Revision as of 13:12, 2 February 2018

Monograph

==COMPUTATIONAL FLUID DYNAMICS

INDICATORS TO IMPROVE

CARDIOVASCULAR PATHOLOGIES DIAGNOSIS

==

by

Eduardo Soudah

Eugenio Oñate

Miguel Cervera

CIMNE Biomedical Engineering Department

Barcelona, December 2016

Dedicated to

my father...

Acknowledgments

First I would like to deeply thank Anne-Beatrice, Victoria and Anna for their patience, encouragement and support throughout my PhD. I would like also to express my gratitude to my parents, Ibrahim and Ana, my brother and sister, Yusef and Sara for their constant moral support. From the start, you always trust in me. Muchas gracias por apoyarme, comprenderme y animarme durante todos estos años fuera de casa.

I would like also to express my sincere gratitude to my supervisor, Prof.Eugenio Oñate for giving me the opportunity to join the International Center for Numerical Methods in Engineering (CIMNE) and work under his supervision. He has strongly supported my work with deep enthusiasm and worthy ideas. His aptitude to go straight to the point together with his talent in studying and solving problems have made a crucial contribution to the outcome of this work. I wish also to thank Prof.Miguel Cervera and Prof.Raimon Jané for their continuous interest, support and guidance during this monograph.

I would like to warmly thank all of the former and present members of CIMNE who made the work place a friendly environment and who are now, more friends than colleagues: to Jorge S.Pérez with whom we had a great collaboration and interesting technical, for his continuous support in my research (Jorge, eres un crack y lo sabes); to Carlos Labra, Enrique Ortega, Ariel Eijo, Temo, Giovana Gavidia, Ilaria and Maurizio Bordone with whom I shared the office during years for his endless good mood, kindness and all of the discussions we had; to Riccardo Rossi, Pooyan Dadvand, Carlos Roig and Jordi Cotela for their support in the integration of the 1D implementation and reduced model into KRATOS and for their kindness and all of the interesting technical discussions we had; to Joaquin (Cartagena) for his endless confidence, friendship and 'salad-shared'; Adrian Silisque, Elke Pahl and Steffen Mohr who made Barcelona a place to enjoy; to GID Department for always being so nice, ready to help and supportive; to Miguel Angel y Lucia for your really good mood, to Alberto for listen and support me during his stage at Barcelona, you showed me a new way of living and Joaquin Arteaga, thanks to you I am writing this. Further thanks to all of the people who work at CIMNE. It was great working with you all. Thank you for your work and your motivation.

Sincere thanks to CIMNE-TIC department: Jordi Jimenez who is always positive and always enthusiastic and motivated by new projects, to Sergio Valero, maestro Sensei, maestro programador, Angel Priegue (Alias Cruchi) 'tranquilo Edu no pasa nada', Cruchi thanks for your patience and thank for the 'Cafe-relaxing time in plaza mayor' on Friday evening, Aleki who always bring a great atmosphere to the office; Claudio Zinggerling for your professionalism and all of the discussions we had on different subjects and his friendly support; Andy for his friendliness; Pedro for his vitality, Francesc Campá for his good mood and friendship and Alberto Tena for his kindness,...por cierto, la undécima ESTA DADA!!!

Many great collaborations in the clinical setting were possible during the project. To those who contributed, thank you. Special thanks go to Dr.Francesc Carreras, Dr.Pedro Hion-Li at the Hospital Sant Pau i Creu Blanca, Dr.Xavier Alomar at Clinica Creu Blanca and Jean-Paul Aben at Pie Medical Imaging, all of whom gave me the chance to discover the clinical environment and freely shared with me their inestimable knowledge.

Last, but not least, my gratitude also goes to Administration and Project Management departments, Anna, Cecilia, Maëlle, Merce, Sandra, Irene, Marí Carmen, Francisco, Cristina, among others, for your help in making easy the administrative processes.

Further thanks to all of co-authors have contributed to the achievement of this monograph, thank you very much. This work would not be possible without your support.

Most of all, I would like to warmly thank all the friends from Palencia and Valladolid. You always have been there.

Eduardo Soudah Prieto

Barcelona, 2016

Presentation

The studies included in the monograph belong to the same research line, leading to three papers already published in international journals.

Paper 1.

Title: A Reduced Order Model based on Coupled 1D/3D Finite Element Simulations for an Efficient Analysis of Hemodynamics Problems.

Authors: E.Soudah, R.Rossi, S.Idelsohn, E.Oñate.

Journal: Journal of Computational Mechanics. (2014) 54:1013-1022.

DOI: 10.1007/s00466-014-1040-2

Paper 2.

Title: CFD Modelling of Abdominal Aortic Aneurysm on Hemodynamic Loads using a Realistic Geometry with CT.

Authors: E.Soudah, E.Y.K. Ng, T.H Loong, M.Bordone, P. Uei and N.Sriram.

Journal: Computational and Mathematical Methods in Medicine. Volume 2013 - 472564, 01/06/2013.

DOI: 10.1155/2013/472564

Paper 3.

Title: Mechanical stress in abdominal aortic aneurysms using artificial neural networks.

Authors: Eduardo Soudah, José F. Rodríguez, Roberto López

Journal of Mechanics in Medicine and Biology. Vol. 15, No. 3 (2015) 1550029

DOI: 10.1142/S0219519415500293

Articles are reprinted with permission, and have been reformatted to fit the layout of the Monogrpah.

List of Figures List of Figures List of Figures

List of Tables List of TablesList of Tables

Abstract

In recent years, the study of computational hemodynamics within anatomically complex vascular regions has generated great interest among clinicians. The progress in computational fluid dynamics, image processing and high-performance computing have allowed us to identify the candidate vascular regions for the appearance of cardiovascular diseases and to predict how this disease may evolve. In this monograph we attempt to introduce into medicine the computational predictive paradigm that has been used in engineering for many years. Several groups have tried to create predictive models for cardiovascular pathologies, but they have not yet begun to use them in clinical practice. Our final aim is to go further and obtain predictive variables to be used in the clinical field.

We try to predict the evolution of aortic abdominal aneurysm, aortic coarctation and coronary artery disease in a personalized way for each patient. We propose diagnostic indicators that can improve the diagnosis and predict the evolution of the disease more efficiently than the methods used until now. In particular, a new methodology for computing diagnostic indicators based on computational hemodynamics and medical imaging is proposed. We have worked with data of anonymous patients to create real predictive technology that will allow us to continue advancing in personalized medicine and generate a more sustainable health systems. The objective of this monograph is therefore to develop predictive models for cardiovascular pathologies by merging medical imaging and computational techniques at a clinical level.

It is expected in the near future that larger databases of patient-specific computational models will be available to doctors. These data can be used with predictive models to improve diagnosis and to define personalized therapies and treatments.

Resumen

Durante los últimos años, el estudio de las enfermedades cardiovasculares mediante el uso técnicas computacionales ha generado muchas expectativas en el campo de la medicina. Los avances realizados en técnicas de procesamiento de imágenes, métodos computacionales y el uso de grandes procesadores de cálculo han permitido identificar y correlacionar variables hemodinámicas con los estados incipientes o de desarrollo de patologías cardiovasculares. Hoy en día la medicina se basa en el diagnóstico, pero en esta monografía queremos tratar de introducir el concepto de medicina computacional preventiva. El objetivo principal es desarrollar modelos preventivos basados en indicadores de diagnóstico para patologías cardiovasculares combinando procesamiento de imágenes y técnicas computacionales.

En esta monografía, tratamos de predecir la evolución de aneurismas abdominales, la formación del trombo intraluminal en el interior del saco aneurismático, el estudio de la ateroesclerosis y de la coartación de aorta, así como, posibles problemas derivados de la válvula aórtica de manera personalizada. Para entender cómo una patología cardiovascular evoluciona y cuándo va a convertirse en un riesgo para la salud, es necesario desarrollar una metodología eficiente que permita calcular indicadores de diagnóstico. En esta monografía, hemos propuesto indicadores de diagnóstico basados en técnicas computacionales e imágenes médicas que pueden mejorar el diagnóstico y a la vez predecir la evolución de una patología de manera más eficiente que los métodos utilizados hasta ahora. Sin embargo, el objetivo final es llevar dichos indicadores a la práctica clínica. Actualmente estamos trabajando con datos de pacientes anónimos para crear una gran base de datos que nos permita avanzar en la medicina personalizada y en la generación de sistemas de salud más sostenibles. Es de esperar que en el futuro existan estas bases de datos a disposición de los médicos, y que estos datos sirvan para mejorar el diagnóstico y definir tratamientos personalizados.

Resum

En els últims anys, l'estudi de l'hemodinàmica computacional en regions vasculars anatòmicament complexes ha generat un gran interès entre els clínics. El progrés obtingut en la dinàmica de fluids computacional, en el processament d'imatges i en la computació d'alt rendiment ha permès identificar regions vasculars on poden aparèixer malalties cardiovasculars, així com predir-ne l'evolució. En aquesta tesi s'intenta introduir en la medicina el paradigma computacional predictiu utilitzat des de fa molts anys en l'enginyeria. Diversos grups han tractat de crear models predictius per a les patologies cardiovasculars, però encara no han començat a utilitzar-les en la pràctica clínica. El nostre objectiu és anar més enllà i obtenir variables predictives que es puguin utilitzar de forma pràctica en el camp clínic.

Tractem de predir l'evolució de l'aneurisma d'aorta abdominal, la coartació aòrtica i la malaltia coronària de forma personalitzada per a cada pacient. Per entendre com la patologia cardiovascular evolucionarà i quan suposarà un risc per a la salut, cal desenvolupar noves tecnologies mitjançant la combinació de les imatges mèdiques i la ciència computacional. Proposem uns indicadors que poden millorar el diagnòstic i predir l'evolució de la malaltia de manera més eficient que els mètodes utilitzats fins ara. En particular, es proposa una nova metodologia per al càlcul dels indicadors de diagnòstic basada en l'hemodinàmica computacional i les imatges mèdiques. Hem treballat amb dades de pacients anònims per crear una tecnologia predictiva real que ens permetrà seguir avançant en la medicina personalitzada i generar sistemes de salut més sostenibles. Per tant, l'objectiu d'aquesta tesi és el desenvolupament de models predictius de patologies cardiovasculars mitjançant la fusió d'imatges mèdiques i tècniques computacionals a nivell clínic.

Es pot preveure que en el futur tots els metges disposaran de bases de dades de tota la nostra anatomia, fisiologia i models computacionals. Aquestes dades es poden utilitzar en els models predictius per millorar el diagnòstic i definir teràpies o tractaments personalitzats.

1 Cardiovascular physiology

The purpose of this appendix is to introduce which are the basics of the cardiovascular physiology. This brief overview of the cardiovascular physiology is only included for the purpose of providing essential information to scientists without a background in medicine. In this appendix the macroscopic and microscopic structure of arterial walls, blood modeling and cardiovascular system is briefly explained. For a more detailed exposition of the different mechanical/rheological characteristics of cardiovascular system and the overall functioning of the blood vessel see Tortora et al.[1].

1.1 Cardiovascular physiology

Cardiovascular physiology is the study of the cardiovascular system, specifically addressing the physiology of the heart (cardiac physiology) and blood vessels (circulatory physiologic) (see figure 1[2]). The cardiovascular system is a pressurized closed system responsible for transporting nutrients, hormones, and cellular waste throughout the body. From a physical point of view, there are three independent circuits:

Systemic Circulation: The former brings oxygenated blood from the heart, thought arteries and capillaries, to the various organs (systemic arterial system) and then brings it back to heart (systemic venous system). The systemic arterial system is an extensive high-pressure system; hence the structure of its blood vessels reflects the high pressures to which they are subjected. The systemic venous system acts as a collecting system, returning blood from the capillary networks to the heart passively down a pressure gradient.
Pulmonary circulation: The latter pumps the venous blood into the pulmonary artery where it enters the pulmonary system, through the pulmonary veins, get oxygenated and is finally received by the heart, ready to be sent to the systemic circulation (where the blood is pumped through the aortic valve into the aorta).
The coronary circulation arises from the aorta and provides a blood supply to the myocardium, the heart muscle

Figure 1: The cardiovascular system is a close loop. The heart is a pump that circulates blood through the system. Arteries take blood away from the heart (systemic circulation) and veins (pulmonary circulation) carry blood back to the heart.

This monograph will focus on systemic arterial system and coronary circulation.

1.1.1 Blood Vessels

The blood vessels are the part of the circulatory system that transport blood throughout the body. The vascular system is composed of arteries, arterioles, capillaries, venules and veins (see figure 2[3]). The three main types of blood vessels are:

Arteries, which carry blood away from the heart at relatively high pressure,
Veins, which carry blood back to the heart at relatively low pressure and
Capillaries, which provide the link between the arterial and venous blood vessels.

Regarding the small vessels mention that arterioles are the smallest branches of the arterial network. Arterioles vary in diameter ranging from 0.3 mm to 0.4 mm. Any artery with a diameter smaller than 0.5 mm is considered to be an arteriole. Capillaries are specialized for diffusion of substances across their wall. Capillaries are the smallest vessels of the blood circulatory system and form a complex inter linking network. Pressure is essentially lost in the capillaries. As the capillaries begin to thicken and merge, they become venules. Venules eventually become veins and head back to the heart.

In general, arteries are roughly subdivided into two types: elastic (or large arteries) and muscular (or small arteries). Elastic arteries have relatively large diameters and are located close to the heart (for example, the aorta, the carotid and iliac arteries), while muscular arteries are located at the periphery (for example, femoral, celiac, cerebral arteries). The walls of all the blood vessels, except the capillaries which are only one cell thick, have the same basic components but the proportion of the components varies with function. Therefore, the structure of the vessels in the different parts of the circulatory or vascular system varies and the differences relate directly to the function of each type of vessel (see table 1). Arteries are not just tubes through which the blood flows.

Table. 1 Vessel Type
Vessels	Diameter of lumen (mm)	Wall thickness (mm)	Mean pressure (kPa)
Aorta	25	2	12.5
Large arteries	1-10	1	12
Small arteries	0.5-1	1	12
Arteriole	0.01-0.5	0.03	7
Capillary	0.006-0.01	0.001	3
Venule	0.01-0.5	0.003	1.5
Vein	0.5-15	0.5	1
Vein cava	30	1.5	0.5

All blood vessels, except capillaries, are composed of three distinct layers (tunica intima, tunica media and tunica externa or adventicia) surrounding a central blood carrying canal (known as the lumen). The constituents of arterial walls from the mechanical perspective are important to researchers interested in constitutive issues.

Tunica intima. The tunica intima is the innermost layer of the artery. It composed of a lining layer of highly specialized multi-functional flattened epithelial cells termed endothelium. This sits on a basal lamina; beneath this is a very thin layer of fibro-collagenous support tissue.
Tunica media. The tunica media is the middle layer in a blood vessel wall and is a complex three-dimensional network of smooth muscle cells reinforced by organized layers of elastic tissue which form elastic laminae. The tunica media is particularly prominent in arteries, being relatively indistinct in veins and virtually non-existent in very small vessels. From the mechanical perspective, the media is the most significant layer in a healthy artery.
Tunica Adventicia. The tunica adventitia or externa is the outermost layer of blood vessels. It is composed largely of collagen, but smooth muscle cells may be present, particularly in veins. The tunica adventitia is often the most prominent layer in the walls of veins. Within the tunica adventitia of vessels with thick walls (such as large arteries and veins) are small blood vessels which send penetrating branches into the media to supply it with blood.

Veins do not have as many elastic fibers as arteries. Veins do have valves, which keep the blood from pooling and flowing back to the legs under the influence of gravity. When these valves break down, as often happens in older or inactive people, the blood does flow back and pool in the legs. The result is varicose veins, which often appear as large purplish tubes in the lower legs.

Figure 2: The human circulatory system (simplified). Red indicates oxygenated blood (arterial system), blue indicates deoxygenated (venous system).

1.1.2 Blood Modelling

Blood is a suspension of cells into a fluid called plasma. It delivers oxygen and nutrients to the cells and remove COFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _{2}}

and waste products. Blood also enables hormones and other substances to be transported between tissues and organs. The blood makes up about 7% of the weight of a human body, with a volume of about 5 litters in an average adult. Understanding blood physiology depends on understanding the components of blood. Blood is made up of plasma (about 55%) and cellular elements (about 45% These cellular elements include red blood cells (also called RBCs or erythrocytes), white blood cells (also called leukocytes) and platelets (also called thrombocyte) suspended in a plasma. Plasma is essentially a blood aqueous solution containing 92% water, 8% blood plasma proteins, and trace amounts of other materials (i.e albumin or globulin). Plasma has many functions as involving colloid, osmotic effects, transport, signaling, immunity and clotting.

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erythrocytes or red cells (45.0% of blood volume)in a woman 4.800.000 and in a men 5.400.400 erythrocytes per mmFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{3}}
(or microliter) Size: disc biconcave 7 or 7.5 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }

m of diameter. Erythrocytes are responsible for the exchange of oxygen and COFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _{2}}

with the cells.

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leukocytes or white cells(1.0% of blood volume) 4.500 y 11.500 per mmFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{3}}
(or microliter) in the blood. Size: between 8 and 20 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }

m. Leukocytes play a major role in the human immune system.

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thrombocytes (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {>}}

1.0% of blood volume): Platelets are responsible for blood clotting (coagulation).

Blood viscosity is a measure of the resistance of blood to flow. The blood viscosity increases as the percentage of cells in the blood increases: more cells mean more friction, which means a greater viscosity. The percentage of the blood volume occupied by red blood cells is called the haematocrit. With a normal haematocrit of about 40 (that is, approximately 40% of the blood volume is red blood cells and the remainder plasma), the viscosity of whole blood (cells plus plasma) is about 3 times that the viscosity of the water. Other factors influencing blood viscosity include temperature, where an increase in temperature results in a decrease in viscosity. This is particularly important in hypothermia, where an increase in blood viscosity will provoke problems with blood circulation.

Blood compressibility is the relation between all of its components and their volume fraction or a measure of the relative volume change of a fluid as a response to a pressure change. The 92% of the blood is water, and how the water has a high relation of compressibility, blood can be consider an incompressible fluid. Mathematically, it is mean that the mass is conserved within the domain.

Usually, for small arteries (less than 1mm in diameter) blood is consider as Non-Newtonian fluid, however in medium/large arteries blood may be considered as Newtonian fluid. To explain this behaviour it is necessary to explain which the Fahraeus-Lindqvist effect is. Fahraeus-Lindqvist effect is characterized by a decrease in the apparent blood viscosity as the arteries diameter decreases below 500 mm. The minimum apparent viscosity is reached when the tube diameter is higher than 8 mm, upon further decreases in tube diameter, the apparent viscosity increases very rapidly. The physical reason behind the Fahraeus-Lindqvist effect is the formation of a cells-free layer near the wall of the tube[4]. The layer is devoid of RBCs and has a reduced local viscosity. The extent of the cell-free layer, which depends on the vessel size and haematocrit, is a major factor that determinate the apparent viscosity of the blood. The core of the tube, on the contrary, is rich with RBCs and has a higher local viscosity. However, in large arteries with internal diameter > 500 mm, although the blood density depends on the red cells concentration, the blood may be considered a homogeneous fluid with standard behavior (Newtonian fluid)[4]. The viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }

of the fluid is proportional to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau }

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2 Numerical Model

One-Dimensional (1D) models of blood flow have been extensively used to study wave propagation phenomena in arteries. These models allow us to investigate physical mechanisms underlying changes in pressure and flow pulse waveforms that are produced by cardiovascular disease, however these models do not taken into account the effects provoked by the 3D geometry. In this appendix, the 1D mathematical formulation and the reduced model used in paper 1 ('A Reduced Order Model based on Coupled 1D/3D Finite Element Simulations for an Efficient Analysis of Hemodynamics Problems.) are briefly explained.

2.1 1D Mathematical Model

A preliminary basic knowledge about the cardiovascular system was given in appendix 1. We introduce an one-dimensional mathematical model to describe the flow motion in arteries and its interaction with the wall displacement in order to provide a better understanding of the hemodynamics in large vessels. In absence of branching, a short of an artery may be considered as a cylindrical compliant tube, and it can be described by using a curvilinear cylindrical coordinate system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{(r, \Theta , z)}}

with the corresponding base unit vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{(e_{r}, e_{\Theta }, e_{z})}}
radial, circumferential and axil unit vector, respectively, as show in figure 3. The vessel extends from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z=0}}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z=L}}
and this length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{L}}
is constant with time, therefore, the spatial domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\Omega _{c}}}
in cylindrical coordinate is defined as follows:

Section of an artery with the principal geometrical parameters

Figure 3: Section of an artery with the principal geometrical parameters

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(2.1)

Defined our domain, the following assumptions must be taken into account in order to deduce the one-dimensional mathematical model:

Radial displacements. The wall displaces along the radial direction solely, thus at each point on the tube surface we may write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\eta =\eta \cdot e_{r}}}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\eta =R_{z}-R_{0}}}

is the displacement with respect to a reference radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{R_{0}}}

.

Axial symmetry. All quantities are independent from the angular coordinate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\Theta }}

. As a consequence, every axial section, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z=constant}} , remains circular during the wall motion. The arteries radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{R}}

is a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{t}}

. A generic axial section will be indicated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{S=S (z,t)}}

where,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): S(z,t)= \{ (r,\Theta ,z): 0 \leqslant r \leqslant R(z,t); \Theta \epsilon [0,2\Pi ); \Delta t > 0 \}

(2.2)

and, its measure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{A}}

is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A(z,t) = \int _S \,d\sigma =\pi R^{2}(z,t)= \pi \cdot \left[R_{0}(z)+\eta (z,t)\right]^2

(2.3)

Dominance axial velocity, the velocity components orthogonal to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z}}

axis are negligible compared to the component along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z}}

. The latter is indicated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{u_{z}}}

and its expression in cylindrical coordinates reads:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_{z}(r,z,t)=\overline{u}(z,t)\cdot s\cdot \left[\frac{r}{R(z,t)}\right]

(2.4)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{\overline{u}}}

is the mean velocity in each axial section and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{s}}
is a velocity profile.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{u}(z,t)= \frac{1}{A} \cdot \int _S u_{z}\,d\sigma

(2.5)

Constant pressure, we assume that the pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{P}}

is constant on each axial section Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{S}}

, so that it depends only on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{z}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{t}}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{p}(z,t)= \int _S p_{z}\,d\sigma

(2.6)

No body forces. We neglect body forces.

The resulting state variables are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{Q}(z,t) = \int _S(z,t) u_{z}\,d\sigma = A(z,t)\cdot \overline{u}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A(z,t) = \int _{\mathcal{S}(z,t)} d\sigma = \pi R^2(z,t)

(2.7)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}

is the cross-sectional area and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q}
is the volumetric flow rate.

Therefore, we have three independent variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{(A, u, p)}} , or equivalently Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{(A, Q, p)}} . Thus, we require three independent equations to get a solution. These three equations will be provided by equations of conservations of mass and momentum and an algebraic law that link the pressure and area of the artery.

2.1.1 Conservation equations

The conservation equations reflect a certain physical amount of a continuous medium that must always be satisfied and which are not limited in their application to the material. By applying the conservation equations in the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

the body Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta }
leads to an integral relationship. Since the integral relationship must hold for any sub-domain of the body, then the conservation equations can be expressed as partial differential equations. Before continuing with the conservation equations, the material time derivative of an integral relationship to any property space is defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{d}{dt} \int _{\Omega } (\bullet )= \int _{\Omega } (\dfrac{d(\bullet )}{dt}+\mathcal{r}(\bullet ) \cdot \mathbf{v})d\Omega

(2.8)

which is the Reynold's transport theorem.

2.1.2 Conservation of the mass

A fundamental law of Newtonian mechanics is the conservation of the mass, also called continuity equation, contained in a material volume. Considering the vessel shown in figure 3 as our control volume, the principle of mass conservation requires that the rate of change of mass within the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _{t}}

plus the net mass flux out of the control volume is zero.

Denoting the vessel volume as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): V(t)= \int _{0}^L Adz,

(2.9)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

is the length of the vessel and assuming there are no infiltration through the side walls, the mass conservation can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho \dfrac{dV(t)}{dt} + \rho Q(L,t) - \rho Q(0,t) = 0

(2.10)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }

is the blood density. If infiltration does occur we must add a source term to this equation [5]. To determine the one-dimensional equation of mass conservation, we insert the volume into equation 2.10 and, note that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q(L,t)-Q(0,t)=\int _0^L \dfrac{\partial Q}{\partial z} dz,

we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho \dfrac{d}{dt}\int _0^L A(z,t)dz + \rho \int _0^L \dfrac{\partial Q}{\partial z} dz=0.

If we assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

is independent of time we can take the time derivative inside the integral to arrive at

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho \int _0^L \Biggl\{\dfrac{\partial A}{\partial t} + \dfrac{\partial Q}{\partial z} \Biggr\}dz =0

Since we have not specified the length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L} , the control volume is arbitrary and so the above equation must be true for any value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

and so in general we require that the integrand is zero. We therefore obtain the differential one-dimensional mass conservation equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial A}{\partial t} + \dfrac{\partial Q}{\partial z} = \dfrac{\partial A}{\partial t} + \dfrac{\partial (uA)}{\partial z} = 0

(2.11)

2.1.3 Conservation of the momentum

The momentum equation, also called the equation of motion, is a relation equating the rate of change of momentum of a selected portion of the body and the some of all forces acting on that portion. Again we consider the vessel as our control volume and assume that there is no flux through the side walls in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z} -direction. In this case, it states that the rate of change of momentum within the integration domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _{t}}

plus the net flux of the momentum out of the domain itself is equal to the applied forces on the domain and can be expressed over an arbitrary length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}
as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial }{\partial t} \int _{0}^L \rho Q dz + (\alpha \rho Qu)_{L}-(\alpha \rho Qu)_{0}=F

(2.12)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F}

is defined as the applied forces in the z-direction acting on the domain. The equation 2.12 includes the momentum-flux correction coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha }

, also called Coriolis coefficient, which accounts for the fact that the momentum flux calculated with averaged quantities (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{u}} ) does not consider non-linearity of sectional integration of flux momentum. So we may assume

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial }{\partial t} \int _{S} \rho \tilde{u}^2 A \equiv \alpha \rho \tilde{u}^2 A = \alpha \rho Q \tilde{u} \quad \Rightarrow \quad \alpha (z,t)= \dfrac{\int _{S} \tilde{u}^2 d\sigma }{A \tilde{u}^2}=\dfrac{\int _{S} \tilde{s}^2 d\sigma }{A}

(2.13)

In general, the coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha }

vary in time and space, yet in our model it is taken constant as a consequence of 2.5 It is immediate to verify that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha \geq 1}

.

The axial velocity profile s(y) is chosen a priori through the power-law relation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s(y)=\gamma ^{-1}(\gamma +2)(1 - y^\gamma )

(2.14)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

is the radial coordinate and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma }
is a proper coefficient. Commonly accepted approximation are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma =2}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 4/3}

), which corresponds to the Poiseuille solution (parabolic velocity profile), while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma =9}

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 1.1}

) leads to a more physiological flat profile, following the Womersley theory. The blood profile trend with these values are shown in figure 4.

We will see that the choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 1}

, which indicates a completely flat velocity profile, would lead to certain simplification in our analysis.

Blood flow profile adopting different values of γ.

Figure 4: Blood flow profile adopting different values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma

.

To complete the equation 2.12 we need to define the applied forces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}}

which typically involve a pressure and a viscous force contribution,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): F=(PA)_{0}-(PA)_{L}+ \int _{0}^{L}\int _{\partial S} \hat{P}n_{z}dsdz + \int _{0}^{L}fdz

(2.15)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial S}

represents the boundary of the section Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_{z}}

is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

-component of the surface normal and f stands for the friction force per unit of length. The pressure force acting on the side walls, given by the double integral, can be simplified since we assumed both constant sectional pressure and axial symmetry of the vessel; so we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{0}^{L}\int _{\partial S} \hat{P}n_{z}dsdz = \int _{0}^{L} P \dfrac{\partial A}{\partial z}dz

(2.16)

If we finally combine equations 2.12,2.15 and 2.16 we obtain the momentum conservation for the computation domain expressed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{d}{dt} \int _{0}^{L} PQdz + (\alpha \rho Qu)_{L}-(\alpha \rho Qu)_{0} = (PA)_{0}-(PA)_{L}+ \int _{0}^{L} P \dfrac{\partial A}{\partial z}dz + \int _{0}^{L} fdz

(2.17)

To obtain the one-dimensional differential equation for the momentum we note that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\alpha \rho Qu)_{L}-(\alpha \rho Qu)_{0} = \int _{0}^{L} \dfrac{\partial (\alpha \rho Qu)}{\partial z}dz

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (PA)_{0}-(PA)_{L} = - \int _{0}^{L} \dfrac{\partial (PA)}{\partial z}dz

which inserted into 2.17, taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

independent of time and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }
constant, gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho \int _{0}^{L} \Biggl\{\dfrac{\partial Q}{\partial t}+\dfrac{\partial (\alpha Qu)}{\partial z} \Biggr\}dz = \int _{0}^{L} \Biggl\{-\dfrac{\partial (PA)}{\partial z}+ P\dfrac{\partial A}{\partial z}+f \Biggr\}dz

Once again this relationship is satisfied for an arbitrary length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

and therefore can only be true when the integrands are equal. So the one-dimensional equation for the momentum conservation becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial Q}{\partial t}+ \alpha \dfrac{\partial }{\partial z} \Biggl(\dfrac{Q^2}{A} \Biggr)= - \frac{A}{\rho }\dfrac{\partial P}{\partial z} + \frac{f}{\rho }

(2.18)

The viscous term in the equation 2.15 can be taken proportional to the averaged velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{u}} , thus we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{f}{\rho } = K_R \dfrac{Q}{A}

Therefore we finally obtain the equation of the momentum continuity

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial Q}{\partial t} + \alpha \dfrac{\partial }{\partial z} \Biggl(\dfrac{Q^2}{A} \Biggr)= - \frac{A}{\rho }\frac{\partial P}{\partial z} + K_r\bar{u}

(2.19)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_R}

is a strictly positive quantity that represents the viscous resistance of the flow per unit length of tube. It depends on the kinematic viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu = \frac{\mu }{\rho }}
of the fluid and the velocity profile s chosen. For a power law profile Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s(y)}

, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_R=2 \pi \nu (\gamma +2)} . In particular, for a parabolic profile Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_R=8 \pi \nu } , while for a flat profile we obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_R=22 \pi \nu } .

2.1.4 Vessel wall constitutive model

Once we obtained the two governing equations 2.11 and 2.19, it is possible to write the one-dimensional system as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial A}{\partial t} + \dfrac{\partial Q}{\partial z} = 0

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial Q}{\partial t} + \dfrac{\partial }{\partial z}(\alpha \dfrac{Q^2}{A})+ \dfrac{A}{\rho } \dfrac{\partial P}{\partial z} + K_{R} \dfrac{Q}{A} = 0

(2.20)

for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): z Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \in (0,L)}

and t Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle >}
0, where the unknown variables are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P}

. The system of equations 2.20 may be also expressed alternatively in terms of the variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A,\bar{u})}

instead Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A,Q)}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial A}{\partial t} + \dfrac{\partial A \bar{u}}{\partial z} = 0

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \bar{u}}{\partial t} + \bar{u} \dfrac{\partial \bar{u}}{\partial z} + \dfrac{1}{\rho } \dfrac{\partial P}{\partial z} + K_{R} \bar{u} = 0

(2.21)

As we can notice the number of unknown variables is greater than the number of equations (three against two); therefore we must provide another equation in order to close the system. A possibility is to introduce an algebraic relation linking the area of the vessel and pressure to the wall deformation. For the paper 1 we have considered the Generalised string model [6], which is written in the following form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{w}h_{0}\dfrac{\partial ^2 \eta }{\partial t^2} - \tilde{\gamma } \dfrac{\partial \eta }{\partial t} - \tilde{a} \dfrac{\partial ^2 \eta }{\partial z^2} - \tilde{c} \dfrac{\partial ^3 \eta }{\partial t \partial z^2} + \tilde{b} \eta = (P- P_{ext}), \qquad z \in (0,L), t>0

(2.22)

We may identify the physical significance of the various terms:

Inertia term: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{w}h_{0}\dfrac{\partial ^2 \eta }{\partial t^2}

, proportional to the wall acceleration

Voigt viscoelastic term: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\gamma } \dfrac{\partial \eta }{\partial t}

, viscoelastic term, proportional to the radial displacement velocity

Longitudinal pre-stress state of the vessel: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{a} \dfrac{\partial ^2 \eta }{\partial z^2}

,

Viscoelastic term: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{c} \dfrac{\partial ^3 \eta }{\partial t \partial z^2}

,

Elastic term: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{b} \eta

.

Besides Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _w}

is the vessel density, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_0}
is the wall thickness, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{a}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{b}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{c}}
are three positive coefficients. We can develop the last term of 2.22 being

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \eta = R - R_0 \qquad \Rightarrow \qquad \eta = \dfrac{\sqrt{A} - \sqrt{A_{0}}}{\sqrt{\pi }}, \quad \quad A_0 = \pi R_0^2

Elastic model

The elastic response is the dominating effect, while the other terms are less important. Consequently, a first model is obtained by neglecting all derivatives in 2.22. Pressure and area will then be related by the following algebraic law

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{b}= \dfrac{E h_0}{k R_0^2}= \dfrac{\pi E h_0}{k A_0}, \qquad k=1- \nu ^2

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}

is the Young modulus of elasticity and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu }
represents the Poisson ratio, typically taken to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu = 0.5}
(then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=0.75}

) since biological tissue is practically incompressible. We have taken Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P - P_{ext} = \tilde{b} \eta = \beta \dfrac{\sqrt{A} - \sqrt{A_{0}}}{A_{0}}

(2.23)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \beta = E h_{0} \sqrt{\pi }

is in general a function of z trough the Young modulus E. In a more general setting, the algebraic relationship may be expressed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P = P_{ext} + \psi (A;A_{0},\beta )

(2.24)

where we outlined that the pressure will depend not only on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A} , but also on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}

and on a set of coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta = {\beta _1,\beta _2,\dots ,\beta _n}}
which accounts for the physical and mechanical characteristics of the arterial vessel. Both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\beta }}
are given functions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

, but they do not vary in time. It is required that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi }

be at least a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^1}
function of its arguments and be defined for each positive value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}

. In addition we must have, for all the allowable values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta }
that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \psi }{\partial A}>0, \qquad \psi (A_0;A_0,\beta )=0

There are several examples of algebraic pressure-area relationship for one-dimensional models of arterial flow Langewouters_1984,Smith_2003; here we assumed the relationship 2.23, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta = \beta _1}

and, for the sake of simplicity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{ext}=0}

. The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi }

can be written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \psi (A;A_0,\beta _1 ) = \beta _1 \dfrac{\sqrt{A} - \sqrt{A_0}}{A_0}

(2.25)

It is useful introduce the Moens-Korteweg velocity

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_1(A;A_0,\beta )=\sqrt{\dfrac{A}{\rho }\dfrac{\partial \psi }{\partial A}}

which represents the propagation speed of waves along the cylindrical vessels. In our case may be readily computes as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_1=\sqrt{\dfrac{\beta }{2 \rho A_0}}A^{\frac{1}{4}}

(2.26)

Taking into account 2.23 the system 2.20 can be written in the conservation form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \mathbf{U}}{\partial t} + \dfrac{\partial \mathbf{F(U)}}{\partial z} = \mathbf{S(U)}, \quad z \in (0,L), \quad t>0

(2.27)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{U}= \begin{bmatrix}A \\ Q \end{bmatrix}

(2.28)

are the conservative variables,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{F(U)} = \begin{bmatrix}Q \\ \alpha \dfrac{Q^2}{A} + C_1 \end{bmatrix}

the corresponding fluxes, and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{S(U)} = \begin{bmatrix}0 \\ -K_R\dfrac{Q}{A}+\dfrac{\partial C_1}{\partial A_0} \dfrac{dA_0}{dz}+\dfrac{\partial C_1}{\partial \beta } \dfrac{d \beta }{dz} \end{bmatrix}

a source term of the system. In our modelling, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta _1}
are taken constant along the axial direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}
because we assume that both the initial area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}
and the Young modulus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}
do not vary in space; so the expression of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{S}}
accounts only for the friction term depending on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_R}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_1}

is a primitive of the wave speed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_1}

, given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_1(A;A_0,\beta )= \int _{A_0}^A c_1^2(\tau ; A_0, \beta ) d\tau

Applying the relationship 2.26 and 2.25, we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_1=\sqrt{\dfrac{\beta _1}{2\rho A_0}}A^{\frac{1}{4}} \quad \Rightarrow \quad C_1= \frac{\beta _1}{3\rho A_0}A^{\frac{3}{2}}

(2.29)

2.1.5 Characteristic analysis

One of the methods for solving non-linear hyperbolic system of partial differential equations, like the one-dimensional elastic model 2.27 , is the characteristic analysis [7]. After some simple manipulations the system 2.27 may be written in the quasi-linear form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \mathbf{U}}{\partial t} + \mathbf{H(U)}\dfrac{\partial \mathbf{U}}{\partial z} = \mathbf{B} (\mathbf{U}), \qquad z \in (0,L), t>0

(2.30)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{H(U)}= \begin{bmatrix}0 & 1 \\ \dfrac{A}{\rho }\dfrac{\partial \psi }{\partial A} - \alpha \bar{u}^2 & 2\alpha \bar{u} \end{bmatrix} = \begin{bmatrix}0 & 1 \\ c_1^2 - \alpha \Bigg(\dfrac{Q}{A} \Bigg)^2 & 2 \alpha \dfrac{Q}{A} \end{bmatrix}

is the Jacobian matrix. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta }
are constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B} = - \mathbf{S}}

. Considering 2.30, we can calculate the eigenvalues for the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H(U)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda _{1,2}= \alpha \dfrac{Q}{A} \pm c_\alpha

(2.31)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_\alpha =\sqrt{c_1^2 + \alpha ( \alpha -1) \dfrac{Q^2}{A^2}}

Since the Coriolis coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha \geq 1}

(we considered, for simplicity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 1}

), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_\alpha }

is a real number; besides, under the assumption that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A>0}

, indeed a necessary condition to have physical relevant solution, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_1>0}

therefore we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_\alpha >0}

which means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}}
has two real distinct eigenvalues and so, by definition, the system 2.30 is strictly hyperbolic. For typical values of velocity, vessel section and mechanical parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta _1}
encountered in main arteries under physiologically conditions, we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _1>0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _2<0}

, i.e., the flow is sub-critical everywhere. Furthermore, it may be shown [8] that the flow is smooth. Discontinuities, which would normally appear when treating a non -linear hyperbolic system, do not have the time to form on out context because of the pulsatility of the boundary conditions. Afterwards this considerations, from now on we will assume sub-critical regime and smooth solutions.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\mathbf{l}_1,\mathbf{l}_2)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\mathbf{r_1},\mathbf{r_2})}
be two couples of left and right eigenvectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}}

. The matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{R}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{L}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{\Lambda }}
are defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{L} = \begin{bmatrix}\mathbf{l}_1^T \\ \mathbf{l}_2^T \end{bmatrix}, \quad \mathbf{R} = \begin{bmatrix}\mathbf{r}_1 & \mathbf{r}_2 \end{bmatrix}, \quad \mathbf{\Lambda } = \begin{bmatrix}\lambda _1 & 0 \\ 0 & \lambda _2 \end{bmatrix}

(2.32)

Since left and right eigenvalues are mutually orthogonal, we choose them so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{L} \mathbf{R} = \mathbf{I}} , being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{I}}

the identity matrix. Matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}}
may then be decomposed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{H}=\mathbf{R}\mathbf{\Lambda }\mathbf{L}

and the system 2.30 takes the equivalent form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{L} \dfrac{\partial \mathbf{U}}{\partial t} + \mathbf{\Lambda }\mathbf{L}\dfrac{\partial \mathbf{U}}{\partial z} + \mathbf{L}\mathbf{B(U)} = 0

(2.33)

If there exist two quantities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2}
which satisfy

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial W_1}{\partial \mathbf{U}} = \mathbf{l_1}, \quad \dfrac{\partial W_2}{\partial \mathbf{U}} = \mathbf{l_2}

(2.34)

we will call them characteristic variables of the hyperbolic system. By setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{W}=[W_1,W_2]^T}

the system 2.33 may be elaborated into

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \mathbf{W}}{\partial t} + \mathbf{\Lambda }\dfrac{\partial \mathbf{W}}{\partial z} + \mathbf{G} = 0

(2.35)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{G} = \mathbf{L}\mathbf{B} - \dfrac{\partial \mathbf{W}}{\partial A_0}\dfrac{dA_0}{dz}- \dfrac{\partial \mathbf{W}}{\partial \beta }\dfrac{d\beta }{dz}

Under the assumption that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta _1}
are constant in space and taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{B}}
negligible, the equation 2.35 becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \mathbf{W}}{\partial t} + \mathbf{\Lambda }\dfrac{\partial \mathbf{W}}{\partial z} = 0

which is a system of decoupled scalar equation written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial W_i}{\partial t} + \mathbf{\lambda _i}\dfrac{\partial W_i}{\partial z} = 0

(2.36)

From 2.36 we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2}
are constant along two characteristics curves in the (z,t) plane 5 described by the differential equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{dz}{dt}=\lambda _1 \quad , \quad \dfrac{dz}{dt}=\lambda _2

Diagram of characteristics in the (z,t) plane. The solution on the point R is obtained by the superimposition of the two characteristics W₁ and W₂.

Figure 5: Diagram of characteristics in the (z,t) plane. The solution on the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R

is obtained by the superimposition of the two characteristics Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2

.

The expression for the left eigenvectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{l}_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{l}_2}
is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{l}_1 = \xi \begin{bmatrix}c_\alpha - \alpha \bar{u} \\ 1 \end{bmatrix}, \quad \mathbf{l}_2 = \xi \begin{bmatrix}-c_\alpha - \alpha \bar{u} \\ 1 \end{bmatrix},

(2.37)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi =\xi (A,\bar{u})}

is any arbitrary smooth function of its arguments with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi >0}

. Here we have expressed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{l}_1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{l}_2}
as functions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A,\bar{u})}
instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A,Q)}
in order to simplify the next developments.

For an hyperbolic system of two equations is always possible to find the characteristic variables (or, equivalently, the Riemann invariants) locally, that is in a small neighbourhood of any point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}}

[9], yet the existence of global characteristic is not in general guaranteed. However, assuming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha = 1}
the relationship 2.37 take the much simpler form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial W_1}{\partial A}= \xi c_1, \qquad \dfrac{\partial W_1}{\partial \bar{u}}= \xi A	(2.38)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial W_2}{\partial A}= -\xi c_1, \qquad \dfrac{\partial W_2}{\partial \bar{u}}= \xi A	(2.39)

We now show that a set of global characteristic variables do exist for the problem at hand. Since we note, from 2.38, that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_{1,2}}

are exact differentials being

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial ^2 W_i}{\partial A \partial \bar{u}}=\dfrac{\partial ^2 W_i}{ \partial \bar{u}\partial A}

for any values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{u}}

, we also have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_1}

does not depend on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{u}}
and then, from above relationship we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_1 \dfrac{\partial \xi }{\partial \bar{u}}=\xi +A\dfrac{\partial \xi }{\partial A}

In order to satisfy this relation we have to choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g=g(A)}

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g=-A\dfrac{\partial g}{\partial A}}

. To do this we can take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g=A^{-1}} . As a consequence we can write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \partial W_1=\dfrac{c_1}{A} \partial A + \partial \bar{u}, \quad \partial W_2=-\dfrac{c_1}{A} \partial A + \partial \bar{u}

(2.40)

Taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_0,0)}

as a reference state for our variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_0,\bar{u})}
we can integrate the above relationships obtaining

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1=\bar{u} + \int _{A_0}^A \dfrac{c_1(\epsilon )}{\epsilon }d\epsilon , \qquad W_2=\bar{u} - \int _{A_0}^A \dfrac{c_1(\epsilon )}{\epsilon }d\epsilon

(2.41)

Introducing the expression 2.29 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_1}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_{1,2}=\dfrac{Q}{A} \pm 4 \Bigg(\sqrt{\dfrac{\beta _1}{2\rho A_0}}A^{\frac{1}{4}} - c_0 \Bigg)

(2.42)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_0}

is the wave speed related to the reference state.  We finally can write the variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A,Q)}
in terms of the characteristic ones,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A= \Bigg(\dfrac{2 \rho A_0}{\beta _1} \Bigg)^2 \Bigg(\dfrac{W_1 - W_2}{8} \Bigg)^4, \qquad Q=A \dfrac{W_1+W_2}{2}

(2.43)

allowing in particular, the implementation of boundary and compatibility conditions, that we will discuss in the next section.

2.1.6 Boundary conditions

By the characteristic analysis of the one-dimensional model we pointed out the hyperbolic nature of the one-dimensional system of blood flow in arteries; consequently the solution is given by the superimposition of two waves whose eigenvalues Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _{1,2}}

represent the propagation speeds of such waves. As we have seen previously, they always have opposite sign and so blood flow is sub-critical; under this condition, we need an initial condition along all the spatial domain and two boundary conditions to close the governing system: one at the inlet section Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=0}
and the other at the outlet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=L}
(see figure 6).

Boundary and initial conditions of the hyperbolic system.

Figure 6: Boundary and initial conditions of the hyperbolic system.

One-dimensional model with absorbing conditions.

Figure 7: One-dimensional model with absorbing conditions.

Different type of boundary conditions can be imposed. An important class of boundary conditions is represented by the so-called non-reflecting or absorbing boundary conditions [10], which allows the simple wave associated with the characteristics to enter or leave the domain without spurious reflections (see Figure 7). Absorbing boundary conditions can be imposed by defining values for the wave entering the domain; in our case we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _1>0}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _2<0}
so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1}
is the entering characteristic in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2}
the inlet characteristic in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Z=L}

. In Hedstrom [11], non-reflecting boundary conditions for an hyperbolic problem are written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{l}_1 \cdot \Bigg[\dfrac{\partial \mathbf{U}}{\partial t} - \mathbf{B}(\mathbf{U}) \Bigg]_{x=0} =0, \quad \mathbf{l}_2 \cdot \Bigg[\dfrac{\partial \mathbf{U}}{\partial t} - \mathbf{B}(\mathbf{U}) \Bigg]_{x=L} =0

When there is an explicit formulation of the characteristic variables, it is possible impose the boundary conditions directly in terms of incoming characteristics, for example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1(t)=g_1(t), \qquad \hbox{in} \quad z=0, t>0

being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g_1(t)}

a given function. However, the problem rarely have boundary data in terms of variable characteristics, they are normally expressed in terms of physical variables.

In addition to absorbing boundary conditions based on characteristic variables, it may impose a function that describes the temporal trend on the edge of one of the unknown functions of the problem, then the flux flow Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q}

(or the speed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

) or the area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A} . Conditions of this type are typically used on the proximal node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=0}

and can be expressed as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q(0,t) = g_q(t), \quad t>0

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A(0,t) = g_a(t), \quad t>0

The boundary conditions imposed by the knowledge of the physical variables are reflective. Therefore, if we impose such a condition in the proximal node, the incoming characteristic variable, that we denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2} , will be partially reflected in the computational domain. This is a real physical phenomenon.

The initial conditions are the conditions to be imposed by defining the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A(z,t)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q(z,t)}
along the spatial domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z\in (0,L)}
at the initial time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=0}

. For instance if we require the area at the initial time, the initial condition is expressed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A(z,t)= A_0(z), \quad z\in (0,L)

2.1.6.1 Terminals lumped parameter

The assumptions made for the 1-D model become less appropriate with decreasing the size of the arteries; for example, the blood flow in the larger arteries is pulsatile and is dominated by inertia while in the capillaries is almost stable and dominant by the viscosity. Consequently, the 1-D model should be limited until at the distal section of the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (z=L)} . We have seen a first approach which imposes not reflective boundary conditions in the vessels terminals 2.44, but this solution is not adherent to reality. We then introduce the lumped parameter models (0-D) who consider the fact that the pressure waves are physically in part reflected and partly absorbed. These models coupled with the one-dimensional constitutive equation 2.27 leads to a multiscale framework 1-D/0-D. Therefore, the hemodynamic effects of the blood vessels after the distal section limit are generally simulated using a lumped parameter model governed by ordinary differential equations that relate the pressure with the flow at the outlet of the 1-D model [12].

Expressing the system 2.30 in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A,P,Q)}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q=A\bar{u}}
and linearising around the state of reference Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_0,0,0)}

, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta }

an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}
be constant along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

, is obtained.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_{1D}\dfrac{\partial p}{\partial t} + \dfrac{\partial q}{\partial z}=0

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L_{1D}\dfrac{\partial q}{\partial t} + \dfrac{\partial p}{\partial z}= - R_{1D}q

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p = \dfrac{a}{C_{1D}}

(2.44)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q}
are the perturbation variables for area, pressure and volume flux, respectively Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_0+a,p,q)}
and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_{1D} =\dfrac{\rho c_0}{A_0}, \qquad C_{1D} =\dfrac{A_0}{\rho c_0^2}, \qquad L_{1D} =\dfrac{\rho }{A_0}

(2.45)

are the viscous resistance to flow, wall compliance and blood inertia, respectively, per unit of length of vessel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l} . Integrating system 2.44 over the length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l}

yields the lumped parameter model, where the variables are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_{0D}=R_{1D}l}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_{0D}=C_{1D}l} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{0D}=L_{1D}l}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{p}=\frac{1}{l} \int _0^lp dz, \hat{q}=\frac{1}{l} \int _0^lq dz}
are the mean pressure and flow over the whole domain. In physiological conditions pulsatile waves travel at a speed greater compared to that of the blood, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{p}=p_{in}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{q}=q_{out}}

. Therefore, the final 0-D model is the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C_{0D}\dfrac{\partial p_{in}}{\partial t} + q_{out} - q_{in}=0

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L_{0D}\dfrac{\partial q_{out}}{\partial t} + R_{0D}q_{out} + p_{out} - p_{in}= 0

(2.46)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_{in}=q(0,t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_{out}=q(L,t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{in}=p(0,t)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{out}=p(L,t)}
are the flows and pressures at the inlet and outlet of the 0-D domain. As it is represented in Figure 8, the system 2.46 is analogous to an electric circuit, in which the role of the flow and pressure are played by the current and potential, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_{0D}}
corresponds to an electric resistance, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_{0D}}
to a capacitance and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{0D}}

to an inductance [13].

1-D arterial vessel domain (left) and the equivalent 0-D system discretises at first order in space (right).

Figure 8: 1-D arterial vessel domain (left) and the equivalent 0-D system discretises at first order in space (right).

Table. 2 Analogy between hydraulic and electrical network.
Hydraulic	Physiological variables	Electric
Pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P[Pa]}	Blood pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [mmHg]}	Voltage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}
Flow rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q[m^3/s]}	Blood flow rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [L/s]}	Current Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I}
Volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V[m^3]}	Blood volume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [L]}	Charge Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q[C]}
Viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta }	Blood viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu [Pa\cdot s]}	Electrical resistance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R}
Elastic coefficient	Vessel's wall compliance	Capacitance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C}
Inertance	Blood inertia	Inductance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

2.1.7 Implementation

The nonlinear hyperbolic system 2.27 has been discretized using a Taylor-Galerkin scheme [14], which is the finite element equivalent of Lax-Wendroff (based on the expansion in Taylor series) stabilisation for the finite difference method. This method may result in short computational times, and is second order accurate in both time and space.

Considering the equation 2.27 and having Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}=\dfrac{\partial \mathbf{F}}{\partial \mathbf{U}}}

we may write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial \mathbf{U}}{\partial t} = \mathbf{S} - \dfrac{\partial \mathbf{F}}{\partial z}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dfrac{\partial ^2 \mathbf{U}}{\partial t^2} = \dfrac{\partial \mathbf{S}}{\partial \mathbf{U}} \dfrac{\partial \mathbf{U}}{\partial t} - \dfrac{\partial }{\partial z} \Bigg(\mathbf{H} \dfrac{\partial \mathbf{U}}{\partial t} \Bigg)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): = \dfrac{\partial \mathbf{S}}{\partial \mathbf{U}} \Bigg(\mathbf{S} - \dfrac{\partial \mathbf{F}}{\partial z} \Bigg)- \dfrac{\partial \mathbf{H} \mathbf{B}}{\partial z} + \dfrac{\partial }{\partial z} \Bigg(\mathbf{H} \dfrac{\partial \mathbf{F}}{\partial z} \Bigg)

(2.47)

For simplicity the dependence of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{S}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}}
from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}}
is dropped. Starting from the above equations, we now consider the time intervals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (t^n,t^{n+1})}

, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=0,1,\dots ,T}

then we discretize the equation in time using a Taylor series which includes first and second order derivatives if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}}

. Therefore we obtain the following semi-discrete schemes for the approximation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}^{n+1}}

of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}(t^{n+1})}

Taylor-Galerkin scheme:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{u}^{n+1} = \mathbf{u}^n + \Delta t \mathbf{u}_t^{n} + \dfrac{\Delta t^2}{2}\mathbf{u}_{tt}^{n}

(2.48)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{U}^{n+1} = \mathbf{U}^n - \Delta t \dfrac{\partial }{\partial z} \Bigg(\mathbf{F}^n + \dfrac{\Delta t}{2} \mathbf{H}^n \mathbf{S}^n \Bigg)+ \dfrac{\Delta t^2}{2} \Bigg[\mathbf{S}_\mathbf{U}^n\dfrac{\partial \mathbf{F}^n}{\partial z} - \dfrac{\partial }{\partial z} \Bigg(\mathbf{H}^n \dfrac{\partial \mathbf{F}^n}{\partial z}\Bigg)\Bigg]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): + \Delta t \Bigg(\mathbf{S}^n + \dfrac{\Delta t}{2}\mathbf{S}_\mathbf{U}^n\mathbf{S}^n \Bigg), \qquad n=0,1,\dots

(2.49)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{S}_\mathbf{U}^n= \dfrac{\partial \mathbf{S}^n}{\partial \mathbf{U}}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F}^n}

, stands for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{F(U^n)}} , just as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}^n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{S}^n}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{S}_\mathbf{U}^n}

the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}^0}

is given by the initial conditions.

For each time interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (t^n,t^{n+1})}

we apply a spatial discretization carried out using the Galerkin finite element method. To this purpose we subdivide the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \Omega =\{ z:z\in (0,L)\} }

, which is the 1-D counterpart of the 3-D domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _t} , into a finite number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{el}}

of linear elements length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle l}
(Figure 9).

One-dimensional mesh representing a vessel.

Figure 9: One-dimensional mesh representing a vessel.

Moreover we introduce a trial function space, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}} , and a weighting function space, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{W}} . These spaces are both defined to consist of all suitably smooth functions and to be such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{T} = \Big\{\mathbf{U}(z,t) | \mathbf{U}(z,0) = \mathbf{U}^0(z) \hbox{ on } \Omega _t \quad \hbox{at}\quad t=t^0 \Big\}, \qquad \mathcal{W}= \Big\{\mathcal{W}(\mathbf{z}) \Big\}

Considering the scheme, we multiply the equation 2.49 for the weight function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{W}}

and we integrate it over the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _t}
obtaining, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall t >0 t^0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{\Omega } \Bigg(\mathbf{U}^{n+1}-\mathbf{U}^n \Bigg)d\Omega = - \Delta t \Bigg[\int _{\Omega } \dfrac{\partial \mathbf{W}}{\partial z} \mathbf{F}_{LW}^n d\Omega - \int _{\Omega } \mathbf{S}_{LW}^n \mathbf{W} d\Omega \Bigg]-

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): - \dfrac{\Delta t^2}{2} \Bigg[\int _{\Omega } \dfrac{ \partial \mathbf{W}}{\partial z} \mathbf{S_U}^n\mathbf{F}^n - \int _{\Omega } \dfrac{\partial \mathbf{W}}{\partial z} \dfrac{\partial \mathbf{F}^n}{\partial z}\mathbf{H}^n \Bigg]-

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): - \Delta t \Bigg[N_i \bar{\mathbf{F}}_r^n |_{z=L} - N_i \bar{\mathbf{F}}_l^n |_{z=0} \Bigg]

(2.50)

where we have assumed

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{F}_{LW}^n(\mathbf{U}_j) = \mathbf{F}^n + \dfrac{\Delta t}{2}\mathbf{H}^n\mathbf{S}^n

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{S}_{LW}^n(\mathbf{U}_j) = \mathbf{S}^n + \dfrac{\Delta t}{2}\mathbf{S_U}^n\mathbf{S}^n

Starting from the weak form of the problem 2.50 we build the subspaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}^h}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{W}^h}
for the trial and weighting function spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{T}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{W}}
defining them as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{T}^h = \Bigg\{\hat{\mathbf{U}}(z,t)| \hat{\mathbf{U}}(z,t)= \sum _{j=1}^N \mathbf{U}_j(t)N_j(z); \quad \mathbf{U}(t^0)= \bar{\mathbf{U}}(z_j)= \mathbf{U}_j^0 \Bigg\}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{W}^h = \Bigg\{\mathcal{W}(z,t)|\mathcal{W}(z)= \sum _{j=1}^N W_j(t)N_j(z) \Bigg\}

(2.51)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_j}

is the standard linear finite element shape function (Figure 10) associated to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j-th}
node, located at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=z_j}

, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}_j}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\mathbf{U}}}
at the node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j}

. Since we are using the Galerkin method, the base shape functions defined above are used as weighting.

Figure 10: Sketch of a 1D linear shape function.

Adopting the following notation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (W,U)_{\Omega _e} = \int _{\Omega _e} W \cdot U d\Omega ,

and considering the sum of each element contribution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{\Omega } \dots = \sum _{el} \int _{\Omega _e} \dots ,

the equation 2.50 becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sum _{el}(N_i,N_j)_{\Omega _e}( \mathbf{U}_j^{n+1}-\mathbf{U}_j^{n} ) = \Delta t \sum _{el} [ ( N_{i,z},N_j)_{\Omega _e} \mathbf{F}_{LW}^n(\mathbf{U}_j) + N_i,N_j)_{\Omega _e} \mathbf{S}_{LW}^n(\mathbf{U}_j) ] -

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): - \dfrac{\Delta t^2}{2} \sum _ {el} \Bigg((N_i,N_j)_{\Omega _e} \mathbf{S_U}^n(\mathbf{U}_j) \dfrac{\partial \mathbf{F}_j^n}{\partial z} \Bigg)-

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): - \dfrac{\Delta t^2}{2} \sum _ {el} \Bigg((N_{i,z},N_j)_{\Omega _e} \mathbf{H}_j^n(\mathbf{U}_j) \dfrac{\partial \mathbf{F}_j^n}{\partial z}\Bigg)-

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): - \Delta t \Bigg[N_i \bar{\mathbf{F}}_r^n |_{z=L} - N_i \bar{\mathbf{F}}_l^n |_{z=0} \Bigg]

(2.52)

For what concerns the border nodes, we have to consider the boundaries condition. Starting from the equation 2.52, we have the term of boundary conditions represented by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta t \Bigg[N_i \bar{\mathbf{F}}_r^n |_{z=L} - N_i \bar{\mathbf{F}}_l^n |_{z=0} \Bigg], \qquad i=1,2

which implies the knowledge of the flux terms depending from the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Q}
at inlet and outlet sections of the domain. To extract them from the characteristic information Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1(0,t)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2(L,t)}
we need an additional expression for the other characteristic variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2(0,t)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1(L,t)}
to recover Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}(A,Q)}
using the equation 2.43. To this purpose we adopted a technique based on the extrapolation of the outgoing characteristics. Having the friction parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K_r}
small with respect to the other equation terms in 2.27, we assume that at the boundary points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=0}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z=L}
the flow is generated by the characteristic system 2.36. At a generic time step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}^n}
known and we linearise the eigenvalues Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _{1,2}}
of 2.27 by taking their values at respective boundary for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t=t^n}

. The solution corresponding to this linearised problem at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t^{n+1}}

gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2^{n+1}(0)= W_2^n(-\lambda _2^n(0)\Delta t)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1^{n+1}(L)= W_1^n(-\lambda _1^n(L)\Delta t)

which is a first-order approximation of the outgoing characteristic variables from the previous step. By using these information together with the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1^{n+1}(0)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2^{n+1}(L)}
, we are able to compute, through 2.43, the required boundary data.

We choose to use, for time integration, both a second and fourth order explicit Runge-Kutta scheme; such methods are diffused in computational fluid dynamics, and show good properties, e.g ease of programming, simple treatment of boundary conditions and good stability. Regarding the stability, the second order Taylor-Galerkin scheme entails a time step limitations. A linear stability analysis [15] indicates that the following Courant-Friedrichs-Lewy condition should be satisfied

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta t \le \dfrac{\sqrt{3}}{3}\min _{0\le i \le N} \Bigg[\dfrac{h_i}{\max (\lambda _{1,i},\lambda _{1,i+1})} \Bigg]

(2.53)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _{1,i}}

here indicates the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _1}
at mesh node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_i}

. This condition, which is necessary to obtain the stability of a method explicitly imposes a constraint on the choice of the discretization time and space of the method used; it corresponds to a CFL number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\sqrt{3}}{3}} .

2.1.7.1 Branching

The vascular system is characterized by the presence of branching. The flow in a bifurcation is intrinsically 3-D, however may be still described by means of a 1-D model. In order to manage a branching zone, when using a 1-D formulation, we follow a technique called domain decomposition [16]. The numerical solver accounts for the treatment of two types of bifurcation: the bifurcation 2-1 typical of the arterial system and the bifurcation 1-1 which represents two vessel linked together with different mechanical properties.

2.1.7.2 Bifurcation 1-2

The bifurcation 1-2 represents the typical branching of the arterial system. As we have introduced we have used the domain decomposition method to solve this problem. We divide the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

into three partitions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _1}

,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _2}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _3}
as is showed at Figure 11; doing this we have 3 sub-problems which must be coupled imposing adequate boundary conditions. Then we have to evaluate 6 variables, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_i,Q_i)}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1,2,3}

, corresponding to the problem unknowns, area and flow rate for each one of the vessels composing the branching.

Figure 11: Domain decomposition of a bifurcation 1-2.

From the decomposition of the governing system into characteristic variables we know that the system can be interpreted in terms of a forward and backward travelling waves. Considering the model of a splitting bifurcation shown in Figure 11, we denote the parent vessel by an index 1 and its two daughter vessels by the indices 2 and 3, respectively. The simplest condition we can impose to require the mass conservation through the bifurcation and therefore the flow rate balance can be written

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_1=Q_2+Q_3

remembering that the flow moves from the sub-domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _1}

to the sub-domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _3}

. Other two assumptions can be obtained from the requirement of continuity of the momentum flux at the bifurcation. This lead to consider the total pressure term continuous at the boundary. So we may write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_1}{A_1} \Bigg)^2 = P_2 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_2}{A_2} \Bigg)^2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_1}{A_1} \Bigg)^2 = P_3 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_3}{A_3} \Bigg)^2

The remaining three relationship can be derived using the characteristic variables. The parent vessel can only reach the junction by a forward travelling wave. This wave is denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_1^1} , where the superscript is the vessel number while the subscript stands for the forward direction. Similarly, the characteristics variables of daughter vessels, which can reach the bifurcation only by backwards travelling wave, are represent by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2^2}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_2^3}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1^1 = \dfrac{Q_1}{A_1} + 4\sqrt{\dfrac{\beta _1}{2\rho A_{01}}} A_1^{1/4} = u_1 + 4(c_1-c_0^1)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2^1 = \dfrac{Q_2}{A_2} - 4\sqrt{\dfrac{\beta _2}{2\rho A_{02}}} A_2^{1/4} = u_2 + 4(c_2-c_0^2)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2^2 = \dfrac{Q_3}{A_3} - 4\sqrt{\dfrac{\beta _3}{2\rho A_{03}}} A_3^{1/4} = u_3 + 4(c_3-c_0^3)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_0^1} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_0^2 } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_0^3}

are the values of the wave speed evaluated using the area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_0}
in the vessels 1,2 and 3.  In summary, the resulting system which determines the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_1,Q_1)}

,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_2,Q_2)}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_3,Q_3)}
at the bifurcation is the following

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1^1 = \dfrac{Q_1}{A_1} + 4\sqrt{\dfrac{\beta _1}{2\rho A_{01}}} A_1^{1/4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2^1 = \dfrac{Q_2}{A_2} - 4\sqrt{\dfrac{\beta _2}{2\rho A_{02}}} A_2^{1/4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2^2 = \dfrac{Q_3}{A_3} - 4\sqrt{\dfrac{\beta _3}{2\rho A_{03}}} A_3^{1/4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_1=Q_2+Q_3

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_1}{A_1} \Bigg)^2 = P_2 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_2}{A_2} \Bigg)^2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_1}{A_1} \Bigg)^2 = P_3 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_3}{A_3} \Bigg)^2

(2.54)

We can solve it through the Newton-Raphson technique for differential systems of non-linear equations.This type of modelling does not consider the geometry of the junctions. For instance, the angle between the various vessels are not take into account.

2.1.7.3 Bifurcation 1-1

The discontinuity at the interface between arteries with different materials(mechanical behaviour) or geometrical properties is solved with a similar process used in the treatment of the bifurcations 2-1. Following the domain decomposition method adopted before, we proceed by splitting the problem in two sub-domains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _1}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega _2}

, and solving the following non-linear system for the interface variables, namely

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1 = \dfrac{Q_1}{A_1} + 4\sqrt{\dfrac{\beta _1}{2\rho A_{01}}} A_1^{1/4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2 = \dfrac{Q_2}{A_2} - 4\sqrt{\dfrac{\beta _2}{2\rho A_{02}}} A_2^{1/4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_1=Q_2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_1}{A_1} \Bigg)^2 = P_2 + \dfrac{1}{2}\rho \Bigg(\dfrac{Q_2}{A_2} \Bigg)^2

(2.55)

Again, We solve the non-linear system obtained through the Newton-Raphson method. In both systems 2.54 and 2.55, it has been verified that the determinant of the Jacobian is different from zero for all allowable values of the parameters, thus guaranteeing that the Newton iteration is well-posed [17].

Domain decomposition of a bifurcation 1-1.

Figure 12: Domain decomposition of a bifurcation 1-1.

2.1.7.4 Coupling 1-D and 3D-reduced model

In order to consider into the 1D model an external pressure drop, we need to modify the total pressure term (2.55) adding the function fFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{3D}} (kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _1} ,kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _2} ):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_{i} + \frac{1}{2}\rho \frac{{Q_{i}}^{2}}{{A_{i}}^{2}} =P_{j} + \frac{1}{2}\rho \frac{{Q_{j}}^{2}}{{A_{j}}^{2}} + {f^{3D}}_j(k_1,k_2)

(2.56)

where indexes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hbox{i}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hbox{j}}
denote the parent and the daughter vessels respectively and the function fFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{3D}}

(kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _1} ,kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _2} ) denotes the external pressure drop (or energy losses). In our case fFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^{3D}}

is the pressure drop of the 3D model.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {f^{3D}}_j(k_1,k_2)= k_1 Q_j + k_2\mid Q_j\mid Q_j

(2.57)

kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _1}

and kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _2}
are the viscous and turbulent coefficients that should be adjusted according to the pressure drop between the inlet/outlet planes defined in the 3D model.

Personalized 3D reduced order model In our particular case, firstly we need to solve the 3D problem (real case). To estimate the coefficients k1 and k2 of the equation (2.57), at each time step tFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _{n}} , we calculate and store the mean values of the flow and the pressure at the inlet and the outlet of our 3D domain. Using these values we choose kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _1}

and kFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _2}
that minimizes the sum of squared pressure drop by least squares method. In this way we are able to capture the energy losses provoked by the geometry of our 3D model, and take into account in the 1D model.

2.1.8 Coupling 1-D and 0-D models

The existence and uniqueness of the solution of a coupled problem between the 0-D model system 2.46 and the hyperbolic 1-D system 2.35, has been proven by Formaggia [18] for a sufficiently small time so that the characteristic curve leaving the 1-D/0-D interface does not intersect with incoming characteristic curves. Numerically, the coupling problem between a 1-D domain and a 0-D model is established through the solution of a Riemann problem at the interface (Figure 13). An intermediate state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A^*,U^*)}

originates at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t+ \Delta t}
from the states Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_L,U_L)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_R,U_R)}
at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t}

. The state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_L,U_L)}

corresponds to the end point of the 1-D domain, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A_R,U_R)}
is a virtual state selected so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A^*,U^*)}
satisfies the relation between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^*}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U^*}
dictated by system 2.46. The 1-D and 0-D variables at the interface are related through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q_{in}=A^*U^*}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{in}= \frac{\beta }{A_0} (\sqrt{A^*}-\sqrt{A_0})}

, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{out}}

is prescribed as a constant parameter that represents the pressure at which flow to the venous system ceases.

Coupling 1-D/0-D model.

Figure 13: Coupling 1-D/0-D model.

According to the method of characteristics, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G=0}

equation 2.35 leads to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_1(A^,U^) = W_1(A_L,U_L)	(2.58)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): W_2(A^,U^) = W_2(A_R,U_R)	(2.59)

Solving 2.58 for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^*}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U^*}
yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A^*= \Bigg[\sqrt{\dfrac{2 \rho A_0}{\beta }} \dfrac{W_1(A_L,U_L)-W_2(A_R,U_R)}{8} + A_0^{\frac{1}{4}} \Bigg]^4

(2.60)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): U^*= \dfrac{W_1(A_L,U_L)+W_2(A_R,U_R)}{2}

(2.61)

The 1-D outflow boundary condition is imposed by enforcing that either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U_R=U_L} , which reduces Eq. 2.60 to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A_R= \Bigg[2(A^*)^{\frac{1}{4}} - (A_L)^{\frac{1}{4}} \Bigg]^4

(2.62)

or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A_R=A_L} , which reduces Eq. 2.61 to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): U_R = 2U^* - U_L

(2.63)

2.1.8.1 Terminal resistance (R) model

This model simulates the peripheral circulation as a purely resistive load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Rp} , (RFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _{0D}} =Rp, LFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _{0D}} =0, CFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle _{0D}} = 0) (Figure 8) and in which the state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (A^*,U^*)}

satisfies

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A^*U^*=\dfrac{P(A^*)- p_{out}}{R_p}

(2.64)

Combining with 2.46 we leads to a non-linear equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{F}(A^*) = R_p \Bigg[\Big[U_L + 4c(A_L)\Big]A^* -4c(A^*)A^* \Bigg]- \dfrac{\beta }{A_0} \Bigg(\sqrt{A^*}- \sqrt{A_0} \Bigg)+ p_{out} = 0

(2.65)

which is solved using Newton-Raphson method, with the initial value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^*=A_L} . Once Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^*}

has been obtained, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U^*}
is calculated from Eq. 2.61. If we consider both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}
equal to zero, we lead to the single terminal resistance model.

2.1.8.2 Three-element (RCR) Windkessel model

This model accounts for the resistance and the compliance of the peripheral vessels using the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle RCR}

Windkessel model, which accounts for the cumulative effects of all distal vessels (small arteries, arterioles and capillaries). The three-element Windkessel model consists of two resistances Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_2}
and a capacitor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C}

. According to Section 2.1.8.1, we consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1}

to let any incoming wave reach the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle CR_2}
system without being reflected. Waves are reflected by the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle CR_2}
system, which is governed by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C\dfrac{dP_c}{dt} = A^*U^* - \dfrac{p_c-p_{out}}{R_2}

(2.66)

The first resistance, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1} , is introduced in order to absorb the incoming waves and reduces artificial wave reflections. It satisfies

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A^*U^*=\dfrac{P(A^*)-(p_c)^n}{R_1}

(2.67)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (p_C)^n}

is the pressure at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C}
at the time step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

. It is determining by solving at every time step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

the first-order time discretisation of the Eq. 2.66

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p_C^n = p_C^{n-1}- \dfrac{\Delta t}{C} A^*U^*

(2.68)

The coupling is solved as for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R}

terminal resistance model, but with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{out} = p_c}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R_1 = R_p}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathcal{F}(A^*) = R_1 \Bigg[\Big[U_L + 4c(A_L)\Big]A^* -4c(A^*)A^* \Bigg]- \dfrac{\beta }{A_0} \Bigg(\sqrt{A^*}- \sqrt{A_0} \Bigg)+ p_c = 0

(2.69)

Again, once Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A^*}

is obtained by Netwon-Rapshon, we can proceed to calculated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle U^*}

, int his case from 2.67.

2.1.9 Validation

A validation against in-vivo data is very difficult because some of the geometrical and elastic properties of the biological system are very complicated to measure. This is the reason because experimental replicas of the cardiovascular system to assess numerical tool are commonly used. To validate the 1D formulation implemented, the experimental model developed by Matthys et al. has been used [12] (1:1 silicone human arterial network). The silicone network is connected proximal to a pulsatile pump providing for a periodic input flow with the following settings: 70 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle bpm}

and a stroke volume of 70 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ml}

, creating a mean pressure of approximately 100 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle mmHg}

at the aortic root, these are typical values of a normal healthy person. Outflow boundary conditions were set of terminal resistance tubes connected to overflow reservoirs, creating a closed loop hydraulic system which induces a back pressure of 3.2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle mmHg}

. A 65–35Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle %}

water–glycerol mixture, with density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }
= 1050 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle kg\cdot m^3}
and viscosity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }
=5 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle mPa\cdot s}

, was used to simulate the blood. The elastic wall properties of the silicone sample have a constant Young's modulus of 1.2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle MPa} . The properties of the silicone network are summarize in table 3[12], as we can see the measurement report an interval of confidence, which unfortunately will affect the comparison between the experimental data and our results. For the simulations we have used the mean values show in the table 3.

Although the experimental set-up is only an approximation to the human systemic circulation, it is able to reproduce pulse waveforms with significant physiological features in the aortic vessels. Figure 15 shows some of the main physiological features of pulse pressure, velocity and flow rate described previously. We observe that the numerical solver is able to describe the peaking and steepening pulse pressure as we move away from the heart, and a reduction in amplitude of the flow with the distance from the heart.

Figure 14: a) Plan view schematic of the hydraulic model. 1: Pump (left heart); 2: catheter access; 3: aortic valve; 4: peripheral resistance tube; 5: stiff plastic tubing (veins); 6: venous overflow; 7: venous return conduit; 8: buffering reservoir; 9: pulmonary veins. (b) Topology and references labels of the arteries simulated, whose properties are given in table 3. (c) Detail of the pump and the aorta.

Figure 15: Simulated physiological(blank line) versus numerical results(red line) features of pressure and flow rate in difference section of the cardiovascular system.

Table. 3 Properties of the 37 silicon vessels used in the in-vitro model. The interval of confidence of the geometrical measurements is indicated in the heading.
n	Arterial segment	l [cm]	r [cm]	h [mm]	c[Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ms^{-1}} ]	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R_p[GPa\cdot s \cdot m^{-3}]
		Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \pm 2.0%	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \pm 3.5%	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \pm 2.5%
1.	Ascending Aorta	3.6	1.440	0.51	5.21	-
2.	Innonimate	2.8	1.100	0.35	4.89	-
3.	R. Carotid	14.5	0.537	0.28	6.35	2.67
4.	R. Subclavian I	21.8	0.436	0.27	6.87	-
5.	R. Subclavian II	16.5	0.334	0.16	6.00	-
6.	R. radial	23.5	0.207	0.15	6.78	3.92
7.	R. ulnar	17.7	0.210	0.21	8.81	3.24
8.	Aortic arch I	2.1	1.300	0.50	5.41	-
9.	L. Carotid	17.8	0.558	0.31	6.55	3.11
10.	Aortic arch II	2.9	1.250	0.41	4.98	-
11.	L. Subclavian I	22.7	0.442	0.22	6.21	-
12.	L. Subclavian II	17.5	0.339	0.17	6.26	-
13.	L. radial	24.5	0.207	0.21	8.84	3.74
14.	L. ulnar	1.91	0.207	0.16	7.77	3.77
15.	Thoracic Aorta I	5.6	1.180	0.43	5.29	-
16.	Intercostals	19.5	0.412	0.27	7.07	2.59
17.	Thoracic Aorta II	7.2	1.100	0.34	4.84	-
18.	Celiac I	3.8	0.397	0.20	6.20	-
19.	Celiac II	1.3	0.431	1.25	14.9	-
20.	Splenic	19.1	0.183	0.13	7.24	3.54
21.	Gastric	19.8	0.192	0.11	6.73	4.24
22.	Hepatic	18.6	0.331	0.21	6.95	3.75
23.	Abdominal Aorta I	6.2	0.926	0.33	5.19	-
24.	L. renal	12.0	0.259	0.19	7.39	3.46
25.	Abdominal Aorta II	7.0	0.790	0.35	5.83	-
26.	R. renal	11.8	0.255	0.16	6.95	3.45
27.	Abdominal Aorta III	10.4	0.780	0.30	5.41	-
28.	R. iliac-femoral I	20.5	0.390	0.21	6.47	-
29.	R. iliac-femoral II	21.6	0.338	0.15	5.89	-
30.	R. iliac-femoral III	20.6	0.231	0.20	8.04	-
31.	L. iliac-femoral I	20.1	0.402	0.20	6.19	-
32.	L. iliac-femoral II	19.5	0.334	0.16	6.11	-
33.	L. iliac-femoral III	20.7	0.226	0.13	6.67	-
34.	R. anterior tibial	16.3	0.155	0.15	8.47	5.16
35.	R. posterior tibial	15.1	0.153	0.12	7.73	5.65
36.	L. posterior tibial	14.9	0.158	0.11	7.23	4.59
37.	L. anterior tibial	12.6	0.156	0.10	7.01	3.16

3 Python Script

language=Python, basicstyle=, keywordstyle=, commentstyle=, stringstyle=, showstringspaces=false, identifierstyle=, procnamekeys=def,class linewidth=5cm

3.1 Phyton Script

fuentes/annex/wall_shear_Th.py

Appendix

References

[1] Tortora, G.J. and Derrickson, B. (2006) "Principios de anatomía y fisiología". Médica Panamericana

[2] Pearson Education Inc. (2013) "Hypertension and Physiology of the Heart, Pearson Education Inc"

[3] Wikipedia. (2004) "Circulatory system –- Wikipedia, The Free Encyclopedia"

[4] Perktold. (1991) "Pulsatile non-Newtonian Blood Flow in Three-Dimensional Carotid Bifurcation Models: A Numerical Study of Flow Phenomena Under Different Bifurcation Angles.", Volume 13. Journal of Biomedical Engineering 507-515

[5] J. J. Wang and K. H. Parker. (2004) "Wave propagation in a model of the arterial circulation", Volume 37. Journal Biomechanics 457-470

[6] A. Quarteroni and R. Sacco. (2000) "Matematica Numerica"

[7] A. Quarteroni and L. Formaggia. (2002) "Mathematical modelling and numerical simulation of the cardiovascular system"

[8] Canic.S and Kim.H. (2003) "Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axisymmetric vessels". Mathematical Methods in Applied Sciences

[9] E. Godlewski and P. A. Raviart. (1996) "Numerical approximation of hyperbolic systems of conservation laws"

[10] K. Thompson. (1987) "Time dependent boundary conditions for hyperbolic systems", Volume 68. Journal of Computational Physics 1-24

[11] Hedstrom G. W. (1979) "Non-reflecting boundary conditions for nonlinear hyperbolic systems". Journal of Computational Physics

[12] Matthys, KS and Alastruey, J and Peiro, J and Khir, AW and Segers, P and Verdonck, PR and Parker, KH and Sherwin, SJ. (2007) "Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements", Volume 40. Journal of Biomechanics 3476–3486

[13] I. Kokalari and T. Karaja and M. Guerrisi. (2013) "Review on lumped parameter method for modeling the blood flow in systemic arteries". Journal Biomedical Science and Engineering

[14] J. Donea and S. Giuliani and L. Quartapelle. (1984) "Time-accurate solutions of advenction-diffusion problems by finite elements", Volume 71. Computer Methods in Applied Mechanics and Engineering 93-122

[15] L. Quartapelle. (1993) "Numerical Solution of the Incompressible Navier-Stokes Equations"

[16] A. Quarteroni and A. Valli. (1999) "Domain decomposition methods for partial differential equation"

[17] D. Formaggia and D. Lamponi and A. Quarteroni. (1992) "One-dimensional model for blood flow in arteries". Journal of Engineering. Mathematics

[18] M. A. Fernandez and V. Mlsic and A. Quarteroni. (2005) "Analysis of a geometrical multiscale blood flow model based on the coupling of ODE's and hyperbolic PDE's", Volume 4. SIAM Journal Multiscale Modeling. Simulation 215-236

@@ Line 1,566: / Line 1,566: @@
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:draft_Samper_336772589-branching.png|600px|Domain decomposition of a bifurcation 1-2.]]
+|[[Image:Draft_Samper_336772589_9944_branching.png|600px|Domain decomposition of a bifurcation 1-2.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 11:''' Domain decomposition of a bifurcation 1-2.

Revision as of 13:12, 2 February 2018

Acknowledgments

Presentation

List of Figures List of Figures List of Figures

List of Tables List of TablesList of Tables

Abstract

Resumen

Resum

1 Cardiovascular physiology

1.1 Cardiovascular physiology

1.1.1 Blood Vessels

1.1.2 Blood Modelling

2 Numerical Model

2.1 1D Mathematical Model

2.1.1 Conservation equations

2.1.2 Conservation of the mass

2.1.3 Conservation of the momentum

2.1.4 Vessel wall constitutive model

Elastic model

2.1.5 Characteristic analysis

2.1.6 Boundary conditions

2.1.6.1 Terminals lumped parameter

2.1.7 Implementation

2.1.7.1 Branching

2.1.7.2 Bifurcation 1-2

2.1.7.3 Bifurcation 1-1

2.1.7.4 Coupling 1-D and 3D-reduced model

2.1.8 Coupling 1-D and 0-D models

2.1.8.1 Terminal resistance (R) model

2.1.8.2 Three-element (RCR) Windkessel model

2.1.9 Validation

3 Python Script

3.1 Phyton Script

References

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