Large-scale simulation of arterial walls: mechanical modeling

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre-Hadamard condition automatically. When arteries are overstretched, discontinuous damage is observed. For the modeling of this effect we apply a damage model, which basically assumes that the damage occurs mainly in fiber direction. For the numerical simulation we consider an atherosclerotic artery and apply a high internal pressure which is comparable to the pressure applied during a balloon-angioplasty. The 3D-discretization results in a large system of equations, therefore, a parallel algorithm using FETI-DP is applied to solve the boundary value problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling residual stresses in arterial walls based on anisotropic growth

With the aim of obtaining a general local formulation for anisotropic growth in soft biological tissues, a model based on the multiplicative decomposition of the growth tensor is formulated. The two parts of the growth tensor are associated with the main anisotropy directions. Together with an anisotropic driving force, the model enables an effective stress reduction by including growth-induced residual stresses, which is demonstrated in a numerical example of an idealized arterial segment. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of arterial wall shear stress

The arteries can remodel their structure and function to adapt themselves to the mechanical environment. In various factors that lead to vascular remodeling, the shear stress on the arterial wall induced by the blood flow is of great importance. However, there are many technique difficulties in measuring the wall shear stress directly at present. In this paper, through analyzing the pulsatile blood flow in arteries, a method has been proposed that can determine the wall shear stress quantitatively by measuring the velocity on the arterial axis, and that provides a necessary means to discuss the influence of arterial wall shear stress on vascular remodeling.     

The application of densitometry for determining the optical density of the intervertebral disk in man

An experimental study was carried out on the optical density of the intervertebral disks of the lumbar vertebrae and the change in their volume upon axial strain. A definite dependence was found between increasing axial strain and the changes in the optical density of the annulus fibrosus and nucleus pulposus of the disk.Riga Medical Institute. Translated from Mekhanika Polimerov, No. 2, pp. 315–319, March–April, 1975.     

Damping mechanisms and shock-like transitions in the human arterial tree

The unidirectional nature of blood flow in mammalian arteries encourages the modelling of propagating pressure and flow pulses in the arterial tree by means of a one-dimensional mathematical approximation. It has been shown that this approximation yields realistic results in the proximal as well as in the distal regions of a simulated arterial conduit, providing that the damping induced by the viscoelasticity of the vessel walls as well as by the viscosity of the blood are properly taken into account. Often, models were formulated on the basis of an elastic formulation of the arterial wall properties and of an approximation for the damping due to blood viscosity which is derived from parabolic velocity profiles (Poiseuille model). Yet, such models are known to produce shock-like transitions in the propagating pulses which are not observed in man under physiological conditions.The viscoelastic damping characteristics of the vessel walls are such that they reduce the tendency of shock formation in the model. This can be shown with the aid of a wave front expansion, from which criteria for the steepening of wave fronts are derived. The application of the results to the human arterial system shows that shock-like waves are not to be expected under normal conditions. However, in case of a pathologically increased pressure rise at the root of the aorta, shock-like transitions may still develop in the periphery. Such circumstances can occur, e. g., in cases of severe aortic valve insufficiency (without aortic stenosis).Furthermore, the damping characteristics associated with blood viscosity are shown to impede mathematical shock formation in the strict sense a priori. Steepening of a wave front according to the criteria mentioned above still may occur, however, with increasing steepening this process is gradually counterbalanced by the dissipative mechanism induced by viscous friction.Finally, the relative influence of the damping due to wall viscoelasticity and blood viscosity is discussed as a function of the relevant parameters describing the geometry and pulsatility of the typical blood flow conditions.

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Zusammenfassung Die vorwiegend axialen Strömungsverhältnisse in den menschlichen Arterien rechtfertigen es, die Ausbreitung der Fluß- und Druckpulse im arteriellen System auf der Basis eines eindimensionalen mathematischen Strömungsmodelles zu analysieren. Es konnte in früheren Arbeiten gezeigt werden, daß eine solche Näherung realistische Resultate sowohl im proximalen als auch im distalen Abschnitt eines simulierten Arterienastes liefert, falls die Dämpfung aufgrund des viskoelastischen Verhaltens der Blutgefäßwände sowie der Viskosität des Blutes in geeigneter Weise mitberücksichtigt werden. Häufig wurde in bisherigen Modellen eine elastische Formulierung der Gefäßwandeigenschaften verwendet und eine Näherung für den Einfluß der Blutviskosität eingesetzt, welche aus der Annahme parabolischer Geschwindigkeitsprofile folgt (Poiseuille-Näherung). Es ist aber bekannt, daß solche Modelle im Verlauf der Ausbreitung eines Pulses stoßartige Wellenfronten ergeben, welche im Menschen unter natürlichen Bedingungen nicht beobachtet werden.Die viskoelastischen Dämpfungseigenschaften der Gefäßwände vermindern die Tendenz des Modelles, Stoßwellen zu bilden. Dies wird mit Hilfe einer Wellenfrontentwicklung gezeigt, aufgrund welcher Kriterien für das Aufsteilen einer Welle hergeleitet werden. Die Anwendung der Resultate auf das menschliche arterielle System zeigt, daß stoßartige Wellen unter normalen physiologischen Bedingungen nicht zu erwarten sind. Im Falle einer pathologisch erhöhten Druckanstiegsgeschwindigkeit an der Wurzel der Aorta jedoch können stoßartige Übergänge auftreten. Solche Bedingungen ergeben sich beispielsweise bei schweren Aortenklappeninsuffizienzen (ohne gleichzeitige Stenose). Weiterhin wird gezeigt, daß die Dämpfung aufgrund der Blutviskosität Stoßwellenbildung im mathematischen Sinne überhaupt verhindert. Aufsteilen einer Wellenfront im Sinne der erwähnten Kriterien kann zwar auftreten; dieser Prozeß wird jedoch durch die visköse Dissipation allmählich aufgehoben.Schließlich wird der relative Einfluß der Dämpfung aufgrund der Wandviskoelastizität und der Blutviskosität in Funktion der relevanten Parameter diskutiert, welche die Geometrie und Pulsatilität typischer Blutströmungen bestimmen.

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Supported in part by grant No. 3.804.0.82 of the Swiss National Science Foundation.     

Features of biomechanical properties of human coronary arteries

The mechanical properties, biochemical composition, and structure of proximal and distal parts of the right and anterior descending branch of the left coronary arteries were studied experimentally. The vessels were removed during an autopsy of 121 males and 84 females aged from one day to 80 years. The material was divided into six age groups. From the proximal and distal parts, branchless segments of 15–20 mm long vessels were cut out. The mechanical properties of the coronary arteries were determined by passing a fluid at a pressure ranging from 0 to 240 mm Hg. It was found that the part of the wall of the coronary artery adjacent to the myocardium was thicker than the other part of the arterial wall. With increasing age, the mean thickness of the wall of both coronary arteries increased, but the wall thickening process was nonuniform in character in the proximal and distal parts, and individual layers. The changes of the stretch ratio and tangential modulus in the circumferential direction with age and sex were also investigated. The greatest changes in the wall thickness and mechanical parameters were found in the left coronary artery wall for mean over 40 years and in the right coronary artery wall for females over 50 years. The results of the biochemical and densitometric investigations agree well.     

Features of biomechanical properties of human coronary arteries

The mechanical properties, biochemical composition, and structure of proximal and distal parts of the right and anterior descending branch of the left coronary arteries were studied experimentally. The vessels were removed during an autopsy of 121 males and 84 females aged from one day to 80 years. The material was divided into six age groups. From the proximal and distal parts, branchless segments of 15–20 mm long vessels were cut out. The mechanical properties of the coronary arteries were determined by passing a fluid at a pressure ranging from 0 to 240 mm Hg. It was found that the part of the wall of the coronary artery adjacent to the myocardium was thicker than the other part of the arterial wall. With increasing age, the mean thickness of the wall of both coronary arteries increased, but the wall thickening process was nonuniform in character in the proximal and distal parts, and individual layers. The changes of the stretch ratio and tangential modulus in the circumferential direction with age and sex were also investigated. The greatest changes in the wall thickness and mechanical parameters were found in the left coronary artery wall for mean over 40 years and in the right coronary artery wall for females over 50 years. The results of the biochemical and densitometric investigations agree well.Published in Mekhanika Kompozitnykh Materialov, Vol. 35, No. 2, pp. 221–238, March–April, 1999.     

Changes in the mechanical properties of human coronary arteries with age

Conclusions We were the first to experimentally find that the greatest changes in E₂, E_2(K) ^tan, _2(K), and W_(K) of the human cardial coronary arteries are observed in people who have reached the age of 40 years. To compare the experimental data, all the parameters of the mechanical properties of the coronary arteries were determined at one and the same stress level — 0.01 kgf/mm², which on an average corresponds to an increased intravascular pressure of within 200 to 240 mm Hg. Under elinical conditions, this pressure is considered critical for arterial vessels affected by atherosclerosis. The most marked shifts in the changes of the mechanical properties were found in the superior part of the left coronary artery. At the age of 40, a considerable decrease in the deformability of the arterial walls in the circular direction is observed, and the walls become more rigid. At the same time, the vessel wall becomes thicker, especially because of increase in its internal layer — the intima. All this facilitates the development of intravascular thrombosis in people suffering from coronary sclerosis and leads to the development of acute stenosis, frequently exceeding 75% of the clearance area of the vessel, or to a complete obstruction of the superior part of the left coronary artery. If the inferior part of this vessel still retains a fair deformability, it is possible to perform a reconstructive operation, by means of which normal blood circulation is restored from the initial part of the ascending aorta to the inferior section of the corresponding coronary artery. From the experimentally found characteristics of the mechanical properties of the inferior parts of coronary vessels, we can select the best autoplastic material — a vessel segment derived from the patient himself — with properties similar to those of the affected vessel, which ensures the successful performance of such an operation, namely, the reconstruction of the human coronary vessels.Lecture delivered at the Third All-Union Conference on Polymer Mechanics, Riga, November 10–12, 1976.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2,, pp. 277–282, March–April, 1977.     

On the motion of magnetic domain walls

In certain regimes and for relative small applied fields, magnetic domain walls behave as mechanical particles in a viscous fluid. By analogy their dynamic can be described with an evolution equation of the form Mq̈ + β^–1 q̇ = H (1) where M and are the effective wall mass and wall mobility, and H is an applied forcing. These effective parameters depend on the particular geometrical configuration and dynamical model considered. We present a method that can be used to systematically estimate the wall mass and wall mobility in terms of the static wall geometry and various physical constants. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of the ion permeability of vessel walls

A fundamentally new technique has been developed for determining the local ion permeability of the intact vessel wall in the presence of continuous blood flow. This technique has been used to investigate the permeability of the canine femoral artery with respect to Na and Cl ions using radioactive indicators. It is shown that for these two ions the permeability mechanism is different. The quantitative diffusion characteristics of C1 ions in the natural vessel wall are obtained.     

On the boundary layer at perforated walls

Zusammenfassung Für Windkanalversuche im Geschwindigkeitsgebiet der schallnahen Str?mungen werden seit einigen Jahren perforierte oder geschlitzte Kanalw?nde verwendet. Frühere experimentelle Untersuchungen [3] haben angedeutet, dass die Grenzschicht entlang solchen W?nden der turbulenten Mischzone in einer freien Strahlgrenze gleicht. Diese Beobachtung legt die Annahme affiner Geschwindigkeitsprofile und linear anwachsender Mischzonenbreite nahe (Figur 1). Wird die Prandtlsche Mischl?nge für die turbulente Schubspannung der Mischzonenbreite proportional gesetzt, dann resultiert eine konstante Schubspannung entlang der perforierten Wand. Diese kann abgesch?tzt werden, wenn man die perforierte Wand als eine Folge von soliden Wandstücken und freien Strahlgrenzen betrachtet (Figur 4) und damit eine mittlere Wandschubspannung berechnet. Mit dieser ist dann auch die aus dem Kanal entweichende Luftmenge bestimmt. Versuche mit einer perforierten und einer quergeschlitzten Wand best?tigen die theoretischen Voraussagen.

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Part of this research was sponsored by the Air Research and Development Command under Contract AF 18(600)-1330 and part by the ARO, Inc., under Subcontract T-306.     

Motion of magnetic domain walls

We consider the dynamics of one-dimensional micromagnetic domain walls in layers of uniaxial anisotropy. In the regime of bulk materials, i.e. when the thickness is assumed to be infinite, and the magnetostatic interaction terms appear as local quantities, explicit traveling wave solutions for the corresponding Landau-Lifshitz equation, known asWalker exact solutions, can be constructed. A natural question is whether this construction can be perturbed to the non-local regime of layers of finite thickness. Our stability analysis gives an affirmative answer. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the elastic and damping properties of the human femur in bending

The results of an experimental investigation of the deflection of the human femur subjected to a bending moment are presented. The existence of two principal bending planes is established. The equivalent flexural rigidity of the dry bone is calculated for both static and dynamic loading. It is found that the human femur has a nonlinear elastic characteristic. The stiffness and damping properties obtained in static and dynamic testing are compared.Paper presented at the First All-Union Conference on Engineering and Medical Biomechanics, Riga, October, 1975.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 642–646, July–August, 1975.     

Gravitational repulsion of spherical domain walls

It is shown that the equations of motion and Tolman's formula require reverse flow of time on a spherical domain wall. Moreover, the active gravitational mass of the wall becomes negative. The critical mass and radius of a bubble are found for realistic models.Institute of Physics, Georgian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 303–311, August, 1994.