric and hemodynamic data measured on a specific person and validated with noninvasive measurements of flow and pressure on the same person, providing thus a patient-specific model and validation. The systemic arterial tree geometry was obtained from MR angiographic measurements. A nonlinear viscoelastic constitutive law for the arterial wall is considered. Arterial wall distensibility is based on literature data and adapted to match the wave propagation velocity of the main arteries of the specific subject, which were estimated by pressure waves traveling time. The intimal shear stress is modeled using the Witzig-Womersley theory. Blood pressure is measured using applanation tonometry and flow rate using transcranial ultrasound and phase-contrast-MRI. The model predicts pressure and flow waveforms in good qualitative and quantitative agreement with the in vivo measurements, in terms of wave shape and specific wave features. Comparison with a generic one-dimensional model shows that the patient-specific model better predicts pressure and flow at specific arterial sites. These results obtained let us conclude that a patient-specific one-dimensional model of the arterial tree is able to predict well pressure and flow waveforms in the main systemic circulation, whereas this is not always the case for a generic one-dimensional model.     

Underlying mechanisms of symmetric calcium wave propagation in rat ventricular myocytes

Calcium waves in heart cells are mediated by diffusion-coupled calcium-induced calcium release. The waves propagate in circular fashion. This is counterintuitive in view of the accepted ultrastructure of the cardiac myocyte. The density of calcium release sites in the transverse direction is four times higher than in the longitudinal direction. Simulations with release sites localized along Z-lines and isotropic diffusion yielded highly elliptical, nonphysiological waves. We hypothesized that subcellular organelles counteracted the higher release site density along the Z-lines by acting as transverse diffusion barriers and sites of active calcium uptake. We quantified the reduction of transverse diffusion by microinjecting cells with the nonreactive dye fluorescein. The ratio of the radial diffusion coefficient to the longitudinal coefficient was 0.39. Inhibition of mitochondrial uptake by rotenone accelerated the wave in the transverse direction. Simulations with release sites clustered at the Z-lines and a transverse diffusion coefficient 50% of the longitudinal coefficient generated waves of ellipticity 2/1 (major axis along the Z-line). Introducing additional release sites between the Z-lines at a density 20% of that on the Z-lines produced circular waves. The experiments and simulations support the presence of transverse diffusion barriers, additional uptake sites, and possibly intermediate release sites as well.     

On arterial fiber dispersion and auxetic effect

There are two polar contemporary approaches to the constitutive modeling of arterial wall with anisotropy induced by collagen fibers. The first one is based on the angular integration (AI) of the strain energy on a unit sphere for the analytically defined fiber dispersion. The second one is based on the introduction of the generalized structure tensors (GST). AI approach is very involved computationally while GST approach requires somewhat complicated procedure for the exclusion of compressed fibers.We present some middle ground models, which are based on the use of 16 and 8 structure tensors. These models are moderately involved computationally and they allow excluding compressed fibers easily. We use the proposed models to study the role of the fiber dispersion in the constitutive modeling of the arterial wall. Particularly, we study the auxetic effect which can appear in anisotropic materials. The effect means thickening of the tissue in the direction perpendicular to its stretching. Such an effect was not observed in experiments while some simple anisotropic models do predict it. We show that more accurate account of the fiber dispersion suppresses the auxetic effect in a qualitative agreement with experimental observations.     

Propagation of MHD waves in a plasma in a sheared magnetic field with straight field lines

The propagation of MHD plasma waves in a sheared magnetic field is investigated. The problem is solved using a simplified model: a cold plasma is inhomogeneous in one direction, and the magnetic field lines are straight. The waves are assumed to travel in the plane perpendicular to the radial coordinate (i.e., the coordinate along which the plasma and magnetic field are inhomogeneous). It is shown that the character of the singularity at the resonance surface is the same as that in a homogeneous magnetic field. It is found that the shear gives rise to the transverse dispersion of Alfvén waves, i.e., the dependence of the radial component of the wave vector on the wave frequency. In the presence of shear, Alfvén waves are found to propagate across magnetic surfaces. In this case, the transparent region is bounded by two turning points, at one of which, the radial component of the wave vector approaches infinity and, at the other one, it vanishes. At the turning point for magnetosonic waves, the electric and magnetic fields are finite; however, the radial component of the wave vector approaches infinity, rather than vanishes as in the case with a homogeneous field.     

A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca²⁺) in living cells. Modeling Ca²⁺ release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca²⁺ wave speed on physiologically important parameters such as the threshold for Ca²⁺ release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca²⁺ resequestration from the cytosol to the ER, and the total [Ca²⁺] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca²⁺ waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called ‘tango waves’ and the dependence of Ca²⁺ wave speed on the total [Ca²⁺].      

Effect of the ponderomotive force on the development of beam-plasma instability

The amplitude of the wave generated in a plasma during the development of beam-plasma instability is nonuniform in the longitudinal direction. The ponderomotive force associated with this nonuniformity leads to a redistribution of the plasma density; as a result, the wave amplitude and its spatial distribution change. As the beam current grows, the ponderomotive force plays an increasingly important role and radically changes the mechanism by which the beam-plasma instability saturates. Ion acoustic waves generated by the ponderomotive force propagate in the direction opposite to the propagation direction of the beam, thereby ensuring distributed feedback and giving rise to a strong low-frequency self-modulation of the wave amplitude and phase. Results are presented from experimental investigations of the self-modulation regime of the beam-plasma instability in a magnetized plasma waveguide. Theoretical estimates of the parameters of the low-frequency self-modulation agree well with the experimental data.     

Theory of transverse surface electromagnetic waves propagating along a plasma-metal boundary with a finite radius of curvature in a magnetic field

The spectra of electromagnetic waves propagating perpendicular to the axis of a plasma-filled metal waveguide in a magnetic field are studied with allowance for the effects exerted upon the wave frequency by the radial plasma density variation and by the emission of waves through a narrow axial slit in a waveguide wall. The case of wave propagation along the boundary between a plasma and a cylindrical metal waveguide wall with a periodically varying radius of curvature is also considered. The electromagnetic properties of the plasma are described by a dielectric tensor in the hydrodynamic approximation. The spatial distribution of the wave field is determined by the method of successive approximations. Results are presented from both analytical and numerical investigations. Analytical expressions for the corrections to the wave frequency due to the emission of the wave energy from the waveguide and due to the slight corrugation of the waveguide wall are obtained. The rates of wave damping due to the emission of the wave energy through a narrow axial slit and due to collisions between the plasma particles are found. The correction to the frequency that comes from the periodic variation of the radius of curvature of the plasma surface is calculated to within terms proportional to the square of the small parameter describing the azimuthal corrugation of the waveguide wall. The effect of the radial plasma density variation on the dispersion of the surface modes is examined both analytically and numerically.     

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.     

Activation of a cGMP-sensitive calcium-dependent chloride channel may cause transition from calcium waves to whole cell oscillations in smooth muscle cells

In vitro, alpha-adrenoreceptor stimulation of rat mesenteric small arteries often leads to a rhythmic change in wall tension, i.e., vasomotion. Within the individual smooth muscle cells of the vascular wall, vasomotion is often preceded by a period of asynchronous calcium waves. Abruptly, these low-frequency waves may transform into high-frequency whole cell calcium oscillations. Simultaneously, multiple cells synchronize, leading to rhythmic generation of tension. We present a mathematical model of vascular smooth muscle cells that aims at characterizing this sudden transition. Simulations show calcium waves sweeping through the cytoplasm when the sarcoplasmic reticulum (SR) is stimulated to release calcium. A rise in cGMP leads to the experimentally observed transition from waves to whole cell calcium oscillations. At the same time, membrane potential starts to oscillate and the frequency approximately doubles. In this transition, the simulated results point to a key role for a recently discovered cGMP-sensitive calcium-dependent chloride channel. This channel depolarizes the membrane in response to calcium released from the SR. In turn, depolarization causes a uniform opening of L-type calcium channels on the cell surface, stimulating a synchronized release of SR calcium and inducing the shift from waves to whole cell oscillations. The effect of the channel is therefore to couple the processes of the SR with those of the membrane. We hypothesize that the shift in oscillatory mode and the associated onset of oscillations in membrane potential within the individual cell may underlie sudden intercellular synchronization and the appearance of vasomotion.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes.

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.     

Arterial pulsation-driven cerebrospinal fluid flow in the perivascular space: a computational model

This study was conducted to determine whether local arterial pulsations are sufficient to cause cerebrospinal fluid (CSF) flow along perivascular spaces (PVS) within the spinal cord. A theoretical model of the perivascular space surrounding a "typical" small artery was analysed using computational fluid dynamics. Systolic pulsations were modelled as travelling waves on the arterial wall. The effects of wave geometry and variable pressure conditions on fluid flow were investigated. Arterial pulsations induce fluid movement in the PVS in the direction of arterial wave travel. Perivascular flow continues even in the presence of adverse pressure gradients of a few kilopascals. Flow rates are greater with increasing pulse wave velocities and arterial deformation, as both an absolute amplitude and as a proportion of the PVS. The model suggests that arterial pulsations are sufficient to cause fluid flow in the perivascular space even against modest adverse pressure gradients. Local increases in flow in this perivascular pumping mechanism or reduction in outflow may be important in the etiology of syringomyelia.     

Arterial Pulsation-driven Cerebrospinal Fluid Flow in the Perivascular Space: A Computational Model

This study was conducted to determine whether local arterial pulsations are sufficient to cause cerebrospinal fluid (CSF) flow along perivascular spaces (PVS) within the spinal cord. A theoretical model of the perivascular space surrounding a "typical" small artery was analysed using computational fluid dynamics. Systolic pulsations were modelled as travelling waves on the arterial wall. The effects of wave geometry and variable pressure conditions on fluid flow were investigated. Arterial pulsations induce fluid movement in the PVS in the direction of arterial wave travel. Perivascular flow continues even in the presence of adverse pressure gradients of a few kilopascals. Flow rates are greater with increasing pulse wave velocities and arterial deformation, as both an absolute amplitude and as a proportion of the PVS. The model suggests that arterial pulsations are sufficient to cause fluid flow in the perivascular space even against modest adverse pressure gradients. Local increases in flow in this perivascular pumping mechanism or reduction in outflow may be important in the etiology of syringomyelia.     

Surface artery volume pulse wave measurement - verification of a new method

The aim of this paper is to prove the possible reproducibility of measurement with a new developed device for artery elasticity monitoring and determining the standard of major pulse wave parameters. As a measurement sensor, a conic probe with thin convex membrane was used. This technique allows setting an arbitrary pressure to a measured surface artery. We measured pulse waves on the radial arteries of 108 individuals. We expected similar features in arterial wall elasticity. We concentrated primarily on the amount of subcutaneous fat. For the measured waves we evaluated five following pulse wave parameters: relative crest time, elasticity index, dicrotic wave attenuation, dicrotic wave time and interwave distance. There were no significant differences in measured pulse wave parameters among the tested groups of subjects.     

A calcium wave mediated by gap junctions coordinates a rhythmic behavior in C. elegans

Intercellular calcium waves can be observed in adult tissues, but whether they are instructive, permissive, or even required for behavior is predominantly unknown. In the nematode Caenorhabditis elegans, a periodic calcium spike in a pacemaker cell initiates a calcium wave in the intestine. The calcium wave is followed by three muscle contractions that comprise the defecation motor program. Normal wave propagation requires the pannexin gap-junction subunit INX-16 at the interfaces of the intestinal cells. In the absence of this gap-junction subunit, calcium waves are frequently absent. The remaining waves are slow, initiate at abnormal locations, or travel in the opposite direction. Abnormal waves are associated with parallel effects in the first step of the motor program: The contractions of the overlying muscles fail to propagate beyond the pacemaker cell, are slow, initiate in abnormal locations, or are reversed. Moreover, the last two motor steps are predominantly absent. Finally, the absence of this gap-junction subunit also affects the reliability of the pacemaker cell; cycle timing is often irregular. These data demonstrate that pannexin gap junctions propagate calcium waves in the C. elegans intestine. The calcium waves instruct the motor steps and regulate the pacemaker cell's authority and reliability.     

Mechanisms of Propagation of Intercellular Calcium Waves in Arterial Smooth Muscle Cells

In rat mesenteric arteries, smooth muscle cells exhibit intercellular calcium waves in response to local phenylephrine stimulation. These waves have a velocity of ∼20 cells/s and a range of ∼80 cells. We analyze these waves in a theoretical model of a population of coupled smooth muscle cells, based on the hypothesis that the wave results from cell membrane depolarization propagation. We study the underlying mechanisms and highlight the importance of voltage-operated channels, calcium-induced calcium release, and chloride channels. Our model is in agreement with experimental observations, and we demonstrate that calcium waves presenting a velocity of ∼20 cells/s can be mediated by electrical coupling. The wave velocity is limited by the time needed for calcium influx through voltage-operated calcium channels and the subsequent calcium-induced calcium release, and not by the speed of the depolarization spreading. The waves are partially regenerated, but have a spatial limit in propagation. Moreover, the model predicts that a refractory period of calcium signaling may significantly affect the wave appearance.     

The effect of shear stress on solitary waves in arteries

In the present work, we study the propagation of solitary waves in a prestressed thick walled elastic tube filled with an incompressible inviscid fluid. In order to include the geometric dispersion in the analysis the wall inertia and shear deformation effects are taken into account for the inner pressure-cross-sectional area relation. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is examined. It is shown that, contrary to thin tube theories, the present approach makes it possible to have solitary waves even for a Mooney-Rivlin (M-R) material. Due to dependence of the coefficients of the governing Korteweg-deVries equation on initial deformation, the solution profile changes with inner pressure and the axial stretch. The variation of wave profiles for a class of elastic materals are depicted in graphical forms. As might be seen from these illustrations, with increasing thickness ratio, the profile of solitary wave is steepened for a M-R material but it is broadened for biological tissues.     

A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution

Accurate constitutive models are required to gain further insight into the mechanical behavior of cardiovascular tissues. In this study, a structural constitutive framework for cardiovascular tissues is introduced that accounts for the angular distribution of collagen fibers. To demonstrate its capabilities, the model is applied to study the biaxial behavior of the arterial wall and the aortic valve. The pressure-radius relationships of the arterial wall accurately describe experimentally observed sigma-shaped curves. In addition, the nonlinear and anisotropic mechanical properties of the aortic valve can be analyzed with the proposed model. We expect that the current model offers strong possibilities to further investigate the complex mechanical behavior of cardiovascular tissues, including their response to mechanical stimuli.