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  • 1.
    Ghosh, Arpan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Mathematical modelling of flow through thin curved pipes with application to hemodynamics2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The problem of mathematical modelling of incompressible flows with low velocities through narrow curvilinear pipes is addressed in this thesis. The main motivation for this modelling task is to eventually model the human circulatory system in a simple way that can facilitate the medical practitioners to efficiently diagnose any abnormality in the system. The thesis comprises of four articles.

    In the first article, a two-dimensional model describing the elastic behaviour of the wall of a thin, curved,  exible pipe is presented. The wall is assumed to have a laminate structure consisting of several anisotropic layers of varying thickness. The width of the channel is allowed to vary along the pipe. The two-dimensional model takes the interactions of the wall with any surrounding material and the  fluid  flow into account and is obtained through a dimension reduction procedure. Examples of canonical shapes of pipes and their walls are provided with explicit systems of differential equations at the end.

    In the second article, a one-dimensional model describing the blood flow through a moderately curved, elastic blood vessel is presented. The two-dimensional model presented in the first paper is used to model the vessel wall while linearized Navier-Stokes equations are used to model the  flow through the channel. Surrounding muscle tissues and presence of external forces other than gravity are taken into account. The model is again obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow.

    The one-dimensional model described in the second paper is used to derive a simplified one-dimensional model of a false aneurysm which forms the subject of the third article. A false aneurysm is an accumulation of blood outside a blood vessel but confined by the surrounding muscle tissue. Numerical simulations are presented which demonstrate different characteristics associated with a false aneurysm.

    In the final article, a modified Reynolds equation, along with its derivation from Stokes equations through asymptotic methods, is presented. The equation governs the steady flow of a fluid with low Reynolds number through a narrow, curvilinear tube. The channel considered may have large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed. These approximations satisfy a modified Poiseuille equation. A justification for the approximations is provided along with a comparison with a simpler case.

    List of papers
    1. A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE
    Open this publication in new window or tab >>A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE
    2018 (English)In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 71, no 3, p. 349-367Article in journal (Refereed) Published
    Abstract [en]

    We present a two-dimensional model describing the elastic behaviour of the wall of a curved flexible pipe. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the channel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with any surrounding or supporting material and the fluid flow into account and is obtained via a dimension reduction procedure. The curvature and twist of the pipes axis as well as the anisotropy of the laminate wall present the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of pipes and their walls are supplied with explicit systems of differential equations at the end.

    Place, publisher, year, edition, pages
    OXFORD UNIV PRESS, 2018
    National Category
    Other Electrical Engineering, Electronic Engineering, Information Engineering
    Identifiers
    urn:nbn:se:liu:diva-150869 (URN)10.1093/qjmam/hby009 (DOI)000441808700006 ()
    Note

    Funding Agencies|Russian Foundation of Basic Research [18-01-00325]

    Available from: 2018-09-06 Created: 2018-09-06 Last updated: 2019-04-16
    2. A one dimensional model of blood flow through a curvilinear artery
    Open this publication in new window or tab >>A one dimensional model of blood flow through a curvilinear artery
    2018 (English)In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 63, p. 633-643Article in journal (Refereed) Published
    Abstract [en]

    We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier-Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved.

    Place, publisher, year, edition, pages
    ELSEVIER SCIENCE INC, 2018
    Keywords
    Blood flow; Curvilinear vessel; Asymptotic analysis; Dimension reduction; Numerical simulation
    National Category
    Fluid Mechanics and Acoustics
    Identifiers
    urn:nbn:se:liu:diva-151627 (URN)10.1016/j.apm.2018.07.019 (DOI)000444362800034 ()
    Note

    Funding Agencies|Russian Foundation of Basic Research [18-01-00325]

    Available from: 2018-10-09 Created: 2018-10-09 Last updated: 2019-04-16
  • 2.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Karlsson, Matts
    Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
    A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients2018Report (Other academic)
    Abstract [en]

    One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.

    In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.

    The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.

  • 3.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei
    St Petersburg State Univ, Russia; St Petersburg State Polytech Univ, Russia; RAS, Russia.
    Zavorokhin, German
    Steklov Math Inst, Russia.
    A fractal graph model of capillary type systems2018In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 63, no 7-8, p. 1044-1068Article in journal (Refereed)
    Abstract [en]

    We consider blood flow in a vessel with an attached capillary system. The latter is modelled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.

  • 4.
    Ghosh, Arpan
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. St Petersburg State Univ, Russia; RAS, Russia.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE2018In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 71, no 3, p. 349-367Article in journal (Refereed)
    Abstract [en]

    We present a two-dimensional model describing the elastic behaviour of the wall of a curved flexible pipe. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the channel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with any surrounding or supporting material and the fluid flow into account and is obtained via a dimension reduction procedure. The curvature and twist of the pipes axis as well as the anisotropy of the laminate wall present the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of pipes and their walls are supplied with explicit systems of differential equations at the end.

  • 5.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Ghosh, Arpan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, S. A.
    St Petersburg State Univ, Russia; Inst Problems Mech Engn RAS, Russia.
    A one dimensional model of blood flow through a curvilinear artery2018In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 63, p. 633-643Article in journal (Refereed)
    Abstract [en]

    We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier-Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved.

    The full text will be freely available from 2020-07-17 13:46
  • 6.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Karlsson, Matts
    Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey A.
    St Petersburg State University, St Petersburg State Polytechnical University, and Institute of Problems of Mechanical Engineering RAS, Russia..
    A one-dimensional model of a false aneurysm2017In: International Journal of Research in Engineering and Science (IJRES), ISSN 2320-9356, Vol. 5, no 6, p. 61-73Article in journal (Refereed)
    Abstract [en]

     A false aneurysm is a hematoma, i.e. collection ofblood outside of a blood vessel, that forms due to a hole  in the wall of an artery . This represents a serious medical condition that needs to be monitored and, under certain conditions, treatedurgently. In this work a one-dimensional model of a false aneurysm isproposed. The new model is based on a one-dimensional model of anartery previously presented by the authors and it takes into accountthe interaction between the hematoma  and the surrounding musclematerial. The model equations are derived  using rigorous asymptoticanalysis for the case of a simplified geometry.   Even though the model is simple it still supports a realisticbehavior for the system consisting of the vessel and the  hematoma. Using numerical simulations we illustrate the behavior ofthe model. We also investigate the effect  of changing the size of the hematoma. The simulations show that ourmodel can reproduce realistic solutions. For instance we show thetypical strong pulsation of an aneurysm by blood entering the hematoma during the work phase of the cardiac cycle, and the blood returning tothe vessel during the resting phase. Also we show that the aneurysmgrows  if the pulse rate is increased due to, e.g., a higher work load. 

  • 7.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, S. A.
    St Petersburg State University, Russia; St Petersburg Polytech University, Russia; Russian Academic Science, Russia.
    A one-dimensional model of flow in a junction of thin channels, including arterial trees2017In: Sbornik. Mathematics, ISSN 1064-5616, E-ISSN 1468-4802, Vol. 208, no 8, p. 1138-1186Article in journal (Refereed)
    Abstract [en]

    We study a Stokes flow in a junction of thin channels (of diameter O(h)) for fixed flows of the fluid at the inlet cross-sections and fixed peripheral pressure at the outlet cross-sections. On the basis of the idea of the pressure drop matrix, apart from Neumann conditions (fixed flow) and Dirichlet conditions (fixed pressure) at the outer vertices, the ordinary one-dimensional Reynolds equations on the edges of the graph are equipped with transmission conditions containing a small parameter h at the inner vertices, which are transformed into the classical Kirchhoff conditions as h -amp;gt;+ 0. We establish that the pre-limit transmission conditions ensure an exponentially small error O(e(-rho/h)),. amp;gt; 0, in the calculation of the three-dimensional solution, but the Kirchhoff conditions only give polynomially small error. For the arterial tree, under the assumption that the walls of the blood vessels are rigid, for every bifurcation node a (2x2)-pressure drop matrix appears, and its influence on the transmission conditions is taken into account by means of small variations of the lengths of the graph and by introducing effective lengths of the one-dimensional description of blood vessels whilst keeping the Kirchhoff conditions and exponentially small approximation errors. We discuss concrete forms of arterial bifurcation and available generalizations of the results, in particular, the Navier-Stokes system of equations.

  • 8.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, S. A.
    St Petersburg State University, Russia; St Petersburg State Technical University, Russia; Russian Academic Science, Russia.
    Effective one-dimensional images of arterial trees in the cardiovascular system2017In: Doklady physics (Print), ISSN 1028-3358, E-ISSN 1562-6903, Vol. 62, no 3, p. 158-163Article in journal (Refereed)
    Abstract [en]

    An exponential smallness of the errors in the one-dimensional model of the Stokes flow in a branching thin vessel with rigid walls is achieved by introducing effective lengths of the one-dimensional image of internodal fragments of vessels. Such lengths are eluated through the pressure-drop matrix at each node describing the boundary-layer phenomenon. The medical interpretation and the accessible generalizations of the result, in particular, for the Navier-Stokes equations are presented.

  • 9.
    Kozlov, Vladimir A
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St.Petersburg State University, Mathematics and Mechanics Faculty, St. Petersburg, Russia.
    A simple one-dimensional model of a false aneurysm in the femoral artery2016In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 214, no 3, p. 287-301Article in journal (Refereed)
    Abstract [en]

    Using the dimension reduction procedure, a one-dimensional model of a periodic blood flow in the artery through a small hole in a thin elastic wall to a spindle-shaped hematoma, is constructed. This model is described by a system of two parabolic and one hyperbolic equations provided with mixed boundary and periodicity conditions. The blood exchange between the artery and the hematoma is expressed by the Kirchhoff transmission conditions. Despite the simplicity, the constructed model allows us to describe the damping of a pulsating blood flow by the hematoma and to determine the condition of its growth. In medicine, the biological object considered is called a false aneurysm.

  • 10.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Karlsson, Matts
    Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia; RAS, Russia.
    A one-dimensional model of viscous blood flow in an elastic vessel2016In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 274, p. 125-132Article in journal (Refereed)
    Abstract [en]

    In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel. (C) 2015 Elsevier Inc. All rights reserved.

  • 11.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey
    Department of Mathematics and Mechanics, St Petersburg State University, Universitetsky prospect, 28, 198504, Petergof, St, Petersburg, Russia.
    Asymptotic Models of Anisotropic Heterogeneous Elastic Walls of Blood Vessels2015Report (Other academic)
    Abstract [en]

    Using the dimension reduction procedure in the three-dimensional elasticity system, we derive a two-dimensional model for elastic laminate walls of a blood vessel. The wall of arbitrary cross-section consists of several (actually three) elastic, anisotropic layers. Assuming that the wall’s thickness is small compared with the vessel’s diameter and length, we derive a system of the limit equations. In these equations, the wall’s displacements are unknown given on the two-dimensional boundary of a cylinder, whereas the equations themselves constitute a second order hyperbolic system. This system is coupled with the Navier–Stokes equations through the stress and velocity, i.e. dynamic and kinematic conditions at the interior surface of the wall. Explicit formulas are deduced for the effective rigidity tensor of the wall in two natural cases. The first of them concerns the homogeneous anisotropic laminate layer of constant thickness like that in the wall of a peripheral vein, whereas the second case is related to enforcing of the media and adventitia layers of the artery wall by bundles of collagen fibers. It is also shown that if the blood flow stays laminar, then the describing cross-section of the orthotropic homogeneous blood vessel becomes circular.

  • 12.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St. Petersburg State University, St. Petersburg State Polytechnical University, Institute for Problems in Mechanical Engineering RAS, 61, Bolshoi pr. V.O., St. Petersburg, Russia.
    One-Dimensional Model of Viscoelastic Blood Flow Through a Thin Elastic Vessel2015In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 207, no 2, p. 249-269Article in journal (Refereed)
    Abstract [en]

    Based on the three-dimensional Oldroyd viscoelastic fluid model, we develop a simple linear one-dimensional model of blood flow through a thin blood vessel with an elastic multilayer cylindrical wall. Unlike known models, the obtained system of integrodifferential equations with respect to the variables z and t (the longitudinal coordinate ant time) includes the Volterra operator in t, which takes into account the relaxation effect of stresses in a pulsating flow of blood regarded as a many-component viscoelastic fluid. We construct a simplified differential model corresponding to “short-term memory.” We study the effect of high-amplitude longitudinal oscillations of wall under a dissection (lamination). Bibliography: 24 titles. Illustrations: 1 figure.

  • 13.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei Aleksandrovich
    St. Petersburg State University, St. Petersburg State Polytechnical University, Russian Academy of Sciences, Russia.
    Transmission conditions in a one-dimensional model of bifurcating blood vessel with an elastic wall2015In: Zapiski Nauchnykh Seminarov POMI, ISSN 0132-6678, Vol. 438, p. 138-177Article in journal (Other academic)
    Abstract [en]

    We derive transmission conditions at a bifurcation point in a one-dimensional model of blood vessels by using a three-dimensional model. Both classical Kirchhoff conditions ensuring the continuity of pressure and zero flux flow in the node has to be modified in order to reflect properly the elastic properties of blood vessels and the nodes themselves. A simple approximate calculation scheme for the new physical parameters in the transmission conditions is proposed. We develop a simplified model of straight fragments of arteries with localized defects such as lateral micro-aneurysms and cholesterol plaques – these models also require setting transmission conditions.

  • 14.
    Kozlov, Vladimir A.
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St. Petersburg State University, St. Petersburg, Russia .
    Asymptotic Models of the Blood Flow in Arteries and Veins2013In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 194, no 1, p. 44-57Article in journal (Refereed)
    Abstract [en]

    Asymptotic analysis is applied for obtaining one-dimensional models of the blood flow in narrow, thin-walled, elastic vessels. The models for arteries and veins essentially distinguish from each other, and the reason for this is the structure of their walls, as well as the operationing conditions. Although the obtained asymptotic models are simple, they explain various effects known in medical practice, in particular, describe the mechanism of vein-muscle pumping of blood.

  • 15.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Nazarov, S A
    Russian Academic Science, Russia .
    Asymptotic model of interaction of blood flow with vein walls and the surrounding muscular tissue2012In: Doklady physics (Print), ISSN 1028-3358, E-ISSN 1562-6903, Vol. 57, no 10, p. 411-416Article in journal (Refereed)
    Abstract [en]

    n/a

  • 16.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Nazarov, Sergei
    Institute for Problems in Mechanical Engineering, St Petersburg.
    Surface Enthalpy and Elastic Properties of Blood Vessels2011In: Doklady physics (Print), ISSN 1028-3358, E-ISSN 1562-6903, Vol. 56, no 11, p. 560-566Article in journal (Refereed)
    Abstract [en]

    1. Blood vessels form one of the most complicated and important functional systems of humans and other mammals subjected to many risks and poorly amenable to medicinal treatment. The mathematical simulation of blood flow through the aorta, arteries, veins, peripheral vessels, and capillaries both sepa rately and as a whole still remains an important problem. At the same time, the published bloodflow models give no possibility to take into account the complex (composite and anisotropic) structure of walls of vessels. Moreover, the formulation itself of experiments on determining elastic characteristics of walls of ves sels is at the stage of development (see [1, 2] and the review of literature in monograph [3, sect. 8]).

    This study is devoted to derivation of certain relations for deformable bloodvessel walls on the basis of the conventional representation of their laminar structure (see, for example, [1, 4, 3]). By means of the dimensionreduction procedure and on the basis of the concept of surface enthalpy, we derived a simple formula for the effective elasticmodulus tensor. Contrary to the habitual formulation of the mathematical problem, the blood vessel is not beforehand assumed as a circular cylinder, and the occurrence of a variable curvature, which does not complicate principally the asymptotic analysis, enables us to investigate the deformation of walls of arteries due to damage, such as inhomogeneous calcification (hyalinosis or calcinosis), oblong atherosclerotic deposits (cholesterolplaques), or various surgical treatments (clipping, stenting, and stitching).

1 - 16 of 16
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