1. Universit`a degli Studi di Pavia
Facolt`a di Ingegneria
Corso di Laurea Magistrale in Bioingegneria
A numerical tool for the simulation of the
blood flow
Supervisor:
Ferdinando Auricchio
Tutor:
Eduardo Soudah
Co - supervisor: Author:
Michele Conti Diego Corradini
UIN 413875
Academic year 2013/2014
5. Acknowledgements
I would like to express my sincere gratitude to my supervisors: Prof. Ferdinando Auricchio and Michele
Conti, for their continuous interest, support and availability during this project. I am also very grateful to
them for providing me the opportunity to do this thesis in collaboration with the Universitat Politècnica de
Catalunya (UPC).
Equally I would like to thank Eng. Eduardo Soudah who has been my tutor under the Erasmus program.
I am also really grateful to Prof. Paolo Magni for his help and guidance during the master course.
Finally, I wish to thank all the friends that have shared with me these tough but unforgettable years,
especially Villa Rosselli that has been my second family.
This project has been realized at CIMNE, Ciéntro Internátional de Metodos Numéricos en Ingéneria, UPC,
Campus Nord, building C1, C.Jordi Girona. Barcelona, Spain.
I
7. Abstract
The purpose of this project is to develop a numerical tool based on solving the one-dimensional gov-
erning equations of blood flow, able to predict general hemodynamic features of the human cardiovascular
system. Blood is simulated as an homogeneous, incompressible, and Newtonian fluid, while arteries are
modeled as a straight thin tubes. Different differential equations problems arise depending on the simplifi-
cations made on the mathematical model.
Starting from an existing 1-D code developed in FORTRAN90 at CIMNE [1], which considers a sim-
plified assumption of pure elastic behaviour of the vessel wall, the model is improved by including: an
additional term, the Kelvin-Voigt viscoelastic term, into the constitutive equation of the wall vessel, and
more realistic boundary conditions. The additional viscoelastic term leads the model to an hyperbolic-
parabolic system. An Operator Splitting technique, which consists in solve the system in two different
steps, was proposed to solve the augmented model. The first step, corresponding to the elastic part of the
system, is solved using Taylor-Galerkin scheme, while the second step, corresponding to the viscoelastic
part of the system, is solved using Crank-Nicolson. At the outlet boundary conditions, different lumped
parameter models, such as the R terminal model and the Windkessel RCR model, are coupled in the time
domain to the one-dimensional equations of blood flow in order to simulate the micro-circulation. More-
over, the numerical solver describes the branching of the arterial network: 1-2 connection typical of the
arterial system, 2-1 connection common of the venous system and 1-1 connection which simulates two
adjacent vessels with different mechanical characteristics, for example an healthy artery and one with a
stent.
Several test cases were carried out for the assessment of the proposed model against well-defined exper-
imental taken from literature. Specifically, these experiment regarding the simulation of an human aorta, a
stent implantation and a simplified arterial network of 37 arteries.
The main aim of the thesis has been the development of a finite element code programmed in FOR-
TRAN90 and its validation. This numerical solver is a valid tool for the study of the hemodynamic of the
blood flow for academic purposes and, beside, has the potential to assist in the diagnosis and treatment of
cardiovascular disease with a reasonable computational time.
III
13. List of Figures
1.1 Simplified diagram of the human Circulatory system in anterior view [6]. Red indicates
oxygenated blood (arterial system), blue indicates deoxygenated (venous system). . . . . . 3
1.2 The human heart (left) . Systolic and diastolic part of a cardiac cycle (right) [7]. . . . . . . 4
1.3 Artery, vein and capillaries characteristics and linking. . . . . . . . . . . . . . . . . . . . 4
1.4 Relationship between shear and stress in fluids. . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Qualitative plots of pressure and velocity in various segments of the human aorta. Pressure
waves increase in amplitude and steepness with increasing distance from the heart, whereas
velocity waves decrease in amplitude and increase in width [2]. . . . . . . . . . . . . . . . 8
1.6 Waveforms and mean value of pressure in the systemic circulation [2]. . . . . . . . . . . . 9
2.1 Difficulties of the Lagrangian (left) and Eulerian (right) formulation to follow the fluid-
structure interaction in a single vessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 A simple model of a section of an artery. Γw
0 indicates the initial configuration used for the
derivation of the model. Γw
t indicates the current configuration at time t. Ωt is the domain
occupied by the blood. The vessel wall Γw
t is moving. The location along the z axis of Γin
t
and Γout
t are fixed [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 ALE formulation in a single vessel [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 A model of a "realistic" section of an artery with the principal geometrical parameters [9]. 13
2.5 (a) Shell model of the vessel. (b) Radial wall displacement of the vessel. . . . . . . . . . . 14
2.6 Blood flow profile adopting different values of γ. . . . . . . . . . . . . . . . . . . . . . . 16
2.7 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. . . . . . . . . . . . . . . . . . . . . 22
2.8 Boundary and initial conditions of the hyperbolic system. . . . . . . . . . . . . . . . . . . 24
2.9 One-dimensional model with absorbing conditions. . . . . . . . . . . . . . . . . . . . . . 24
2.10 1-D arterial vessel domain (left) and the equivalent 0-D system discretises at first order in
space (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1 One-dimensional mesh representing a vessel. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Sketch of a 1D linear shape function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Coupling 1-D/0-D model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Lumped parameter models: Terminal resistance model (left) and Windkessel RCR model
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Peripheral circulation considering the 1-D formulation coupled to the Windkessel model. . 34
3.6 Domain decomposition of a bifurcation 1-2. . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Notation for a splitting (left) and merging (right) flow bifurcation. . . . . . . . . . . . . . 36
3.8 Domain decomposition of a bifurcation 1-1. . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1 Flowchart describing the execution of a GiD problem type. . . . . . . . . . . . . . . . . . 39
4.2 Vessel schematized as a straight line representing the vessel longitudinal axis. . . . . . . . 40
4.3 GiD windows material properties. Left window: vessel type belonging data from Reymond
[10]. Right window: customizable vessel with user-defined properties. . . . . . . . . . . . 41
IX
14. X LIST OF FIGURES
4.4 Schematic representation of the arterial tree adapted from Reymond [10]. A: main systemic
arterial tree. B: detail of the aortic arch. C: detail of the principal abdominal aorta branches.
D: schematic of the detailed cerebral arterial tree, which is connected via the carotids and
the vertebrals to the main arterial tree shown in A . L: left; R: right. . . . . . . . . . . . . . 41
4.5 GiD windows for inlet (left) and outlet (right) boundary conditions assignment. . . . . . . 42
4.6 GiD window for bifurcation assignment. Left: splitting junctions. Right: merging junctions. 42
4.7 Inlet and outlet boundary conditions assignment (left window) and bifurcation assignments
(Right window) of the simplified arterial tree simulated in this thesis. . . . . . . . . . . . . 43
4.8 GiD output window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.9 Transformation from the 1-D representation of the model geometry, adopted during the
pre-process, into a 3-D mesh, used in the post-process. . . . . . . . . . . . . . . . . . . . 45
4.10 GiD development control window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.1 Inlet flow rate imposed at the inlet of the aorta. . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Area time histories in the middle of the aorta coupled to an RCR lumped parameter model
to represent the rest of systemic vessel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Flow rate, velocity, pressure and area during 15 heart cycles in the middle of the aorta
obtained with absorbing boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4 Wall displacement of the aorta at step time t = 0.60 (left) and t = 0.61 (right) coupled to
an absorbing boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5 Pressure, velocity and flow rate in the middle of the aorta obtained with the R terminal model. 51
5.6 Periodic pressure (left) and velocity (right) time histories at three locations (x=0 m, x=0.2
m and x=0.4 m) along the 1-D artery coupled to a terminal resistance model. . . . . . . . 51
5.7 Pressure, velocity, flow rate and area in the middle of the single vessel obtained with the
Windkessel RCR model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.8 Periodic pressure (Left), velocity (Right) and flow rate (Bottom) time histories at three
locations (x=0 m, x=0.2 m and x=0.4 m) along the 1-D artery coupled to a RCR lumped
parameter model. [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.9 Effect of the changing of the aortic radius by 10% at x=0.2 m. Solid black line indicates
the control case while the dashed lines the results obtained with the changes. . . . . . . . . 54
5.10 Effect of the changing the aortic length by 10% at x=0.2 m. Solid black line indicates the
control case while the dashed lines the results obtained with the changes. . . . . . . . . . . 55
5.11 Effect of the changing of the aortic thickness by 10% at x=0.2 m. Solid black line indicates
the control case while the dashed lines the results obtained with the changes. . . . . . . . . 55
5.12 Effect of the changing the Young’s modulus by 10% at x=0.2 m. Solid black line indicates
the control case while the dashed lines the results obtained with the changes. . . . . . . . . 56
5.13 Effect of the changing the blood viscosity by 10% at x=0.2 m. Solid black line indicates
the control case while the dashed lines the results obtained with the changes. . . . . . . . . 56
5.14 Effect of the changing the peripheral resistance by 10% at x=0.2 m. Solid black line indi-
cates the control case while the dashed lines the results obtained with the changes. . . . . . 57
5.15 Effect of the changing the peripheral compliance by 10% at x=0.2 m. Solid black line
indicates the control case while the dashed lines the results obtained with the changes. . . . 57
5.16 Effect of the changing the venous pressure by 10% at x=0.2 m. Solid black line indicates
the control case while the dashed lines the results obtained with the changes. . . . . . . . . 57
5.17 Layout of the stent configuration (Top). Representation of the stent implantation developed
in GiD (Bottom). The red tubes indicate the healthy vessels while the blue tube indicate
the stent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.18 Time history at three specific points along the vessel: distal (D), medial (M) and proximal
(P), without stent (Top) and with a stent implanted (Bottom). . . . . . . . . . . . . . . . . 59
5.19 The experimental flow rate measured at the ascending aorta which is enforced as the inflow
boundary condition of the simulations [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
15. LIST OF FIGURES XI
5.20 a) Planview 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 over-
flow; 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 5.2. (c)
Detail of the pump and the aorta [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.21 The color in the image represents the flow rate field. . . . . . . . . . . . . . . . . . . . . . 63
5.22 Simulated physiological features of pressure (top), velocity (middle) and flow rate (bottom)
in difference section of the cardiovascular system. . . . . . . . . . . . . . . . . . . . . . . 64
5.23 Experimental and simulated elastic pressure (Top) and flow rate (Bottom) waveforms in
the middle point of the Aortic Arch. Blue dashed line represents the elastic model while
blue dashed line represents the viscoelastic model of the arterial wall. . . . . . . . . . . . 65
5.24 Experimental and simulated elastic pressure (Top) and flow rate (Bottom) waveforms in
the middle point of the Thoracic Aorta. Blue dashed line represents the elastic model while
blue dashed line represents the viscoelastic model of the arterial wall. . . . . . . . . . . . 66
5.25 Experimental and simulated elastic pressure (Top) and flow rate (Bottom) waveforms in
the middle point of the Splenic. Blue dashed line represents the elastic model while blue
dashed line represents the viscoelastic model of the arterial wall. . . . . . . . . . . . . . . 67
5.26 Experimental and simulated elastic pressure (Top) and flow rate (Bottom) waveforms in
the middle point of the Left Subclavia. Blue dashed line represents the elastic model while
blue dashed line represents the viscoelastic model of the arterial wall. The green dashed
line indicates the viscoelastic model coupled to an RCR Windkessel model. . . . . . . . . 69
5.27 Experimental and simulated elastic pressure (Top) and flow rate (Bottom) waveforms in
the middle point of the Right Anterior Tibial. Blue dashed line represents the elastic model
while blue dashed line represents the viscoelastic model of the arterial wall. The green
dashed line indicates the viscoelastic model coupled to an RCR Windkessel model. . . . . 70
B.1 Main program flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.2 Numerical solver flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
17. Nomenclature
A Cross-sectional area [m2
]
A0 Unstressed cross-sectional area [m2
]
c Wave speed [m/s]
E Young’s Modulus [Pa]
h Vessel wall thickness [cm]
L Vessel length [m]
n Current time step
P Internal Pressure [Pa]
Pext External Pressure [Pa]
Q Volume Flow [ml/s]
r Radial direction [cm]
R Vessel radius [cm]
R0 Unstressed vessel radius [cm]
t Time [s]
∆t Time step [s]
t Time [s]
u Fluid velocity [cm/s]
¯u Mean velocity over a cross-section [m/s]
z Axial direction [m]
H Jacobian matrix of the characteristic system
Λ Vector of eigenvalues
l Left Eigenvector
L Left Eigenmatrix
M Mass matrix
K Stiffness matrix
α Coriolis coefficient
β Parameter involving vessel material properties
∆t Time step [s]
λab Eigenvalues of the characteristic system
ρ Blood viscosity [poise]
µ Kinematic viscosity
ν Poisson ratio of vessel wall
Ω Domain
Γ Boundary
η Vessel wall displacement [cm]
XIII
19. Chapter 1
Introduction to the thesis
«Biomechanics aims to explain the mechanics of life and living. From molecules to organism every-
thing must obey the laws of mechanics.»[11]
Nowadays cardiovascular diseases represent the major cause of death in developed countries and con-
tributes to almost 30% of all global deaths [12]. Those pathologies are strictly related to the blood circu-
lation conditions, therefore an understanding of the local hemodynamics can then have useful applications
for the medical research. The direct extrapolation of hemodynamic parameters is difficult since the blood
circulation is within the living body and so there must be a compromise between the accuracy of measure-
ments and their invasive level. For this reason, numerical simulation of blood flow in the cardiovascular
system has become increasingly important in improving the diagnosis and treatment of cardiovascular dis-
eases.
The blood flow can be model, with different levels of accuracy, including: lumped models (0-D), one-
dimensional models (1-D), and three-dimensional models (3-D) [13].
Lumped parameter models are based on the analogy between blood circulatory system and an electrical
circuit, in which the blood pressure is considered as voltage and the blood flow is treated as current [3].
It is the simplest class of models, it may be used to simply representing global characteristics such as the
heart, the network of capillaries and the venous bed in short time. However, it has the disadvantage that
can not captured wave propagation phenomena due to the lack of explicit spatial representation.
On the other hand, three-dimensional models are the most sophisticated and allow to capture complex
hemodynamic factors. However, they are computationally expensive and hence are not suitable for the
study of the global circulatory system and rapid evaluation of surgical treatments.
One-dimensional models are a good compromise between computational complexity and accuracy.
These models are capable of capturing propagation phenomena and reproducing realistic flow and pressure
waveforms. They are based on the assumption that the dominant direction of blood flow is along the vessel
axis and that the pressure over the cross section of the vessel is constant.
In order to compute global effects, multi-scale framework, which are models of different level of
complexity, are coupled together with the aim of simulating the whole cardiovascular system. Three-
dimensional models are used to simulate a region of interest and reduced-order models (e.g., 1-D, 0-D)
to account for the influence of the rest of the circulation on a region of interest [14][15]. This coupling
method permits us to account for the complex feedback mechanism of the whole cardiovascular system,
while keeping the overall computational costs at a reasonable level [16].
The purpose of the thesis is to develop a numerical tool able to accurately predict time-varying flow rate
and pressure wave forms within a short computational time. For this reason, we have chosen to develop
a numerical code that couples 1-D/0-D model. Recent progress in imaging technology such as nuclear
magnetic resonance, digital angiography, tomography has open greater possibilities for the application
of 1-D modelling. We are now able to provide patient-specific information on vessel geometry, velocity
profiles and pulse wave speeds. This information permit the use of the 1-D formulation to simulate patient
specific arterial networks.
The study of cardiovascular mechanics is only possible with some knowledge of cardiovascular phys-
1
20. 2 1. Introduction
iology, for this reason a brief description of the cardiovascular system is given. This chapter starts with
a brief historical review of the mathematical models of the blood flow. Afterwards, some general aspects
of the fluid mechanics of the heart, the arterial system, the venous system as well as the most important
properties of the vascular tree that determine the pressure and flow characteristics in the cardiovascular
system will be dealt with.
1.1 Human circulatory system
The circulatory system also called the cardiovascular system, is an organ system which primary func-
tion is mass transport, that is, the transport of oxygen, carbon dioxide, hormones, waste products, etc.,
within the body. This system consists primarily of the heart, which serves as the pump, the vascular
system, which serves as the conduit through which the blood flows, and the blood, which serves as the
conducting medium [17].
The cardiovascular system is divided in systemic, pulmonary and coronary circulation:
• The systemic circulation carries oxygenated blood away from the heart to the body, and returns
deoxygenated blood back to the heart. It is pumped by the left heart.
• The pulmonary circulation carries deoxygenated blood away from the heart, to the lungs, and returns
oxygenated blood back to the heart. It is pumped by the right heart.
• The coronary circulation arises from the aorta and provides a blood supply to the myocardium, the
heart muscle.
1.1.1 The heart
The heart is the muscular organ of the circulatory system that acts as a double pump driving the blood
along the two circulations. In order to maintain these two circuits separate, heart is divided in two distinct
parts, each one having two chambers (Figure 1.2). Blood pumping is provided through the alternation of
a contraction phase, called systole, with a relaxation phase, the diastole; the repetition of these two phases
represent the cardiac cycle. During systole blood is pumped into the ascending aorta and during diastole
the aortic valve shunts and blood flows into the ventricular cavity from the atrium. The common duration
of the cardiac cycle is approximately 0.85 s (70 bpm) and each ventricle ejects about 70 ml of blood per
stroke [18], which yields a cardiac output of 5 l/min.
The developing of mathematical models of its mechanics, which involves, among other things, the
study of the electro-chemical activation of the muscle cells, is very complex. We will not cover this aspect
in this work, where we concentrate on vascular flow in arteries.
1.1.2 Vascular system
The vascular system or systemic circulation can be divided into three parts: the arterial system, the
capillary system and the venous system. Vessels can be considered as hollow tubes with strongly variable
diameters and can also be subdivided, according to their diameter, into: large or elastic (7 mm-3 cm),
medium or muscular (0.1 mm- 7mm), arterioles and capillaries (10 µm- 100 µcm) [2].
The arterial system is responsible to carry oxygenated blood to the tissues. Besides the transport
function of the arterial system the pulsating flow produced by the heart is also transformed to an approxi-
mately steady flow in the smaller arteries. Another important function of the arterial system is to maintain
a relatively high arterial pressure. This is of importance for a proper functioning of the brain and kidneys.
This pressure can be kept at this relatively high value because the distal end of the arterial system strongly
bifurcates into vessels with small diameters (arterioles) and hereby forms a large peripheral resistance. The
smooth muscle cells in the walls are able to change the diameter and hereby the resistance of the arterioles.
In this way the circulatory system can adopt the blood flow to specific parts in accordance to momentary
needs (vasoconstriction and vasodilatation). The wall of the arteries is not homogeneous, it consists of
three distinct layers of tunics, which, from inside to outside are called: tunica interna or intima, tunica
21. 1.1. Human circulatory system 3
Figure 1.1: Simplified diagram of the human Circulatory system in anterior view [6]. Red indi-
cates oxygenated blood (arterial system), blue indicates deoxygenated (venous system).
22. 4 1. Introduction
Figure 1.2: The human heart (left) . Systolic and diastolic part of a cardiac cycle (right) [7].
media and tunica externa or adventitia; which are separated by elastic membranes. Elastic arteries have a
large number of collagen and elastin filaments in the tunica media, which gives it the ability to stretch in
response to each pulse. This elasticity helps to maintain a relatively constant pressure in the arteries despite
the pulsating nature of the blood flow.
Leaving the arterioles the blood flows into the capillary system, a network of small vessels. The walls
consist of a single layer of endothelial cells lying on a basement membrane. Here an exchange of nutrients
with the interstitial liquid in the tissues takes place. In physiology, capillary blood flow is mostly referred
to as micro-circulation.
Finally the blood is collected in the venous system in which the vessels rapidly merge into larger
vessels transporting the blood back to the heart. The total volume of the venous system is much larger then
the volume of the arterial system. On the other hand, mean velocities and pressure amplitudes are lower
than in the arterial system.
Figure 1.3: Artery, vein and capillaries characteristics and linking.
Each segment of the cardiovascular system has an ideal combination of size, composition of the wall,
thickness and cross-sectional area in order to perform his task at best (Table 1.1). For instance, aorta
and large arteries have thick wall with an higher content of collagen and elastin as due their primary
task is to delivery blood under high pressure in all the regions of the body. The wall artery belong at
the category of the soft tissues. Its properties depend from the concentration and distribution of smooth
23. 1.1. Human circulatory system 5
muscle cells, collagen and elastin. The structural arrangement of the fibers of collagen and elastin and
their quantitative ratios are responsible for many properties of connective tissues such as density, elasticity,
anisotropy. Elastin has a very high elasticity and in general mechanical properties inferior to those of
collagen, as shown in Table 1.2. This behaviour is fundamental in all those tissues to which the request
is a large deformation with recovery of the original size after the removal of the load, or at least for those
tissues subjected to mechanical stress.
Vessels Diameter of lumen (mm) Wall thickness (mm) Mean pressure (kPa)
Aorta 25 2 12.5
Large arteries 1-10 1 12
Small arteries .5-1 1 12
Arteriole .01-.5 0.03 7
Capillary .006-.01 0.001 3
Venule .01-.5 0.003 1.5
Vein .5-15 0.5 1
Vein cava 30 1.5 0.5
Table 1.1: Properties of the human systemic vessels. (Adapted from Caro [2])
Young’s Modulus E (MPa) Ultimate tensile strength (MPa) Elongation at break (%)
Collagen 1000 50-100 10
Elastin 0,6 1 100
Table 1.2: Average mechanical properties of collagen and elastin.
1.1.3 The blood
Blood is a bodily fluid that delivers necessary substances such as nutrients and oxygen to the cells and
transports metabolic waste products away from those same cells. It is composed of suspension of particles,
called formed elements (about 45%), in a fluid called plasma (about 55%). The most important blood
particles are
• red cells (erythrocytes), responsible for the exchange of oxygen and carbondioxide with cells;
• white blood cells (leukocytes), which play a major role in the human immune system;
• platels (thrombocytes), main responsible of blood coagulation
The plasma is a diluted electrolyte solution containing several different proteins and substances dissolved
within. Plasma also transports nutrients, sugars, fats, hormones, gases, and waste material contained within
blood.
From a mechanical point of view, the blood is composed by flexible red cells suspended in the plasma,
which is a Newtonian fluid. Therefore, red cells makes the blood non-Newtonian. Because of its het-
erogeneous composition, blood rheology, that is the relation between the strain and the stress, is hard to
define. In fact rheologic behaviour of blood depends on several factors like pressure, temperature, and
vessel geometry which values can vary in time. The red blood cells are the main responsible for the special
mechanical propriety, in particular the behaviour of this cells may explain the principal features of blood
like shear-thinning. Shear-thinning means that the fluid viscosity decreases with the increase of the rate of
deformation. This effect is stronger in smaller vessels. Below a critical vessel calibre (about 1 mm), blood
viscosity becomes dependent on the vessel radius and decreases. Instead, increase of speed deformation
produces a decrease of viscosity because into the vessel the red blood cells move to the central part of
the capillary, whereas the plasma stays in contact with the vessel wall. This layer of plasma facilitates
24. 6 1. Introduction
Figure 1.4: Relationship between shear and stress in fluids.
the movement of the red cells, thus causing a decrease of the viscosity [8]. For these reasons the rheo-
logical newtonian model of the blood for middle and big arteries, in physiologic conditions, is considered
acceptable.
Consequently, we can separate models for blood flow in:
• Newtonian model, which neglects shear thinning so is suitable in larger vessels;
• Non-Newtonian model.
As discussed above, when the diameter of the vessel is less that 1 mm, the assumption that blood is a
Newtonian fluid is hard to apply.
A fluid is considered incompressible if is not reduced in volume on increase in pressure. This assump-
tion is valid because the compressibility of liquids, and then the blood, is generally very low, and negligible
for fluid dynamic problems.
Since we consider the blood as an incompressible and homogeneous fluid, we can model it using
Navier-Stokes equations that can be written in this way:
∂u
∂t
+ (u · )u +
1
ρ
P − div[µ( u + (divu)−T
)] = 0
div(u) = 0
(1.1)
where u = (ux, uy, uz) is the fluid velocity, P the pressure, µ the kinematic viscosity and ρ the blood density.
Another important thing is that is not necessary to introduce a turbulence model because blood flow,
most often, it is laminar. In normal physiological situations, the values of the Reynold’s number 1
reached
in the cardiovascular system do not allow the formation of turbulence [19], as it can be seen in Table 1.3.
Vessel Diameter [cm] Velocity [cm/s] Reynold’s Number
Aorta 2.5 48 3400
Arteries 0.4 45 500
Arterioles 0.005 5 0.7
Capillaries 0.0008 0.1 0.002
Venules 0.002 0.2 0.01
Veins 0.5 10 140
Vena Cava 3 38 3300
Table 1.3: Blood vessels types and their main characteristics [2].
1Reynold’s number is an adimensional quantity defined as Re = ρUL
µ where L is a characteristic length (in our case the diameter
of the vessel). Usually if Re < 4 ∗ 103 it can be assumed that the flow is not turbulent.
25. 1.2. Historical review of the 1D model 7
1.2 Historical review of the 1D model
The functioning of the cardiocirculatory system is a subject that has caught the attention of impor-
tant mathematicians and scientists in the past. The first one-dimensional problem of the blood flow was
proposed by Leonhard Euler in 1775 [20]. He derived the partial differential equations expressing the
conservation of mass and momentum of an inviscid fluid. However, he did not recognise the wave-like
nature of the blood flow and was not able to find a solution to his system of equations. Thomas Young first
described the wave nature of the flow and proposed an intuitive relation between pulse wave speed and the
elasticity of the arterial wall based on Newton’s theory of sound in gases [21]. Riemann, in 1860, provided
the analytical solution for the method of characteristics [22], which was crucial to solve the system of
hyperbolic partial differential equations that governs wave propagation in 1-D.
Other important contribution came Daniel Bernoulli, who established the equation of potential and
kinematic energy of a fluid, Fourier, who provided the tools to analyse periodic functions, and Hagen and
Poiseuille, with their experimental work relating mean flow and pressure gradient.
The system of equations derived by Euler is composed by non-linear partial differences. Under physio-
logical conditions, such equations are only weakly non-linear and therefore many characteristics of the flow
may be captured using a linearised system. This is the approach of Womersley [23] who linearised the two
dimensional equations for the flow in straight and obtained the wave solution by Fourier techniques. This
linear analysis has become the "standard" model of waves in arteries and is found in most hemodynamics
books. Noordegraaf [24] constructed an analogue model of the largest human systemic arteries, in which
arteries were represent by interconnected segments consisting of a resistor in series with an inductor and
a capacitor, the left ventricle was represented by a waveform synthesiser and the neglected vessels beyond
the terminal branches were modelled by resistors. This model was improved by Westerhof in 1969 [25].
Afterwards, computers were used to solve blood propagation in the cardiovascular system. In the recent
years have been developed non-linear simulations solved in time-domain [26, 27, 28]. The key aspects of
these models are the effects of non-linearities and viscous term, the treatment of bifurcation, the inflow and
outflow boundary conditions, and the validation of results.
1.3 Pressure and flow waves in the circulatory system
Pressure and flow waves are generated by the ventricular ejection and its interaction with the distensible
arterial walls. In section 1.1.3 we have seen that the blood can be considered as an incompressible fluid,
therefore the elastic arteries must distend to accommodate to sudden increase in arterial blood volume
during systole. The elasticity of the arterial walls is responsible for transforming the discontinuous left
ventricular pressure and cardiac output into a more steady smooth pressure and flow.
The cross sectional area of the vessels depends on the pressure difference over the wall. This pressure
difference is called the transmural pressure and is denoted by p. The relationship between transmural
pressure and cross-sectional area A of the vessel is described by the compliance, that is written as follow:
C =
∂A
∂p
(1.2)
while the distensibility, which describes the ability of the vessel to become stretched, as
D =
1
A
∂A
∂p
=
C
A
(1.3)
Arterial pressure evolves between a maximum value, the systolic pressure, and a minimum value, the
diastolic pressure; the pulse pressure is the difference between them. Pressure and velocity waveforms
depend on both the performance of the heart and characteristics of the arterial network. As we can seen
from fig 1.5, pulse pressure increases as we move away from the heart, while mean pressure decreases.
Furthermore, we can observed that the velocity wave is characterized by a reduction in both amplitude and
mean value due to the flow division in branching site of the capillary system.
Figure 1.6 shows qualitative values of mean pressure in the arterial circulation. During diastole the
26. 8 1. Introduction
Figure 1.5: Qualitative plots of pressure and velocity in various segments of the human aorta.
Pressure waves increase in amplitude and steepness with increasing distance from the heart,
whereas velocity waves decrease in amplitude and increase in width [2].
pressure in the left ventricle is close to zero and the aortic valve is held shut by a pressure difference
of about 90 mmHg. In the ascending aorta, mean blood pressure is about 100 mmHg, with a systolic
pressure of about 120 mmHg and a diastolic pressure of approximately 90 mmHg. In the arterioles, pressure
drops sharply and the diastolic and systolic pressure gradually converge. When blood reaches the capillary
system, its flow is approximately constant and slow, which facilities diffusive exchange of nutrients and
waste products between the blood and the surrounding tissue.
Although the aim of this thesis is to simulate the wave phenomena in the large arterial system, the effect
of the capillary system must to be considered, since these vessels are very important in the characteristics
of pulse waves. Therefore, terminal models will be account for describing the resistance introduced by
capillaries.
1.4 Organization of the thesis
This project is organized as follows:
• In this introductory chapter has been presented the human circulatory system, describing the heart
the vascular system and the blood main characteristics.
• Chapter 2 contains a detailed explanation of the mathematical formulation used to model the one-
dimensional blood flow in arteries. Starting from the governing equations and adding the algebraic
pressure-area relation in order to close such system. First we will include only the elastic term, then
is introduced a more complex mechanical vessel models that account for the viscoelastic effect.
• Chapter 3 illustrates the discretisation of the models obtained, using Taylor-Galerkin method for the
hyperbolic system of the purely elastic model. Then an Operator Splitting approach is introduced
27. 1.4. Organization of the thesis 9
to deal with the more complex model. This method allows to split the problem is into two sub-
problems, an hyperbolic subproblem and a parabolic subproblem, which can be solved separately.
Crank-Nicolson method were used to calculate the approximate solution of the parabolic subprob-
lem. Afterwards, it is explained how to impose the conditions that are required at the boundaries.
Once the mathematical model for one single vessel has been obtained, we will apply it to the case of
a vessel bifurcation.
• Chapter 4 describes the structure of the numerical tool, specifically the coupling of the one-dimensional
model to a pre-post FEM software, GiD (see Appendix A). The numerical solver part was developed
in FORTRAN90 and then imported into a GiD problem type.
• In chapter 5 shows the numerical results obtained with the numerical model implemented. The
capabilities of the formulations are tested by running simulation against well-defined works present
in literature. We consider the simulation of an human aorta, a stent implantation and a simplify
arterial network containing 37 arteries.
• Chapter 6 summarises the thesis achievement and suggests directions for future work.
• At the end, there are some appendices that give some useful information about the code developed
and the software used in this project.
Figure 1.6: Waveforms and mean value of pressure in the systemic circulation [2].
29. Chapter 2
Mathematical modelling
A preliminary basic knowledge about the cardiovascular system was given in chapter 1. In this chapter
we will see the mathematical formulations that describe the blood flow and its interaction with the wall
displacement in the human arterial system. These models can be used as an alternative to the more complex
three dimensional fluid-structure models, at the price of providing just averaged information. Their low
computational cost makes it possible not only to study wave propagation on isolated arterial segment, but
also global circulation system.
2.1 Fluid-structure interaction
A comprehensive study of the hemodynamics quantities must necessarily consider the interaction be-
tween the wall vessel and the blood flow. This aspect is particularly relevant in the larger arteries; in an
healthy adult aorta, for example, the radius can vary from 5% to 10% during the phases of systole and
diastole [8]. In arterioles and capillaries, however, the movement of the wall can be neglected.
We will illustrate a framework for fluid-structure interaction problem particularly convenient for the
analysis and for the set up of numerical solution methods.
2.1.1 ALE formulation
In the description of the continuum mechanics two approaches can be used. The Lagrangian viewpoint,
also called material description, consists of following the material particles in their motion. To this end
one introduces a computational grid which follows the continuum in its motion, the grid nodes being
permanently connected to the same material points. The difficulties caused by an excessive distortion of
the finite element grid are overcome in the Eulerian formulation, or spatial description. The basic idea in
the Eulerian formulation consists in examining as time evolves the physical quantities associated with the
fluid particles passing through a fixed region of space. In an Eulerian description the finite element mesh
is thus fixed and the continuum moves and deforms with respect to the computational grid. Therefore, the
Eulerian description of motion only involves variables.
In general, the Lagrangian point of view is used for the description of the mechanics of solids, it allows
an easy tracking of free surfaces and interfaces between different materials. Instead, the Eulerian widely
used in fluid dynamics.
Since the Eulerian formulation dissociates the mesh nodes from the material particles, convective effects
appear due to the relative motion between the deforming material and the computational grid this presents
numerical difficulties, but permits an easy treatment of complex material motion. By contrast with the
Lagrangian description, serious difficulties are now found in following deforming material interfaces and
mobile boundaries. In Figure 2.1 are represented the problems of these approaches to simulate the fluid-
structure interaction problem of the blood flow in a single vessel.
When considering the flow inside a portion of a compliant artery, we have to compute the flow solution
in a computational domain Ωt varying with time. The boundary of Ωt may in general be subdivided into
11
30. 12 2. Mathematical modelling
Figure 2.1: Difficulties of the Lagrangian (left) and Eulerian (right) formulation to follow the
fluid-structure interaction in a single vessel.
two parts. The first part coincides with the physical fluid boundary, i.e. the vessel wall Γw
t (see Figure. 2.2),
which is moving under the effect of the flow field. The other part of ∂Ωt corresponds to artificial boundaries
which delimit the region of interest from the remaining part of the cardiovascular system. The "artificial"
boundaries are the inlet and outlet (or, using the medical terminology, the proximal and distal) sections,
here indicated by Γin
t and Γout
t , respectively. The location of these boundaries is fixed a priori.
Figure 2.2: A simple model of a section of an artery. Γw
0 indicates the initial configuration used
for the derivation of the model. Γw
t indicates the current configuration at time t. Ωt is the domain
occupied by the blood. The vessel wall Γw
t is moving. The location along the z axis of Γin
t and
Γout
t are fixed [8].
A more convenient situation is the one when, even if the wall is moving, one keeps the inlet and outlet
boundaries at the same spatial location along the vessel axis. With that purpose, we introduce the Arbitrary
Lagrangian Eulerian (in short ALE) mapping. The ALE description is an hybrid approach, which was
developed in an attempt to combine the advantages of the above classical kinematical descriptions, while
minimizing as far as possible their respective drawbacks.
ALE methods was adopted in the finite element context in the work of Donea [29]. In the ALE descrip-
tion of motion, neither the material nor the spatial configuration is taken as reference. Thus, a third domain
is needed.
31. 2.2. Derivation of the 1-D model of blood flows in arteries 13
Figure 2.3: ALE formulation in a single vessel [7].
2.2 Derivation of the 1-D model of blood flows in arteries
One dimensional models provide a simplified description of the flow motion in arteries and its interac-
tion with the vessel wall displacement. Although being inappropriate to provide details on the flow field
(such as recirculation or oscillating shear stress), they can however describe the propagative phenomena
due to the wall compliance.
The geometry of a section of a vessel, as shown in Figure 2.4, may be described by a curvilinear cylin-
der. We employ a cylindrical coordinates, denoted by (r, θ, z), by coordinate system with the corresponding
base unit, where ez is aligned with the axis of the artery [9]. In the absence of branching, a short vessel
Figure 2.4: A model of a "realistic" section of an artery with the principal geometrical parameters
[9].
may be considered as a cylindrical straight tube. As it is represented in Figure 2.2,we take as reference
configuration Γw
0 the one assumed by the vessel at rest when filled with fluid with zero velocity and whose
pressure is equal to the pressure Pext exerted by the tissues external to the vessel.
Γw
0 = (r, θ, z) : r = R0(z), θ ∈ [0, 2π), z ∈ [0, L]
The reference configuration Γw
t at time t of the vessel surface and the cylindrical domain, which represents
the space fills by the blood, are define as follow
Γw
t = (r, θ, z) : r = R(r, θ, z), θ ∈ [0, 2π), z ∈ [0, L]
Ωt = (r, θ, z) : 0 ≤ r ≤ R(z, t), θ ∈ [0, 2π), z ∈ [0, L]
The domain is comprised between z = 0 and z = L, L being the vessel length, which is assumed time-
invariant.
32. 14 2. Mathematical modelling
The 1-D model is deduced by making the following simplifying assumptions:
• Small wall thickness. The vessel wall thickness h is sufficiently small to allow a shell model, and
describe the wall geometry through the average surface. In addition, we well also suppose that is
constant in the reference configuration.
• Axial symmetry. All quantities are independent from the angular coordinate θ. As a consequence,
every axial section z = cost remains circular during the wall motion. The tube radius R is a function
of z and t.
• Radial displacements. The wall displacements are only in the radial direction.
η = ηer = R − R0er (2.1)
• Constant pressure. We assume the pressure P is constant on each section, so that it depends only on
z and t.
• Body forces. We neglect body forces (the inclusion of the gravity force, if needed, is straightforward);
thus we put f=0 in the momentum equation.
• Incompressibility. The vessel maintains its volume constant during the motion
• Dominance of axial velocity. the velocity components orthogonal to the z axis are negligible com-
pared to the component along z. The latter is indicated by uz while the mean velocity as ¯u:
uz(z, t) = ¯u(t, z)s
r
R(z, t)
, ¯u(z, t) =
S(z,t)
uzdσ (2.2)
where s is a velocity profile function. We assumed this profile does not vary in time, thinking s as represen-
tative of an average flow configuration. The assumption of axial symmetry and wall displacement implies
Figure 2.5: (a) Shell model of the vessel. (b) Radial wall displacement of the vessel.
that each axial section S remains circular at all times
S = S(z, t) = {(r, θ, z) : 0 ≤ r ≤ R(z, t), 0 ≤ θ < 2π}
The resulting state variables are
A(z, t) =
S(z,t)
dσ = πR2
(z, t) (2.3)
Q(z, t) =
S(z,t)
uzdσ = A¯u (2.4)
where A is the cross-sectional area and Q is the volumetric flow rate.
33. 2.2. Derivation of the 1-D model of blood flows in arteries 15
There are three ways of deriving the one-dimensional model of blood flows in a single artery. The first
model is derived from the incompressible Navier-Stokes equations with constant viscosity and performs
an asymptotic analysis assuming that the ratio R0/L is small. The second one consists of integrating the
Navier-Stokes equations on a generic section S. The third approach, which is the one that we will show in
details in the next sections, derives model from the basic conservation laws [7].
2.2.1 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 Ω the body β 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
d
dt Ω
(•) =
Ω
(
d(•)
dt
+ (•) · v)dΩ (2.5)
which is the Reynold’s transport theorem.
2.2.2 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 2.2 as our control
volume, the principle of mass conservation requires that the rate of change of mass within the domain Ωt
plus the net mass flux out of the control volume is zero.
Denoting the vessel volume as
V(t) =
L
0
Adz, (2.6)
where L is the length of the vessel and assuming there are no infiltration through the side walls, the mass
conservation can be written as
ρ
dV(t)
dt
+ ρQ(L, t) − ρQ(0, t) = 0 (2.7)
where ρ is the blood density. If infiltration does occur we must add a source term to this equation [30]. To
determine the one-dimensional equation of mass conservation, we insert the volume into equation 2.7 and,
note that
Q(L, t) − Q(0, t) =
L
0
∂Q
∂z
dz,
we obtain
ρ
d
dt
L
0
A(z, t)dz + ρ
L
0
∂Q
∂z
dz = 0.
If we assume L is independent of time we can take the time derivative inside the integral to arrive at
ρ
L
0
∂A
∂t
+
∂Q
∂z
dz = 0
Since we have not specified the length L, the control volume is arbitrary and so the above equation must
be true for any value of L and so in general we require that the integrand is zero. We therefore obtain the
differential one-dimensional mass conservation equation:
∂A
∂t
+
∂Q
∂z
=
∂A
∂t
+
∂(uA)
∂z
= 0 (2.8)
34. 16 2. Mathematical modelling
2.2.3 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
z-direction. In this case, it states that the rate of change of momentum within the integration domain Ωt
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 L as
∂
∂t
L
0
ρQdz + (αρQu)L − (αρQu)0 = F (2.9)
where F is defined as the applied forces in the z-direction acting on the domain. The equation 2.9 includes
the momentum-flux correction coefficient α, also called Coriolis coefficient, which accounts for the fact
that the momentum flux calculated with averaged quantities (¯u) does not consider non-linearity of sectional
integration of flux momentum. So we may assume
∂
∂t S
ρ˜u2
A ≡ αρ˜u2
A = αρQ˜u ⇒ α(z, t) = S
˜u2
dσ
A˜u2
= S
˜s2
dσ
A
(2.10)
In general, the coefficient α vary in time and space, yet in our model it is taken constant as a consequence
of 2.2. It is immediate to verify that α ≥ 1.
The axial velocity profile s(y) is chosen a priori through the power-law relation
s(y) = γ−1
(γ + 2)(1 − yγ
) (2.11)
where y is the radial coordinate and γ is a proper coefficient. Commonly accepted approximation are γ = 2
(α = 4/3), which corresponds to the Poiseuille solution (parabolic velocity profile), while γ = 9 (α = 1.1)
leads to a more physiological flat profile, following the Womersley theory. The blood profile trend with
these values are shown in Figure 2.6.
We will see that the choice of α = 1, which indicates a completely flat velocity profile, would lead to
certain simplification in our analysis.
Figure 2.6: Blood flow profile adopting different values of γ.
To complete the equation 2.9 we need to define the applied forces F which typically involve a pressure
35. 2.2. Derivation of the 1-D model of blood flows in arteries 17
and a viscous force contribution,
F = (PA)0 − (PA)L +
L
0 ∂S
ˆPnzdsdz +
L
0
fdz (2.12)
where ∂S represents the boundary of the section S , nz is the 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
L
0 ∂S
ˆPnzdsdz =
L
0
P
∂A
∂z
dz (2.13)
If we finally combine equations 2.9,2.12 and 2.13 we obtain the momentum conservation for the computa-
tion domain expressed as
d
dt
L
0
PQdz + (αρQu)L − (αρQu)0 = (PA)0 − (PA)L +
L
0
P
∂A
∂z
dz +
L
0
fdz (2.14)
To obtain the one-dimensional differential equation for the momentum we note that
(αρQu)L − (αρQu)0 =
L
0
∂(αρQu)
∂z
dz
(PA)0 − (PA)L = −
L
0
∂(PA)
∂z
dz
which inserted into 2.14, taking L independent of time and ρ constant, gives
ρ
L
0
∂Q
∂t
+
∂(αQu)
∂z
dz =
L
0
−
∂(PA)
∂z
+ P
∂A
∂z
+ f dz
Once again this relationship is satisfied for an arbitrary length L and therefore can only be true when the
integrands are equal. So the one-dimensional equation for the momentum conservation becomes
∂Q
∂t
+ α
∂
∂z
Q2
A
= −
A
ρ
∂P
∂z
+
f
ρ
(2.15)
The viscous term in the equation 2.12 can be taken proportional to the averaged velocity ¯u, thus we write
f
ρ
= KR
Q
A
Therefore we finally obtain the equation of the momentum continuity
∂Q
∂t
+ α
∂
∂z
Q2
A
= −
A
ρ
∂P
∂z
+ Kr ¯u (2.16)
where KR is a strictly positive quantity that represents the viscous resistance of the flow per unit length of
tube. It depends on the kinematic viscosity ν = µ
ρ of the fluid and the velocity profile s chosen. For a power
law profile s(y), we have KR = 2πν(γ + 2). In particular, for a parabolic profile KR = 8πν, while for a flat
profile we obtain KR = 22πν.
36. 18 2. Mathematical modelling
2.2.4 Vessel wall constitutive model
Once we obtained the two governing equations 2.8 and 2.16, it is possible to write the one-dimensional
system as
∂A
∂t
+
∂Q
∂z
= 0
∂Q
∂t
+
∂
∂z
(α
Q2
A
) +
A
ρ
∂P
∂z
+ KR
Q
A
= 0
(2.17)
for all z ∈ (0, L) and t > 0, where the unknown variables are A, Q and P. The system of equations 2.17 may
be also expressed alternatively in terms of the variables (A, ¯u) instead (A, Q).
∂A
∂t
+
∂A¯u
∂z
= 0
∂¯u
∂t
+ ¯u
∂¯u
∂z
+
1
ρ
∂P
∂z
+ KR ¯u = 0
(2.18)
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. In this work we
have considered the Generalised string model [31], which is written in the following form
ρwh0
∂2
η
∂t2
− ˜γ
∂η
∂t
− ˜a
∂2
η
∂z2
− ˜c
∂3
η
∂t∂z2
+ ˜bη = (P − Pext), z ∈ (0, L), t > 0 (2.19)
We may identify the physical significance of the various terms:
• Inertia term: ρwh0
∂2
η
∂t2
, proportional to the wall acceleration
• Voigt viscoelastic term: ˜γ
∂η
∂t
, viscoelastic term, proportional to the radial displacement velocity
• Longitudinal pre-stress state of the vessel: ˜a
∂2
η
∂z2
,
• Viscoelastic term: ˜c
∂3
η
∂t∂z2
,
• Elastic term: ˜bη.
Besides ρw is the vessel density, h0 is the wall thickness, ˜a, ˜b and ˜c are three positive coefficients. We can
develop the last term of 2.19 being
η = R − R0 ⇒ η =
√
A −
√
A0
√
π
, A0 = πR2
0
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 2.19. Pressure and area will then be related by the
following algebraic law
˜b =
Eh0
kR2
0
=
πEh0
kA0
, k = 1 − ν2
where E is the Young modulus of elasticity and ν represents the Poisson ratio, typically taken to be ν = 0.5
(then k = 0.75) since biological tissue is practically incompressible. In this work, we taken k = 1.
P − Pext = ˜bη = β
√
A −
√
A0
A0
(2.20)
37. 2.2. Derivation of the 1-D model of blood flows in arteries 19
where
β = Eh0
√
π
is in general a function of z trough the Young modulus E. In a more general setting, the algebraic relation-
ship may be expressed as
P = Pext + ψ(A; A0, β) (2.21)
where we outlined that the pressure will depend not only on A, but also on A0 and on a set of coefficients
β = β1, β2, . . . , βn which accounts for the physical and mechanical characteristics of the arterial vessel.
Both A0 and β are given functions of z, but they do not vary in time. It is required that ψ be at least a C1
function of its arguments and be defined for each positive value of A and A0. In addition we must have, for
all the allowable values of A, A0 and β that
∂ψ
∂A
> 0, ψ(A0; A0, β) = 0
There are several examples of algebraic pressure-area relationship for one-dimensional models of arterial
flow [32, 33]; here we assumed the relationship 2.20, where β = β1 and, for the sake of simplicity, Pext = 0.
The function ψ can be written as
ψ(A; A0, β1) = β1
√
A −
√
A0
A0
(2.22)
It is useful introduce the Moens-Korteweg velocity
c1(A; A0, β) =
A
ρ
∂ψ
∂A
which represents the propagation speed of waves along the cylindrical vessels. In our case may be readily
computes as
c1 =
β
2ρA0
A
1
4 (2.23)
Taking into account 2.20 the system 2.17 can be written in the conservation form
∂U
∂t
+
∂F(U)
∂z
= S(U), z ∈ (0, L), t > 0 (2.24)
where
U =
A
Q
(2.25)
are the conservative variables,
F(U) =
Q
α
Q2
A
+ C1
the corresponding fluxes, and
S(U) =
0
−KR
Q
A
+
∂C1
∂A0
dA0
dz
+
∂C1
∂β
dβ
dz
a source term of the system. In our modelling, A0 and β1 are taken constant along the axial direction z
because we assume that both the initial area A0 and the Young modulus E do not vary in space; so the
expression of S accounts only for the friction term depending on KR.
C1 is a primitive of the wave speed c1, given by
C1(A; A0, β) =
A
A0
c2
1(τ; A0, β)dτ
38. 20 2. Mathematical modelling
Applying the relationship 2.23 and 2.2.4, we obtain
c1 =
β1
2ρA0
A
1
4 ⇒ C1 =
β1
3ρA0
A
3
2 (2.26)
Voigt viscoelastic term
The vessel model that we use used was according to a perfectly elastic law. Although, the real mechan-
ical behaviour of arterial wall under blood flow is much more complex. Improving the structural model
may serve several purposes, for instance to verify the relevance of different physical terms such as wall
inertia, viscoelasticity, pre-stress-state.
In this project we neglect the second viscoelastic term (˜c), the inertial term and the longitudinal pre-
stress state of the vessel (˜a) in the generalised string model . To obtain an equation of the form A = A0+πη2
,
we linearise the time derivatives as follows
∂η
∂t
≈
1
2
√
A0
√
π
∂A
∂t
(2.27)
Therefore, the adopted model may be written as
−γ
∂A
∂t
+ β
(
√
A −
√
A0)
A0
= P (2.28)
where we have taken, as before Pext = 0 and
γ =
˜γ
2
√
π
√
A0
where β is still given by 2.2.4.
The coupling between the system 2.17 and 2.28 is through the pressure term A
ρ
∂
∂z (P − Pext) in the
momentum equation, while the mass conservation equation remain unaltered. Following Formaggia [34],
we will use it to replace the time derivative of A with the space derivative of Q.
The generelized string model has two viscoelastic components. In this work we have considered only
the term in the form ˜γ∂η
∂t . The contribution of the viscoelastic effect in the momentum equation can be
written as
A
ρ
∂
∂z
− γ
∂A
∂t
= −
Aγ
ρ
∂2
Q
∂z2
(2.29)
Then the wall mechanical system of the vessel augmented by the Voigt viscoelastic term would then read
∂A
∂t
+
∂Q
∂z
= 0
∂Q
∂t
+
∂
∂z
F2(A, Q) −
Aγ
ρ
∂2
Q
∂z2
= S2(A, Q)
(2.30)
This analysis puts into evidence that the Voigt-type viscoelastic has a diffusive (second order derivatives)
term into the momentum equation. In the numerical experiments carried out we set ˜γ = 3000Kg/m3
s [27].
2.3 Characteristic analysis
One of the methods for solving non-linear hyperbolic system of partial differential equations, like the
one-dimensional elastic model 2.24 , is the characteristic analysis [9]. After some simple manipulations
the system 2.24 may be written in the quasi-linear form:
∂U
∂t
+ H(U)
∂U
∂z
= B(U), z ∈ (0, L), t > 0 (2.31)
39. 2.3. Characteristic analysis 21
where
H(U) =
0 1
A
ρ
∂ψ
∂A
− α¯u2
2α¯u
=
0 1
c2
1 − α
Q
A
2
2α
Q
A
is the Jacobian matrix. If A0 and β are constant B = −S. Considering 2.31, we can calculate the eigenvalues
for the matrix H(U)
λ1,2 = α
Q
A
± cα (2.32)
where
cα = c2
1 + α(α − 1)
Q2
A2
Since the Coriolis coefficient α ≥ 1 (we considered, for simplicity, α = 1), cα is a real number; besides,
under the assumption that A > 0, indeed a necessary condition to have physical relevant solution, c1 > 0;
therefore we have cα > 0 which means H has two real distinct eigenvalues and so, by definition, the system
2.31 is strictly hyperbolic. For typical values of velocity, vessel section and mechanical parameter β1
encountered in main arteries under physiologically conditions, we find that λ1 > 0 and λ2 < 0, i.e., the flow
is sub-critical everywhere. Furthermore, it may be shown [35] 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 (l1, l2) and (r1, r2) be two couples of left and right eigenvectors of H. The matrices R, L and Λ are
defined as
L =
lT
1
lT
2
, R = r1 r2 , Λ =
λ1 0
0 λ2
(2.33)
Since left and right eigenvalues are mutually orthogonal, we choose them so that LR = I, being I the
identity matrix. Matrix H may then be decomposed as
H = RΛL
and the system 2.31 takes the equivalent form
L
∂U
∂t
+ ΛL
∂U
∂z
+ LB(U) = 0 (2.34)
If there exist two quantities W1 and W2 which satisfy
∂W1
∂U
= l1,
∂W2
∂U
= l2 (2.35)
we will call them characteristic variables of the hyperbolic system. By setting W = [W1, W2]T
the system
2.34 may be elaborated into
∂W
∂t
+ Λ
∂W
∂z
+ G = 0 (2.36)
where
G = LB −
∂W
∂A0
dA0
dz
−
∂W
∂β
dβ
dz
Under the assumption that A0 and β1 are constant in space and taking B negligible, the equation 2.36
becomes
∂W
∂t
+ Λ
∂W
∂z
= 0
which is a system of decoupled scalar equation written as
∂Wi
∂t
+ λi
∂Wi
∂z
= 0 (2.37)
40. 22 2. Mathematical modelling
From 2.37 we have W1 and W2 are constant along two characteristics curves in the (z,t) plane 2.7
described by the differential equations
dz
dt
= λ1 ,
dz
dt
= λ2
Figure 2.7: 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.
The expression for the left eigenvectors l1 and l2 is given by
l1 = ξ
cα − α¯u
1
, l2 = ξ
−cα − α¯u
1
, (2.38)
where ξ = ξ(A, ¯u) is any arbitrary smooth function of its arguments with ξ > 0. Here we have expressed
l1 and l2 as functions of (A, ¯u) instead of (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 U [36], yet the
existence of global characteristic is not in general guaranteed. However, assuming α = 1 the relationship
2.38 take the much simpler form
∂W1
∂A
= ξc1,
∂W1
∂¯u
= ξA (2.39)
∂W2
∂A
= −ξc1,
∂W2
∂¯u
= ξA (2.40)
We now show that a set of global characteristic variables do exist for the problem at hand. Since we
note, from 2.39, that W1,2 are exact differentials being
∂2
Wi
∂A∂¯u
=
∂2
Wi
∂¯u∂A
for any values of A and ¯u, we also have that c1 does not depend on ¯u and then, from above relationship we
obtain
c1
∂ξ
∂¯u
= ξ + A
∂ξ
∂A
In order to satisfy this relation we have to choose g = g(A) such that g = −A
∂g
∂A
. To do this we can take
41. 2.3. Characteristic analysis 23
g = A−1
. As a consequence we can write
∂W1 =
c1
A
∂A + ∂¯u, ∂W2 = −
c1
A
∂A + ∂¯u (2.41)
Taking (A0, 0) as a reference state for our variables (A0, ¯u) we can integrate the above relationships obtaining
W1 = ¯u +
A
A0
c1( )
d , W2 = ¯u −
A
A0
c1( )
d (2.42)
Introducing the expression 2.2.4 for c1 we have
W1,2 =
Q
A
± 4
β1
2ρA0
A
1
4 − c0 (2.43)
with c0 is the wave speed related to the reference state. We finally can write the variables (A, Q) in terms
of the characteristic ones,
A =
2ρA0
β1
2
W1 − W2
8
4
, Q = A
W1 + W2
2
(2.44)
allowing in particular, the implementation of boundary and compatibility conditions, that we will discuss
in the next section.
42. 24 2. Mathematical modelling
2.4 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 λ1,2 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 z = 0 and the other at the outlet z = L (see Figure 2.8).
Figure 2.8: Boundary and initial conditions of the hyperbolic system.
Figure 2.9: 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 [37], which allows the
simple wave associated with the characteristics to enter or leave the domain without spurious reflections
(see Figure 2.9). Absorbing boundary conditions can be imposed by defining values for the wave entering
the domain; in our case we have λ1 > 0 and λ2 < 0 so W1 is the entering characteristic in z = 0 and W2
the inlet characteristic in Z = L. In the work of Hedstrom [38], non-reflecting boundary conditions for an
hyperbolic problem are written as
l1 ·
∂U
∂t
− B(U)
x=0
= 0, l2 ·
∂U
∂t
− B(U)
x=L
= 0
When there is an explicit formulation of the characteristic variables, it is possible impose the boundary
43. 2.4. Boundary conditions 25
conditions directly in terms of incoming characteristics, for example
W1(t) = g1(t), in z = 0, t > 0
being g1(t) a given function. However, the problem rarely have boundary data in terms of variable charac-
teristics, 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 Q (or the speed u) or the area A. Conditions of this type are typically used on the proximal node z = 0
and can be expressed as follows:
Q(0, t) = gq(t), t > 0
A(0, t) = ga(t), 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
W2, 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 A(z, t) and Q(z, t) along
the spatial domain z ∈ (0, L) at the initial time t = 0. For instance if we require the area at the initial time,
the initial condition is expressed as
A(z, t) = A0(z), z ∈ (0, L)
2.4.1 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 (z = L). We have seen a first approach which imposes not reflective
boundary conditions in the vessels terminals 2.4, 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
2.24 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 [39].
Expressing the system 2.31 in terms of (A, P, Q) with Q = A¯u and linearising around the state of
reference (A0, 0, 0), with β an A0 be constant along z, is obtained.
C1D
∂p
∂t
+
∂q
∂z
= 0
L1D
∂q
∂t
+
∂p
∂z
= −R1Dq
p =
a
C1D
(2.45)
where a,p and q are the perturbation variables for area, pressure and volume flux, respectively (A0 +a, p, q)
and
R1D =
ρc0
A0
, C1D =
A0
ρc2
0
, L1D =
ρ
A0
(2.46)
are the viscous resistance to flow, wall compliance and blood inertia, respectively, per unit of length of
vessel l. Integrating system 2.45 over the length l yields the lumped parameter model, where the variables
are R0D = R1Dl, C0D = C1Dl, L0D = L1Dl and ˆp = 1
l
l
0
pdz, ˆq = 1
l
l
0
qdz are the mean pressure and flow
over the whole domain. In physiological conditions pulsatile waves travel at a speed greater compared to
44. 26 2. Mathematical modelling
Hydraulic Physiological variables Electric
Pressure P[Pa] Blood pressure [mmHg] Voltage V
Flow rate Q[m3
/s] Blood flow rate [L/s] Current I
Volume V[m3
] Blood volume [L] Charge q[C]
Viscosity η Blood viscosity µ[Pa · s] Electrical resistance R
Elastic coefficient Vessel’s wall compliance Capacitance C
Inertance Blood inertia Inductance L
Table 2.1: Analogy between hydraulic and electrical network [3].
that of the blood, then ˆp = pin and ˆq = qout. Therefore, the final 0-D model is the following:
C0D
∂pin
∂t
+ qout − qin = 0
L0D
∂qout
∂t
+ R0Dqout + pout − pin = 0
(2.47)
where qin = q(0, t), qout = q(L, t), pin = p(0, t) and pout = p(L, t) are the flows and pressures at the inlet and
outlet of the 0-D domain. As it is represented in Figure 2.10 ,the system 2.47 is analogous to an electric
circuit, in which the role of the flow and pressure are played by the current and potential, R0D corresponds
to an electric resistance, C0D to a capacitance and L0D to an inductance.
Figure 2.10: 1-D arterial vessel domain (left) and the equivalent 0-D system discretises at first
order in space (right).
45. Chapter 3
Numerical approximation of the
problem
The mathematical models formulated in the context of scientific computing are mostly systems de-
scribed by partial differential equations (PDEs). They are almost never solved analytically, except in cases
purely academic. The theorems and principles derived from analytical mathematics allow us to study some
characteristics of a general problem, like if the problem is well-define, but rarely allow to arrive at an
explicit representation of the solution. Therefore, it is necessary to develop methods of approximation of
the analytical solution that can be implemented by specific algorithms and then carried through computers
[40, 31].
In Chapter 2 we have deduced different analytical models for blood flow. For the simplest case that
uses an elastic wall model, a nonlinear hyperbolic system of equations is obtained that may be solved using
the second-order Taylor-Galerkin method. When a viscoelastic wall model is used instead, the hyperbolic
nature of the equations is lost due to the additional term in the pressure-area relationship.
In this case, the approaches for the numerical solution of the equations may be divided into two main
categories: first, methods that do not exploit the original hyperbolic nature of the equations (e.g., a discon-
tinuous finite element Galerkin method with stabilization terms; an implicit finite difference/ spectral ele-
ment method wherein non-linear terms are solved iteratively at each time-step using the Newton-Raphson
method; etc.). The assumption that the additional terms are less importance than the elastic term allows the
use a second category of methods called Operator Splitting. These methods recover the original hyperbolic
nature of the equations by employing an operator-splitting scheme for the momentum equation. The second
approach is computationally faster than the first one since exploits the hyperbolic nature of the governing
equations of the model. However, it introduces an approximation through the operator-splitting approach.
3.1 Operator splitting
The idea of the Operator Splitting technique is to decompose the system of PDEs into simpler problems,
this allows to treat each problem with specific algorithms. Particularly, explicit or implicit methods can be
used. Different time and space discretization can be choose for the various subproblems, this option is
particularly interesting if operators have very different time step restrictions for explicit schemes.
Let us consider the time intervals (tn
, tn+1
), for n = 0, 1, . . . , T with tn
= n∆t. In every time interval,
the hyperbolic problem is solved to get a predictor ˆU, which is used as the initial condition of the second
problem. The second step can be viewed as a corrector. The original problem is approximated by a
sequential application of the two subproblems in a certain order.
First step: elastic response ˆU
By removing all the component from the formulation 2.19 we can write the closed system of equations
2.34 as
∂ ˆU
∂t
+
∂F(U)
∂z
+ S(U) = 0 (3.1)
27
46. 28 3. Numerical approximation of the problem
where U = [A, Q]T
and ˆU = [A, ˆQ]T
are the total and elastic conservative variables, respectively, F =
[Q, F2]T
are the corresponding fluxes and S = [0, S2]T
accounts for the friction and the non-uniformity of
the geometry and the material, with
F2 = α
Q2
A
+ C1, S2 = KR
Q
A
Second step: viscoelastic correction ˜U
Starting from the prediction ˆU, we correct it with the evolution over ∆t of the parabolic subproblem.
Un e∆tH
−−−→ ˆU
e∆tP
−−−→ Un+1
where Un+1
= ˆU + ˜U, e∆tH
means to solve the hyperbolic subproblem over ∆t while e∆tP
to solve the
parabolic subproblem.
Because the system is dominated by the hyperbolicity, it must be driven mainly by the boundary con-
ditions through the first subproblem.
3.1.1 First step: Elastic model
The nonlinear hyperbolic system 2.24 has been discretized using a Taylor-Galerkin scheme [41], 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 2.24 and having H =
∂F
∂U
we may write
∂U
∂t
= S −
∂F
∂z
∂2
U
∂t2
=
∂S
∂U
∂U
∂t
−
∂
∂z
H
∂U
∂t
=
∂S
∂U
S −
∂F
∂z
−
∂HB
∂z
+
∂
∂z
H
∂F
∂z
(3.2)
For simplicity the dependence of S and F from U is dropped. Starting from the above equations, we now
consider the time intervals (tn
, tn+1
), for n = 0, 1, . . . , T; then we discretize the equation in time using a
Taylor series which includes first and second order derivatives if U. Therefore we obtain the following
semi-discrete schemes for the approximation Un+1
of U(tn+1
):
• Taylor-Galerkin scheme:
un+1
= un
+ ∆tun
t +
∆t2
2
un
tt (3.3)
Un+1
=Un
− ∆t
∂
∂z
Fn
+
∆t
2
Hn
Sn
+
∆t2
2
Sn
U
∂Fn
∂z
−
∂
∂z
Hn ∂Fn
∂z
+ ∆t Sn
+
∆t
2
Sn
USn
, n = 0, 1, . . .
(3.4)
where Sn
U =
∂Sn
∂U
and Fn
, stands for F(Un
), just as Hn
, Sn
and Sn
U; the value U0
is given by the initial
conditions.
For each time interval (tn
, tn+1
) we apply a spatial discretization carried out using the Galerkin finite
element method. To this purpose we subdivide the domain Ω = {z : z ∈ (0, L)}, which is the 1-D counterpart
of the 3-D domain Ωt, into a finite number Nel of linear elements length l (Figure 3.1).
47. 3.1. Operator splitting 29
Figure 3.1: One-dimensional mesh representing a vessel.
Moreover we introduce a trial function space, T , and a weighting function space, W. These spaces are
both defined to consist of all suitably smooth functions and to be such that
T = U(z, t)|U(z, 0) = U0
(z) on Ωt at t = t0
, W = W(z)
Considering the scheme, we multiply the equation 3.4 for the weight function W and we integrate it
over the domain Ωt obtaining, for ∀t > 0t0
Ω
Un+1
− Un
dΩ = − ∆t
Ω
∂W
∂z
Fn
LWdΩ −
Ω
Sn
LWWdΩ −
−
∆t2
2 Ω
∂W
∂z
SU
n
Fn
−
Ω
∂W
∂z
∂Fn
∂z
Hn
−
− ∆t Ni
¯Fn
r |z=L − Ni
¯Fn
l |z=0
(3.5)
where we have assumed
Fn
LW(Uj) = Fn
+
∆t
2
Hn
Sn
and
Sn
LW(Uj) = Sn
+
∆t
2
SU
n
Sn
Starting from the weak form of the problem 3.5 we build the subspaces T h
and Wh
for the trial and
weighting function spaces T and W defining them as
T h
= ˆU(z, t)| ˆU(z, t) =
N
j=1
Uj(t)Nj(z); U(t0
) = ¯U(zj) = U0
j
Wh
= W(z, t)|W(z) =
N
j=1
Wj(t)Nj(z)
(3.6)
where Nj is the standard linear finite element shape function (Figure 3.2) associated to the j − th node,
located at z = zj, and Uj and ˆU at the node j. Since we are using the Galerkin method, the base shape
functions defined above are used as weighting. Adopting the following notation
(W, U)Ωe
=
Ωe
W · UdΩ,
and considering the sum of each element contribution
Ω
· · · =
el Ωe
. . . ,
48. 30 3. Numerical approximation of the problem
Figure 3.2: Sketch of a 1D linear shape function.
the equation 3.5 becomes
el
(Ni, Nj)Ωe
(Un+1
j − Un
j) = ∆t
el
[(Ni,z, Nj)Ωe
Fn
LW(Uj) + Ni, Nj)Ωe
Sn
LW(Uj)]−
−
∆t2
2 el
(Ni, Nj)Ωe
SU
n
(Uj)
∂Fn
j
∂z
−
−
∆t2
2 el
(Ni,z, Nj)Ωe
Hn
j(Uj)
∂Fn
j
∂z
−
− ∆t Ni
¯Fn
r |z=L − Ni
¯Fn
l |z=0
(3.7)
For what concerns the border nodes, we have to consider the boundaries condition. Starting from the
equation 3.7, we have the term of boundary conditions represented by
∆t Ni
¯Fn
r |z=L − Ni
¯Fn
l |z=0 , i = 1, 2
which implies the knowledge of the flux terms depending from the values of A and Q at inlet and outlet
sections of the domain. To extract them from the characteristic information W1(0, t) and W2(L, t) we need an
additional expression for the other characteristic variables W2(0, t) and W1(L, t) to recover U(A, Q) using
the equation 2.44. To this purpose we adopted a technique based on the extrapolation of the outgoing
characteristics. Having the friction parameter Kr small with respect to the other equation terms in 2.24, we
assume that at the boundary points z = 0 and z = L the flow is generated by the characteristic system 2.37.
At a generic time step n we have Un
known and we linearise the eigenvalues λ1,2 of 2.24 by taking their
values at respective boundary for t = tn
. The solution corresponding to this linearised problem at time tn+1
gives
Wn+1
2 (0) = Wn
2 (−λn
2(0)∆t)
Wn+1
1 (L) = Wn
1 (−λn
1(L)∆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 Wn+1
1 (0) and Wn+1
2 (L) , we are able to compute, through
2.44, 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 pro-
gramming, 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 [42] indicates that
49. 3.1. Operator splitting 31
the following Courant-Friedrichs-Lewy condition should be satisfied
∆t ≤
√
3
3
min
0≤i≤N
hi
max(λ1,i, λ1,i+1)
(3.8)
where λ1,i here indicates the value of λ1 at mesh node zi. 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
√
3
3 .
3.1.2 Second step: Viscoelastic term
As we discussed previously, the parabolic part of model which account for the viscoelastic behaviour
of the arterial wall will be solved by the second step of an Operator Splitting strategy. After introducing
the diffusive term Aγ
ρ
Q2
∂z2 in the momentum equation, we obtain the following system:
∂A
∂t
+
∂Q
∂t
= 0
∂ ˆQ
∂t
+
∂
∂z
F2(A, Q) = S2(A, Q)
∂ ˜Q
∂t
−
Aγ
ρ
∂2
Q
∂z2
= 0
(3.9)
We just need to correct the flux by adding the last equation.
∂ ˜Q
∂t
=
Aγ
ρ
∂Q
∂z2
=
Aγ
ρ
∂2 ˆQ
∂z2
+
Aγ
ρ
∂ ˜Q
∂z2
Then a reasonable solution is to use a splitting procedure that use an explicit method for the convection
and an implicit method for the diffusion. The first subproblem is still solve with characteristic method,
while for the parabolic subproblem Crank-Nicolson is very suitable. It belongs to the θ − family time
discretization with θ = 1
2 .
• θ - family:
un+1
− un
∆t
= θun+1
t + 1 − θ un
t (3.10)
For our parabolic problem the resulting time and spatial scheme is the following
˜Qn+1
i − ˆQi
∆t
=
Aγ
2ρ
˜Qn+1
i+1 − 2 ˜Qn+1
i + ˜Qn+1
i−1
∆x2
+
ˆQi+1 − 2 ˆQi + ˆQi−1
∆x2
(3.11)
˜Qn+1
i
∆t
−
Aγ
2ρ
˜Qn+1
i+1 − 2 ˜Qn+1
i + ˜Qn+1
i−1
∆x2
=
Aγ
2ρ
ˆQi+1 − 2 ˆQi + ˆQi−1
∆x2
+
ˆQi
∆t
50. 32 3. Numerical approximation of the problem
Where ˆQ is the solution of the first hyperbolic subproblem.
1
∆t
+
Aγ
∆x2ρ
−
Aγ
2∆x2ρ
0
−
Aγ
2∆x2ρ
1
∆t
+
Aγ
∆x2ρ
−
Aγ
2∆x2ρ
0 −
Aγ
2∆x2ρ
1
∆t
+
Aγ
∆x2ρ
−
Aγ
2∆x2ρ
...
...
...
˜Q1
˜Q2
...
˜Qn
=
=
Aγ
2∆x2ρ
−2 ˆQ1 + ˆQ2
ˆQ1 + −2 ˆQ2 + ˆQ3
...
ˆQn−1 − 2 ˆQn
+
1
∆t
ˆQ1
ˆQ2
...
ˆQn
(3.12)
The matrix of the resulting algebraic system is tridiagonal, which is quite cheap to invert. This scheme is
second order accurate both on time and space. Moreover, it is unconditionally stable.
We note that a second order implicit Finite Element method can also be applied here. But since this
sub-problem is linear and in 1-D, it would be exactly equivalent with this Finite Differences method.
3.2 Coupling 1-D and 0-D models
The existence and uniqueness of the solution of a coupled problem between the 0-D model system 2.47
and the hyperbolic 1-D system 2.36, has been proven by Formaggia [16] 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 3.3). An intermediate state (A∗
, U∗
) originates at
time t + ∆t from the states (AL, UL) and (AR, UR) at time t. The state (AL, UL) corresponds to the end point
of the 1-D domain, and (AR, UR) is a virtual state selected so that (A∗
, U∗
) satisfies the relation between A∗
and U∗
dictated by system 2.47. The 1-D and 0-D variables at the interface are related through qin = A∗
U∗
and pin = β
A0
(
√
A∗ −
√
A0), and pout is prescribed as a constant parameter that represents the pressure at
which flow to the venous system ceases.
Figure 3.3: Coupling 1-D/0-D model.
According to the method of characteristics, if G = 0 equation 2.36 leads to
W1(A∗
, U∗
) = W1(AL, UL) (3.13)
W2(A∗
, U∗
) = W2(AR, UR) (3.14)
51. 3.2. Coupling 1-D and 0-D models 33
Solving 3.13 for A∗
and U∗
yields
A∗
=
2ρA0
β
W1(AL, UL) − W2(AR, UR)
8
+ A
1
4
0
4
(3.15)
U∗
=
W1(AL, UL) + W2(AR, UR)
2
(3.16)
The 1-D outflow boundary condition is imposed by enforcing that either UR = UL, which reduces Eq. 3.15
to
AR = 2(A∗
)
1
4 − (AL)
1
4
4
(3.17)
or AR = AL, which reduces Eq. 3.16 to
UR = 2U∗
− UL (3.18)
3.2.1 Terminal resistance (R) model
This model simulates the peripheral circulation as a purely resistive load Rp, in which the state (A∗
, U∗
)
satisfies
A∗
U∗
=
P(A∗
) − pout
Rp
(3.19)
Combining with 2.47 we leads to a non-linear equation
F (A∗
) = Rp UL + 4c(AL) A∗
− 4c(A∗
)A∗
−
β
A0
√
A∗ − A0 + pout = 0 (3.20)
which is solved using Newton-Raphson method, with the initial value of A∗
= AL. Once A∗
has been
obtained, U∗
is calculated from Eq. 3.16. If we consider both C and L equal to zero, we lead to the single
terminal resistance model.
Figure 3.4: Lumped parameter models: Terminal resistance model (left) and Windkessel RCR
model (right).
3.2.2 Three-element (RCR) Windkessel model
This model accounts for the resistance and the compliance of the peripheral vessels using the RCR
Windkessel model (see Figure 3.4 right), which accounts for the cumulative effects of all distal vessels
(small arteries, arterioles and capillaries). The three-element Windkessel model consists of two resistances
R1 and R2 and a capacitor C. According to Section 3.2.1, we consider R1 to let any incoming wave reach
the CR2 system without being reflected. Waves are reflected by the CR2 system, which is governed by
C
dPc
dt
= A∗
U∗
−
pc − pout
R2
(3.21)
52. 34 3. Numerical approximation of the problem
The first resistance, R1, is introduced in order to absorb the incoming waves and reduces artificial wave
reflections. It satisfies
A∗
U∗
=
P(A∗
) − (pc)n
R1
(3.22)
where (pC)n
is the pressure at C at the time step n. It is determining by solving at every time step n the
first-order time discretisation of the Eq. 3.21
pn
C = pn−1
C −
∆t
C
A∗
U∗
(3.23)
The coupling is solved as for the R terminal resistance model, but with pout = pc and R1 = Rp
F (A∗
) = R1 UL + 4c(AL) A∗
− 4c(A∗
)A∗
−
β
A0
√
A∗ − A0 + pc = 0 (3.24)
Again, once A∗
is obtained by Netwon-Rapshon, we can proceed to calculated U∗
, int his case from 3.22.
Figure 3.5: Peripheral circulation considering the 1-D formulation coupled to the Windkessel
model.
3.3 Branching
The vascular system is characterized by the presence of branching. The flow in a bifurcation is intrinsi-
cally 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 [43]. 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 prop-
erties.
3.3.1 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. showed in Figure 3.6, we divide the
domain Ω into three partitions Ω1,Ω2 and Ω3; doing this we have 3 sub-problems which must be coupled
imposing adequate boundary conditions. Then we have to evaluate 6 variables, (Ai, Qi) with i = 1, 2, 3,
corresponding to the problem unknowns, area and flow rate for each one of the vessels composing the
branching. 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
53. 3.3. Branching 35
Figure 3.6: Domain decomposition of a bifurcation 1-2.
a splitting bifurcation shown in Figure 3.7, 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
Q1 = Q2 + Q3
remembering that the flow moves from the sub-domain Ω1 to the sub-domain Ω2 and Ω3. 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
P1 +
1
2
ρ
Q1
A1
2
= P2 +
1
2
ρ
Q2
A2
2
P1 +
1
2
ρ
Q1
A1
2
= P3 +
1
2
ρ
Q3
A3
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 W1
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
54. 36 3. Numerical approximation of the problem
represent by W2
2 and W3
2 .
W1
1 =
Q1
A1
+ 4
β1
2ρA01
A1/4
1 = u1 + 4(c1 − c1
0)
W1
2 =
Q2
A2
− 4
β2
2ρA02
A1/4
2 = u2 + 4(c2 − c2
0)
W2
2 =
Q3
A3
− 4
β3
2ρA03
A1/4
3 = u3 + 4(c3 − c3
0)
where c1
0,c2
0and c3
0 are the values of the wave speed evaluated using the area A0 in the vessels 1,2 and 3. In
summary, the resulting system which determines the values (A1, Q1),(A2, Q2) and (A3, Q3) at the bifurcation
is the following
W1
1 =
Q1
A1
+ 4
β1
2ρA01
A1/4
1
W1
2 =
Q2
A2
− 4
β2
2ρA02
A1/4
2
W2
2 =
Q3
A3
− 4
β3
2ρA03
A1/4
3
Q1 = Q2 + Q3
P1 +
1
2
ρ
Q1
A1
2
= P2 +
1
2
ρ
Q2
A2
2
P1 +
1
2
ρ
Q1
A1
2
= P3 +
1
2
ρ
Q3
A3
2
(3.25)
We can solve it through the Newton-Raphson technique for differential systems of non-linear equations.
For the case of merging, which is typical in veins or may represent a surgical operation like by-pass,
we proceed in the same way. 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.
Figure 3.7: Notation for a splitting (left) and merging (right) flow bifurcation.
3.3.2 Bifurcation 1-1
When a stent or a prosthesis is implanted to heal vascular pathologies, it causes a sharp variation
of the elastic properties along the artery. This discontinuity at the interfaces between the healthy and the
prosthetic artery can be solve 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
55. 3.3. Branching 37
Ω1 and Ω2, and solving the following non-linear system for the interface variables, namely
W1 =
Q1
A1
+ 4
β1
2ρA01
A1/4
1
W2 =
Q2
A2
− 4
β2
2ρA02
A1/4
2
Q1 = Q2
P1 +
1
2
ρ
Q1
A1
2
= P2 +
1
2
ρ
Q2
A2
2
(3.26)
Again, We solve the non-linear system obtained through the Newton-Raphson method. In both systems
3.25 and 3.26, 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 [27].
Figure 3.8: Domain decomposition of a bifurcation 1-1.
57. Chapter 4
Numerical solver
In Chapter 2 we have introduced the one-dimensional model of blood flow in arteries, while in Chapter
3 we have defined the numerical methods that provide an approximated solution of the one-dimensional
blood flow in arteries problem that we are dealing.
In this chapter we show the creation process of a numerical solver, or problem type, able to solve our
hemodynamic problem and display the obtained results.
The problem type is composed of two parts: a computational part represented by a finite element
code developed in FORTRAN90, and a graphical user interface which is provide by GiD. The graphical
interface allows to define, prepare and visualize all the data related to a numerical solution; these data
include the definition of the geometry, materials, conditions, solutions information and other parameters.
The software can also generate a mesh for finite element, finite volume or finite difference analysis and
write the information for a numerical simulation program in its desired format.
Figure 4.1: Flowchart describing the execution of a GiD problem type.
Figure 4.1 shows a general flowchart for the preparation and the execution of a standard problem type.
The pre- and post-process phases, respectively for what concerns the creation of the INPUT FILE and
the visualization of the OUTPUT FILE, are both supported by GiD. Instead the box SOLVER, which
represents the computation section where the numerical solution is calculated, is an external component of
GiD. Therefore it is possible, on one hand, to customize the pre-post process interface, modifying the files
39
