Politenio di Milano
Imperial College London
Scuola di Ingegneria Industriale e dell’Informazione
Corso di Laurea Specialistica in Ingegneria Matematica
Numerical simulation of blood
flow in venous networks
Relatore:
Prof. Ing. Luca Formaggia
Correlatore:
Dr. Joaquim Peir´o
Marco Cermatori
matr. 739265
Anno Accademico 2012-2013
To Linda and Raffaella
Introduzione
Le vene e ele arterie sono canali essenziali per il trasporto di calore e sostanze
chimiche come ossigeno e nutrienti. Gran parte della ricerca emodinamica si `e
concentrata sulle arterie, poich´e sembra che ci sono pi`
u malattie arteriose letali
rispetto a quelle venose. Nonostante ci`o le vene contengono circa il 70% del
sangue totale nel corpo [15]. Malattie potenzialmente letali come la trombosi
venosa acuta e l’insufficienza venosa cronica sono rispettivamente collegate a
regioni di basso flusso o di alta pressione nelle vene [2]. Per queste ragioni, questo
lavoro presena un modello monodimensionale per il flusso sanguigno nella rete
venosa.
Osservazioni sperimentali del network venoso mostrano che le vene sono pi`
u
distendibili delle arterie. le vene tendono quindi a collassare ad esembio quando
sono contratte dalla pressione muscolare. Inoltre le vene sane delle gambe hanno
valvole che permettono il ritorno venoso del sangue al cuore. Per queste ragioni
`e importante sviluppare metodi numerici efficienti per capire i fenomeni fisiologici. In questo lavoro consideriamo un flusso non stazionario, incomprimibile
e monodimensionale in accordo con il modello in [39].
Il legame area-pressione regola l’interazione fluido-struttura. Questo legame
insieme alle equazioni di conservazione di massa e quantit`
a di moto formano un
sistema non lineare iperbolico.
Instabilit`
a numeriche possono verificarsi. Quando la vena collassa, l’area `e
vicina allo zero. Inoltre possono esserci salti nella velocit`
a prodotti da onde di
shock (see [8], [30]). Quindi occorrono metodi numerici stabili e accurati.
In questo lavoro usiamo il metodo Discontinous-Galerkin con una discretizzazione in tempo di Adam-Bashforth al secondo ordine [39].
Nel capitolo 1 `e presentata un’introduzione istologica, anatomica e fisiologica del sistema venoso. Questi concetti introducono al modello del capitolo 2,
dove figurano le ipotesi del modello, le equazioni che lo governano, e la sua discretizzazione via Discontinuous-Galerkin. Infine i risultati numerici che validano
il modello sono presenti nel capitolo 3.
2
Introduction
Veins and arteries are essential channels for the transport of heat and chemicals, such as oxigen, nutrients, waste products and hormones, within the human
body. Most of the research on haemodynamics has been focused on arteries,
since it seems there are more life-threatening arterial diseases than venous diseases. Nevertheless veins normally contain about 70% of the total blood in the
systemic vascular system [15]. Potentially lethal pathologies such as lower-limb
deep venous trombosis (DVT) and chronic venous insufficiency (CVI) are respectively related to regions of low flow and high blood pressure in veins [2].
There has been very little research on venous haemodynamics. For these reasons, this work presents a one-dimensional model for the blood flow in venous
networks.
Experimental observations of the venous network show anatomical and physiological differences with the arterial one: the veins are more compliant so the
rigid or quasi-rigid hypothesis used in arterial research does not apply. Therefore, modelling flows in collapsible tubes is an important issue in biomechanics.
Veins above the level and outside the skull of a man in an upright position collapse due to hydrostatic reduction of blood transmural pressure (internal minus
external pressure). Vessel collapse is also observed in veins being squeezed by
contracting skeletal muscle. An example of such a muscle is the calf muscle that
pumps the blood from the feet to the heart. The pulmonary system can also exhibit collapse during coughing or sneezing and during forced or rapid expiration
[30], [8].
Muscular contractions play an important role in venous return because they
compress the deep veins of the calf. Moreover, healthy veins of the leg have valves
to keep the blood circulation oriented towards the heart, preventing backflows
[14]. When compression occurs the distal valve of the deep vein as well as the
valves of the perforating veins closes. Blood is then ejected towards the heart.
When the compression relaxes a reduction of pressure takes place in the deep
veins and blood is aspirated into these veins through the perforating veins [14].
For these reasons, it is important to develop general and efficient numerical haemodynamic tools to help clinicians and researchers in understanding of
physiological flow phenomena. Several models have been used in the past to
describe the blood flow. In this work, the flow through the veins is assumed to
be unsteady, incompressible and one-dimensional, in accordance with the model
proposed in [39].
The fluid–structure interaction between the flow and the wall is modelled by
a relation between tube area and the difference between internal and external
pressures. This relation, called the tube law, can be given a priori in analytical
form or fitted from experimental data. Provided with a correct tube law, the
3
4
one-dimensional numerical system can represent flow dynamics in a vessel. The
governing equations form a system of nonlinear hyperbolic partial differential
equations.
The numerical venous network needs to include a model of muscular activity
and a model of valve activation as well. On the contrary, these features are not
needed in a model of the arterial network. Furthermore numerical instabilites
may occur when modelling in venous network. For example when the tube collapse the cross-sectional area may be very close to zero. Moreover, the analysis
of the blood flow in the jugular vein of a giraffe (see [8], [30]) have shown that
the flow may abruptly decelerate from supercritical to subcritical through an
elastic jump. Besides, valves impose important jump interface conditions. Therefore stable, accurate numerical method is needed to solve the equations governing one-dimensional unsteady flow in a collapsible tube. In this work we use
the Discontinous-Galerkin method along with a second order Adams-Bashforth
scheme for time discretization, similarly to the arterial model proposed in [39].
Numerical models for the venous networks are very well suited to the investigations. Since these flows are difficult to analyze via in vivo techniques
numerical predictions play a very important role. Indeed several clinical problems such as venous hypertension can be investigated. For instance, relevant
results may improve the prevention and treatment of CVI and the design of
compressive stockings [2].
This work is organized as follows. Chapter 1 presents an introductory overview
of the venous system. The most relevant histologic, anatomic and patho-physiological
features are illustrated. These concepts introduce to the mathematical model
explained in chapter 2. The main assumptions, the governing equations and the
discretization with the Discontinuous Galerkin method are desscribed. Finally
chapter 3 we validate our model and our code with significant test cases.
Contents
1 Overview of the venous network
1.1 The cardiovascular system . . . . .
1.2 Histology of the venous system . .
1.3 Anatomy of the venous system . .
1.3.1 The veins of the lower limb
1.4 Physiology of the venous system .
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2 Mathematical model
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2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Numerical discretization . . . . . . . . . . . . . . . . . . . . . . . 30
3 Numerical Results
3.1 Verification of the code .
3.2 Riemann solutions . . .
3.3 Calf muscle pump . . .
3.4 Inclined limb . . . . . .
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4 Conclusions and future work
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5
Chapter 1
Overview of the venous
network
In section 1.1 we present an overview of the cardiovascular system. Then
section 1.2 explains the main histological features of blood vessels and veins in
particular in order to illustrate the crucial differences with arteries. In section
1.3 a brief anatomy of veins can be found. More specifically our primary focus
will be on the veins of the lower limb, due to their key role in the venous return.
Finally, section 1.4 illustrates the fundamentals of venous physiology. All of
these aspects will be considered as the building blocks for the development of
the mathematical model of the venous network.
1.1
The cardiovascular system
The chief function of the cardiovascular system (CVS) is the rapid convective
transport of chemicals such as oxygen, glucose, amino acids, fatty acids, vitamins
and water to the tissues; and the rapid washout of metabolic waste products such
as carbone dioxide. The CVS is also crucial for body temperature regulation. It
transports heat from deep organs to the skin surface for dissipation.
Figure 1.1 shows a stylized arrangement of the circulation. The heart consists
of two intermittent muscular pumps, the right and left ventricles. Each pump
is filled with a contractile reservoir, the right or left atrium. The right heart
perfuses the pulmonary circulation. Venous blood enters the right atrium
from the superior and the inferior venae cavae, then flows through the tricuspid
valve into the right ventricule. The ventricle is composed mainly of cardiac
muscle and it receives the blood while the muscle is relaxed. Cardiac relaxation is
called diastole. Contraction then follows, called systole. Systole expels some of
the blood at a low pressure into the pulmonary artery which divides and supplies
the lungs. Inhaled oxygen diffuses into the blood. The oxygenated blood returns
through the pulmonary veins and passes through the left atrium into the left
ventricle.
The left heart is responsible for the systemic circulation. The left ventricle
contracts simultaneously with the right ventricle and ejects the same volume
of blood, but at a much higher pressure. The blood flows through the aorta.
Repetead arterial branching leads, ultimately, to the formation of microscopic
6
1.1.
The cardiovascular system
7
Figure 1.1: Stylized arrangement of the circulation. The red and black lines
represent the arterial and venous blood paths respectively.
thin-walled tubes called capillaries (figure 1.2). Dissolved gases and metabolites
diffuse between the capillary blood and the cells of the body. The deoxygenated
blood returns through a convergent system of veins that drain into the superior
and inferior venae cavae.
The cardiac output is an essential value in blood circulation. It is the
product of stroke volume (the volume of blood ejected per contraction) and
heart rate (the number of contractions per minute). Thus it represents the volume of blood ejected by one ventricule in one minute. The distribution of the
cardiac output can be actively adjusted to match changing regional demands.
For example, in exercise the proportion of the output going to skeletal muscle
can increase to ∼80%. The redistribution of the output is brought about by a
widening of tiny arterial vessels called arterioles inside the active muscle. This
vasodilatation allows blood to flow more easily into the active muscle.
The chief factor that drives blood along a vessel is the gradient of blood
pressure. Thus the vascular pressure profile is fundamental to understanding
the circulation (figure 1.3). Ventricular ejection raises the aortic blood pressure
to ∼100 mmHg above atmospheric pressure, whereas the pressure of great veins
is closer to atmospheric pressure. The pressure difference drives the blood from
artery to vein. Units of “mmHg above atmosferic pressure” are used because
human blood pressure is usually measured with a mercury column, taking atmospheric pressure as the reference or zero level.
Arterial pressure is pulsatile, because the heart ejects blood intermittently.
Between successive ejections the systemic arterial pressure decays from a peax
of ∼120 mmHg to a trough of ∼80 mmHg. At the same time pulmonary pressure
decays from 25 mmHg to 10 mmHg. This is conventionally written as 120/80
1.1.
The cardiovascular system
8
Figure 1.2: Thickness of the wall relative to the diameter of the lumen in different
blood vessels. The ratio varies, however, with blood pressure and vascular tone.
Figure 1.3: Profile of pressure and blood velocity in systemic circulation of a
resting human. The same cardiac output passes each dashed line per minute.
The mean blood velocity is the flow divided by the cross-sectional area of the
vascular bed (from [28]).
1.1.
The cardiovascular system
9
Figure 1.4: Distribution of blood volume in a resting man.
mmHg and 25/10 mmHg respectively.
The largest arteries such as the aorta and the iliac arteries (diameter 1-3 cm
in humans) have very distensible walls because they are rich in elastin. Elastin
is a protein that enables the elastic arteries to expand by ∼ 10% during each
heart beat and thereby accomodate the ejected blood. For this reason the largest
arteries are also referred to as elastic arteries. The recoil of these elastic vessels
during diastole converts the stop-go flow in the ascending aorta into a continuous
flow through the more distal arteries. These medium to small arteries such as
the radial, cerebral and coronary arteries (diameter 0.1-1.0 cm in humans) are
made of a smooth muscle coat that is thicker, relative to the lumen, than in
elastic arteries. Their main function is to deliver blood to the organs. For this
reasons they are referred to as muscular arteries or conduit arteries.
Mean blood pressure falls very little along the elastic and conduit arteries
because the large lumen offers little resistance to flow. The drop in mean pressure from the ascending aorta to radial artery is only 2 mmHg or so. The major
drop in pressure occurs along the smallest, terminal arteries (diameter 100-500
µm) and arterioles (diameter 10-100 µm). The terminal arteries and arterioles
are therefore called the resistance vessels. Since resistance vessels dominate
the resistance to flow, they serve as taps of the circulation; they can turn local
blood low up or down to match local demand. When the resistance vessels dilate (vasodilatation), resistance falls and local blood flow increases. Conversely,
vasoconstriction raises local resistance and reduces local blood flow. The terminal arterioles also adjust the number of capillaries (diameter 4-7 µm) perfused
with blood. The thin wall of a capillary (0.5µm) facilitates the rapid passage
of gases and metabolites. Along with the postcapillary venules (diameter 15-50
µm) they act as the exchange vessels of the CVS.
Venules (diameter 50-200 µm) differ chiefly in size and number. Venules and
small veins are more numerous than the corresponding arterioles and arteries
and their resistance is low. A pressure drop of 10-15 mmHg is sufficient to drive
the cardiac output from the venules to the right atrium. Because of their large
number and size, veins contain about two third of the circulating blood at any
one instant (figure 1.4). They are therefore called capacitance vessels. Due to
their low wall-lumen ratio (figure 1.2) they are easily distended or collapsed, so
1.2.
Histology of the venous system
10
they act as a variable reservoir of blood. Moreover, many peripheral veins are
innervated by vasoconstrictor nerve fibres, so the volume of blood in the venous
reservoir can be actively controlled. For instance, at times of physiological stress
the capacitance vessels constrict and displace blood into the heart and arteries
(refer to [28]).
1.2
Histology of the venous system
We shall now sharpen our focus on veins. We first describe their hystological
composition and structural differences with arteries.
The walls of arteries and veins are composed of three layers called tunics.
The three layers of the vascular wall, from the lumen outward, are the tunica
intima, tunica media and tunica adventitia.
The intima is the innermost coat. It consists of three elements: a single
layer of squamous epithelial cells, the endothelium; the basal lamina of the
endothelial cells; the subendothelial layer consisting of loose connective tissue.
The endothelial layer is the main barrier to the escape of plasma.
The media is the middle layer. It consists primarily of circumferential layers
of smooth muscle cells arranged in a matrix of elastin and collagene fibres. It
supplies mechanical strength and contractile power.
The adventitia is the outermost coat. It mainly consists of longitudinally
arranged collagene tissue and a few elastic fibres. These connective tissue elements gradually merge with the loose connective tissue surrounding the vessel.
The tunica adventitia ranges from relatively thin in most of the arterial system
to quite thick in the venules and veins, where it is the major component of
the vessel wall. The tunica adventitia of large arteries and veins contains also a
system of vessels called vasa vasorum that supply blood to the vascular walls
themselves, as well as a network of autonomic nerves, called nervi vascularis,
that control contraction of the smooth muscle in the vessel walls.
Histologically, the various types of arteries and veins are distinguished from
each other on the basis of the wall thickness and differences in composition
of the layers. Veins can be classified on the basis of size as small (venules),
medium and large veins. Although medium and large veins have three layers,
these layers are not as distinct as in arteries (figure 1.5).
Small veins are further subclassified as postcapillary and muscular venules.
Postcapillary venules receive blood from capillaries and have a diameter as small
as 0.2 mm or slightly larger. They possess an endothelial lining. Muscular venules
are located distal to postcapillary venules in the returning venous network and
have a diameter up to 1 mm. Whereas postcapillary venules have no true tunica
media, the muscular venules have one or two layers of smooth muscle that
constitute a tunica media. These vessels also have a thin tunica adventitia.
Medium veins have a diameter of up to 10mm. Most named veins are in
this category. Valves are a characteristic feature of these vessels and are most
numerous in the inferior portion of the body, particularly the legs, to prevent
retrograde movement of blood because of gravity. The three tunics of the venous
wall are most evident in medium-sized veins. The tunica intima consists of an
endothelium with its basal lamina, a thin subendothelial layer with occasional
smooth muscle cells scattered in the connective tissue elements, and, in some
cases, a thin internal elastic membrane. The tunica media of medium-sized veins
1.2.
Histology of the venous system
11
Figure 1.5: Comparison between artery (on left) and vein (on right). The tunica
media in veins is much thinner than the tunica media in arteries. Veins low
wall-thickness makes them much more compliant than arteries. Therefore the
three layers are less distinct in veins than in arteries (from [31]).
is much thinner than the same layer in medium-sized arteries. It contains several
layers of circularly arranged smooth muscle cells with interspersed collagen and
elastic fibres. The tunica adventitia is typically thicker than the tunica media
and consists of collagen fibres and networks of elastic fibres.
Veins with a diameter greater than 10 mm are classified as large veins.
Often, the boundary between the tunica intima and tunica media is not clear,
and it is not always easy to decide if the smooth muscle cells close to the intimal
endothelium belong to the tunica intima or to the tunica media. The tunica
media is relatively thin and contains circumferentially arranged smooth muscle
cells, collagen fibers, and some fibroblasts. The tunica adventitia of large veins
(e.g., the subclavian veins and the venae cavae) is the thickest layer of the vessel
wall. Along with the usual collagen and elastic fibres and fibroblasts, the tunica
adventitia also contains longitudinally disposed smooth muscle cells (for more
details see [38]).
Histology of the venous valves
One-way valves are present in most veins. They are folds of the tunica intima,
which protrude into the venous lumen like sails (leaflet valve, velum). The
leaflet valve can be in shape of a half-moon or a sickle (figure 1.7). Their shape
often reminds a martin’s nest. Most valves have two flexible leaflets, which freely
move with the bloodstream. Valve leaflets are covered on both faces by a layer of
endothelium with an underlying network of intersecting collagene fibres. This
fibre network is anchored in the venous wall. The space between the venous
wall and the outer edge of the velum (leaflet) is called valve sinus. During
normal blood flow through the vein, the leaflets lay against the venous wall and
1.3.
Anatomy of the venous system
12
Figure 1.6: Medium (on left) and large (on right) vein. The tunica media and
the tunica intima in large veins are not as distinct as in medium veins (from
[38]).
the valve is open. The leaflet valve prevents the back flow of blood. The large
central veins and the veins of the head and neck are not endowed with functional
valves (see [25]).
1.3
Anatomy of the venous system
In this section we shall continue our overview of the venous system with the
basic anatomy and classification of human veins (unless otherwise stated, refer
to [21] for a more detailed illustration).
The veins commence by minute plexuses which receive the blood from the
capillaries. The branches arising from these plexuses unite together into trunks,
and these, in their passage toward the heart, constantly increase in size as they
receive tributaries, or join other veins. The veins are larger and altogether more
numerous than the arteries; hence, the entire capacity of the venous system is
much greater than that of the arterial; the capacity of the pulmonary veins,
however, only slightly exceeds that of the pulmonary arteries.
The veins are cylindrical like the arteries; their walls, however, are thin and
they collapse when the vessels are empty, and the uniformity of their surfaces is
interrupted at intervals by slight constrictions, which indicate the existence of
valves in their interior.
They communicate very freely with one another, especially in certain regions
of the body; and these communications exist between the larger trunks as well
as between the smaller branches. Thus, large and frequent anastomoses, i.e.
reconnections of two streams that previously branched out, are found between
the venous sinuses of the cranium, and between the veins of the neck, where
obstruction would be attended with imminent danger to the cerebral venous
system. The same free communication exists between the veins throughout the
whole extent of the vertebral canal, and between the veins composing the various
1.3.
Anatomy of the venous system
13
Figure 1.7: Valve of a vein from the orbital sinus. 1: leaflet valve; 2: valve sinus;
3: endothelium of the orbital sinus (see [25]).
venous plexuses in the abdomen and pelvis, e.g., the spermatic, uterine, vesical,
and pudendal.
Veins consist of two distinct sets of vessels, the pulmonary and systemic.
The pulmonary veins, unlike other veins, contain arterialized blood, which
they return from the lungs to the left atrium of the heart (figure 1.8). They are
four in number, two from each lung, and are destitute of valves. They commence
in a capillary network upon the walls of the air sacs, where they are continuous
with the capillary ramifications of the pulmonary artery, and, joining together,
form one vessel for each lobule. These vessels uniting successively, form a single
trunk for each lobe, three for the right, and two for the left lung. The vein from
the middle lobe of the right lung generally unites with that from the upper lobe,
so that ultimately two trunks from each lung are formed.
The systemic veins return the venous blood from the body generally to
the right atrium of the heart. They may be arranged into three groups: the veins
of the heart; the veins of the upper extremities, head, neck, and thorax, which
end in the superior vena cava; the veins of the lower extremities, abdomen, and
pelvis, which end in the inferior vena cava.
The systemic venous channels are subdivided into three sets: superficial and
deep veins, and venous sinuses. The superficial veins (cutaneous veins) are
found between the layers of the superficial fascia immediately beneath the skin;
they return the blood from these structures, and communicate with the deep
veins by perforating the deep fascia. The deep veins accompany the arteries.
With the smaller arteries —as the radial, brachial, tibial, peroneal— they exist
generally in pairs, one lying on each side of the vessel, and are called venae
comitantes. The larger arteries —such as the axillary, subclavian, popliteal,
and femoral— have usually only one accompanying vein. In certain organs of
the body, however, the deep veins do not accompany the arteries; for instance,
1.3.
Anatomy of the venous system
14
Figure 1.8: Pulmonary vessels, seen in a dorsal view of the heart and lungs (from
[21]).
the veins in the skull and vertebral canal, the hepatic veins in the liver, and
the larger veins returning blood from the bones. Venous sinuses are channels
which drain the blood from the brain; they are devoid of valves, and are situated
in the dura mater, the outermost layer of the meninges that surround the brain
and spinal cord.
1.3.1
The veins of the lower limb
The veins of the lower extremity are subdivided, like those of the upper, into
two sets, superficial and deep; the superficial veins are placed beneath the
integument between the two layers of superficial fascia; the deep veins accompany the arteries. Both sets of veins are provided with valves, which are more
numerous in the deep than in the superficial set. Valves are also more numerous
in the veins of the lower than in those of the upper limb. Superficial and deep
veins are joined by the perforating veins.
The superficial veins
On the dorsum of the foot the dorsal digital veins receive tributaries from
the plantar cutaneous venous arch and join to form short common digital veins
which unite in the dorsal venous arch. Proximal to this arch is an irregular
venous network which receives tributaries from the deep veins and is joined at
the sides of the foot by the medial and the lateral marginal vein (figure
1.9). The upward continuations of these two veins are the main superficial veins
of the lower extremity, i.e. the great and small saphenous veins and their
tributaries [13].
The great saphenous vein (left of figure 1.10), the longest vein in the
body, begins in the medial marginal vein of the dorsum of the foot. It ascends
1.3.
Anatomy of the venous system
15
Figure 1.9: Anatomy of the superficial and perforating veins of the foot.
in front of the tibial malleolus, then runs upward along the medial side of the
thigh and ends in the femoral vein about 3 cm below the inguinal ligament.
The small saphenous vein (right of figure 1.10) begins behind the lateral
malleolus as a continuation of the lateral marginal vein; it first ascends along
the lateral margin of the tendocalcaneus, and then crosses it to reach the middle
of the back of the leg. Running directly upward, it ends in the popliteal vein.
The small saphenous vein possesses from nine to twelve valves, one of which is
always found near its termination in the popliteal vein.
The deep veins
The deep veins of the lower extremity accompany the arteries and their
branches; they possess numerous valves.
The posterior tibial veins (right of figure 1.11) accompany the posterior
tibial artery, and are joined by the peroneal veins. The anterior tibial veins
(figure 1.11, on left ) are the upward continuation of the dorsal venous arch.
They leave the front of the leg by passing between the tibia and fibula and
unite with the posterior tibial, to form the popliteal vein.
The popliteal vein is formed by the junction of the anterior and posterior
tibial veins. It ascends to reach the adductor magnus muscle, where it becomes
the femoral vein. It receives tributaries corresponding to the branches of the
popliteal artery, and it also receives the small saphenous vein (figure 1.11, right).
The valves in the popliteal vein are usually four in number.
The femoral vein accompanies the femoral artery through the upper twothirds of the thigh. It receives numerous muscular tributaries, and about 4 cm
below the inguinal ligament is joined by the deep femoral vein; near its termination it is joined by the great saphenous vein (refer again to figure 1.11). The
valves in the femoral vein are three in number. The deep femoral vein (profunda femoris) receives tributaries corresponding to the perforating branches of
1.3.
Anatomy of the venous system
16
Figure 1.10: Superficial veins of the lower limb: the great (on left) and small (on
right) saphenous veins and their tributaries (from [21]).
Figure 1.11: Frontal (on left) and dorsal (on right) view of the superficial and
deep veins of the lower limb.
1.4.
Physiology of the venous system
17
Figure 1.12: Frontal (on left) and dorsal (on right) view of the perforating veins
of the lower limb.
the deep femoral artery, and through these establishes communications with the
popliteal vein below.
The external iliac vein, the upward continuation of the femoral vein, begins behind the inguinal ligament, and, passing upward along the pelvis, unites
with the hypogastric vein to form the common iliac vein which ultimately ends
in the inferior vena cava to return blood to the rest of the body (figure 1.11,
left). The external iliac vein frequently contains one, sometimes two, valves.
The perforating veins
Perforating veins (figure 1.12) connect the superficial to the deep leg veins.
They are short channels with valves arranged so as to allow blood to flow into
the deep veins but not in a reverse direction. We introduce the most important
groups of perforating veins.
Dodd’s perforating veins or perforating veins of the adductor canal connect the femoral vein to the long saphenous network in the lower third of the
thigh. Boyd’s perforating veins form a communication between the saphenous
network and the popliteal trunks. Medial gastrocnemius perforating veins
communicate between the intramuscular veins of the gastrocnemius muscle and
the saphenous network of the leg. Cockett’s perforating veins connect the posterior tibial veins with the posterior saphenous branches [11].
1.4
Physiology of the venous system
In the previous sections we introduced the essential histology and anatomy
that we need to explore the venous system. We can now illustrate the physiological factors interacting in the venous nerwork.
1.4.
Physiology of the venous system
18
First we introduce the importance of venous flow and pressure. Secondly
we will consider them as terms in the energy equation (Bernoulli’s theorem).
This will allow us to analyze the extremely important effect of gravity in the
venous system. Finally we will present two accessory elements that remarkably
contribute to the venous flow: the skeletal muscle pump (the calf muscle pump
in particular), which cooperates with the valves of the lower limb to boost venous return; and the respiratory pump, particularly important during forced
expiration manoeuvres. We will also mention a pathological condition of retrograde flow due to valvular incompetence. This prevents the muscular pump
from working properly and is responsible for the very common varicose veins.
Unless otherwise noted, the facts cited herein are taken from [28]. This part
will be the background for a more detailed physical-mathematical analysis of
the venous network in chapter 2.
Venous pressure and flow
As we mentioned in sections 1.1 and 1.2, peripheral veins and venules are
thin-walled, voluminous vessels containing roughly two thirds of the circulating
blood. They serve as an adjustable reservoir of blood that can be used to top
up the central veins, raising the central venous pressure (CVP), which is
the pressure at the entrance to the right atrium. This determines the right
ventricular end-diastolic pressure (RVEDP) and resting distension. Changes in
CVP thus represent changes in right ventricular muscle preload and hence in
stroke volume. The volume of blood in peripheral veins depends both on the
venous blood pressure and on the tone of smooth muscle in the tunica media.
Blood enters venules at ∼12-20 mmHg at heart level. The pressure falls to
∼8-10 mmHg in named veins such as the antecubital or femoral vein at heart
level. Venous resistance is small, so 8-10 mmHg is enough to drive the cardiac
output to the right ventricle, where the diastolic pressure is 0-6 mmHg.
As we will see later in this section, venous pressure increases below heart
level and decreases above heart level due to the effect of gravity. This has marked
effects on local venous volume, because the cross-sectional profile of these thinwalled vessels changes with pressure. The change in shape enables the vein
to accomodate a large volume of blood for only a small change in pressure.
The maximum distensibility (compliance) occurs at ∼ 4 mmHg and is ∼ 100
ml/mmHg for the human systemic venous system, or ∼ 50 times greater than
arterial compliance (venous collapse will be discussed further in the following
chapters).
As previously stated, venous smooth muscle tone regulates venous volume as
well. The veins of the gastrointestinal tract, liver, kidneys and skin are subject
to vasoconstricion boosted by a network of nerves. This vasoconstriction greatly
reduces the capacitance of these veins at a given pressure, displacing blood into
the thoracic compartment. In this way the central venous system is able to exert
some control over the filling pressure of the heart (the CVP).
Flow in the great veins is pulsatile due to the cycle of right atrial pressure
changes (figure 1.13. The flow in the superior vena cava shows two spurts per
cycle. Peak flow occurs during the x descent of the atrial pressure wave and is
caused by atrial relaxation. Peak inflow in the superior vena cava is aided by
the right ventricular systole, because the ballistic effect of firing out a mass of
blood propels the ventricle downwards, like the recoil of a gun (Newton’s law of
1.4.
Physiology of the venous system
19
Figure 1.13: Pressure and flow in human superior vena cava over two cycles
(from [42]).
action and reaction). This stretches the atria and helps to suck blood into them.
A second flow-spurt occurs during the y descent of the atrial pressure wave, due
to the opening of the tricuspide vale in diastole. The second spurt is boosted by
the elastic recoil of the ventricular walls, especially when end-systolic volume is
low during exercise. Thus flow through the great veins is driven primarily by
the upstream pressure of ∼8 mmHg (pressure from behind or vis a tergo) but is
boosted by two transient reductions in downstream pressure due to the motion
of the heart (suction from in front or vis a fronte).
Energy considerations
The pressure difference drives the blood from arteries to veins. Blood hydrostatic pressure is usually measured with a mercury column, taking atmospheric
pressure as the zero level. The total pressure energy in a point of a blood vessel depends on three factor: the hydrostatic pressure, the internal (intramural )
pressure distending the vessel and the external (extramural ) pressure compressing it. The second and third factor can be summed up with the transmural
pressure, the difference between the internal and external pressure. The venous
transmural pressure will play a crucial role in our analysis (see chapter 2).
Besides pressure energy (pressure P × volume V ), also gravitational potential energy (height above the heart h × gravitational force g × mass (density
2
ρ × V )) and kinetic energy ( ρV2v , where v is the blood velocity) affect blood
flow, as recognized by Bernoulli. The sum of these three terms gives the overall
mechanical energy E of unit volume blood V :
E
ρv 2
= P + ρgh +
.
(1.1)
V
2
The Bernoulli theory explains some puzzles. Mean arterial pressure is typically 95 mmHg above atmospheric pressure in the aorta and 183 mmHg above
atmospheric pressure in the artery of the dorsum of the foot during standing
(figure 1.14). Nevertheless the difference of gravitational potential energy (∼90
1.4.
Physiology of the venous system
20
Figure 1.14: Effect of gravity on arterial and venous pressure in a standing
human.
mmHg) provides a net energy difference of 2 mmHg, so blood flows from the
aorta to the foot. Futhermore it is clear that hydrostatic pressure energy and
gravitational potential energy are equal but of opposite signs (∓ρgh). While the
former increases moving downward the latter grows in the opposite direction.
Thus these two terms eliminate each other in the global energy equation (1.1),
which is ultimately determined only by transmural pressure and kinetic energy.
Effects of gravity on the venous system
There are marked changes in the human circulation with posture. These are
initiated by the effect of gravity on venous blood distribution (and depend on
the steep, sigmoidal pressure-volume relation of veins).
The adoption of a standing position, orthostasis, increases the pressure in all
the blood pressure below heart level, and reduces the pressure in all the vessels
above heart level, owing to the drag of gravity on the vertical column of fluid
between the heart and the vessel (figure 1.14). This is particularly important in
veins because their volume is highly sensitive to transmural pressure.
Upon tilting a human subject upright there is a transitory closure of the
venous valves in the limbs that prevents any substantial venous back-flow. Pressure in the dependent veins then rises steadily over ∼30 − 60 seconds as blood
flows in from the arterial system. As venous pressure rises, flow recommences up
the limb veins and pushes open the venous valves. This re-establishes an uninterrupted column of blood. The weight of the continuous fluid column between
heart and feet raises venous pressure in the feet nine-fold, from ∼ 10 mmHg
supine to ∼90 mmHg in orthostasis (figure 1.15).
1.4.
Physiology of the venous system
21
Figure 1.15: Displacement of venous blood volume (red) on moving from supine
position (a) to standing (b). Numbers are typical pressures in cmH2 O. The
hydrostatic indifferent point (HIP) is the point where pressure is unaltered by
tilting; (c) shows how immersion in water increases central volume and CVP
(see [18]).
There is no counterbalance rise in extramural pressure during orthostasis
(unless the subjected is immersed in water), so the dependent veins are greatly
distended. This is plainly visible in the back of the hand when the hand is lowered below heart level. In a human adult about 500 ml of blood accumulate in
the distended veins below heart level over ∼45 seconds during orthostasis. This
is widely referred to as venous ‘pooling’, though the imagery here is misleading: a pool is static whereas the venous blood flows continuously in the steady
state. Most of the redistributed blood comes ultimately from the intrathoracic
compartment, via arterial flow (not venous backflow). The loss of blood from
the thoracic veins reduces the CVP and impairs the stroke volume through the
Frank-Starling mechanism, i.e. the smaller the stretch of the ventricle in diastole,
the smaller the stroke work achieved in systole. This causes a transitory arterial hypotension and sometimes dizziness, called postural hypotension. Nearly
all healthy individuals occasionally experience orthostatic dizziness, especially
when warm and venodilated. On the other hand, gravity reduces the blood pressure in vessels above heart level. When the transmural pressure falls to zero or
less, the unsupported superficial veins collapse, as can be easily observed again
in the back of one’s hand. Deeper veins are better supported and do not collapse
completely. Conversely veins within the cranial cavity are a special case. They
do not collapse because gravity also reduces the pressure of the cerebrospinal
fluid around them, so the transmural pressure hardly changes. The relationship
between transmural pressure and cross-sectional area will be further discussed
in chapter 2
Finally it is important to notice that the concept that leg venous blood flow
must decrease during standing ‘because the blood is going uphill against gravity’
is wrong, because it ignores the fact that gravity exerts an equal and opposite
‘downhill’ pull to the arterial column. Gravity acts equally on the venous and
arterial fluid columns, so the pressure differrence between the arteries and veins
at any level is not affected by orthostasis. Since pressure difference drives flow,
blood flow in the steady state is not directly affected by standing.
Venous flow is also assisted by two accessory pumps, the skeletal muscle
1.4.
Physiology of the venous system
22
Figure 1.16: The physiological events during the operation of the calf muscle
pump (results from [10]).
pump during exercise and the respiratory pump.
The skeletal muscle pump
The principal participants in the muscular pumping mechanism are the veins
of the calf within the calf muscles, gastrocnemius and soleus [10]. These contain
as much as 250 ml of blood [29]. Additional muscle pumps are also recognized
in the thigh and foot [16], [17]. The latter has a much smaller capacity (about
25 ml), but assists in the return of blood from the foot. It may have a ‘pump
priming’ effect on the calf muscle pump. The calf pump is often referred to
as the peripheral heart. During calf contraction, as for example during rhytmic exercise, the pressure within the deep fascia becomes raised to as much as
250 mmHg resulting in all the intra-muscular veins becoming completely compressed. Therefore blood that was present in these veins is emptied into the
outflow tract, the first part of which is the popliteal vein, a large bore vein
which offers virtually no resistance to outflow. (figure 1.16). The large veins
within the gastrocnemius and soleus muscles form the main chamber of the
pump but the deep veins of the posterior tibial and peroneal veins and all other
venae comitantes participate as well [10], [17].
With continuous exercise, the calf blood volume is reduced by 1.5–2.0 ml/100
ml mainly as a result of the compression of the veins in the pump chamber and
the average expelled volume is approximately 100–120 ml, that is, about 50%
of all the blood within the pump. The pump will normally expel this volume in
four or five contractions though one single sustained contraction can expel as
much [10]. The muscle pump thereby reduces the venous pressure in the lower
leg from 70-90 mmHg in immobile orthostasis to 20-40 mmHg during walking,
running and cycling. This increases the arterio-venous pressure gradient driving
blood through the calf muscle by 50-60% [28]. As the gradient of 10–15 mmHg
between the small veins and the heart is sufficient to ensure venous blood flow
1.4.
Physiology of the venous system
23
Figure 1.17: The skeletal muscle pump. Pressure of the dorsal vein of the foot
during orthostasis, interrupted by a short period of rythmic contraction of the
calf muscles (black bar). Red line shows effect of failure venous valves as in
varicose veins. (After [28]).
in the supine position, the increase in pressure gradient produced by the calf
muscle pump during contraction is more than enough to ensure an adequate
rapid venous return to the heart during vigorous erect muscle exercise [10].
Furthermore, by redistributing venous blood from the periphery into the central
veins, the muscle pump prevents CVP from falling during exercise [28].
The mechanism described above is made possible by the crucial function of
the venous valves of the lower limb (figure 1.17). Distal deep venous valves
close during contraction in orfer to prevent axial reflux and functional or physiological valves in the communicating veins prevent reflux from the deep to the
superficial venous system [4].
Venous valves also ensure that the emptied segment refill from the periphery
during each relaxation phase [28]. Venous pressure falls in the foot and ankle
because, as the muscle relaxes, the distal valves open and blood drains rapidly
from the distal veins into the empty muscle veins. At the same time, closure of
the proximal valves interrupts the vertical column of blood between limb and
heart.
The pressure within the muscle fascia falls to low levels. At the moment
when the calf muscles relax, their contained veins are empty, at low pressure
and as yet unfilled by arterial inflow. As the veins are collapsed, they are unaffected by hydrostatic pressure. On the other hand, the superficial veins are
full and subjected to hydrostatic pressure plus the remnant of cardiac generated pressure, the “vis a tergo” [10]. The pressure gradient between the two
compartments becomes 100–110 mmHg resulting in blood immediately flowing
1.4.
Physiology of the venous system
24
from the superficial to the deep compartment through the many communicating
veins [6]. This empties the superficial compartment and reduces its pressure [35].
The valves in the communicating veins open and ensure this normal blood flow
from the superficial to the deep venous system [10]. The muscle pump reduces
capillary filtraction pressure in the feet and ankles, because capillary pressure
is close to venous pressure. This greatly reduces the tendency of the feet and
ankles to swell with oedema fluid in the upright position [28]. Finally arterial
inflow from the lower extremity capillary beds re-primes the calf muscle pump
for a subsequent contraction [10].
We can conclude that the combined action of the calf muscle pump and of
physiological venous valves of the lower limb is vital in ensuring venous return
from the lower limbs during exercise and the reduction of the superficial venous pressure. In this way the damaging effects of the hydrostatic pressure ever
present as a result of man’s upright posture are removed. [10].
Valvular reflux and varicose veins
The valvular apparatus also regulates bakflow or reflux, i.e. the flow of blood
in the opposite direction from physiological flow. The duration of the reflux
flow depends on the physiological or pathological state of the valves [10]. The
minimum duration of 0.5 seconds does seem to be universally recognised as
indicative of pathological reflux (see for instance [9]). Kistner proposed four
classes of deep vein reflux, in ascending order of severity: reflux descending to
mid-thigh, to the knee, to the mid-calf and to the ankle respectively [24].
The accurate quantification of venous reflux in individual veins has not yet
been fully established. Several methods of quantifying venous reflux in individual veins have been described. These include measurements of valve closure
time, velocity at peak reflux, and quantitative volume flow measurements by
calculating the cross-sectional area of the vessel and the time-averaged velocity
[10].
A pathological condition related to the retrograde venous flow is the wellknown varicose vein. Varicose veins are defined as palpable, dilated subcutaneous veins usually larger than 4mm [33]. The mechanism(s) responsible for
their development are incompletely understood. However, a number of theories
of etiology have been suggested [20].
Superficial venous reflux is detectable in individuals without prominent superficial lower limb veins (14% in one study) [26]. The incidence of reflux is
higher in patients with obviously dilated veins (77%) but not invariably present
in those with varicose veins (87%) [26]. The site of reflux can be anywhere along
the great or small saphenous veins and frequently is located at a valve distal to
the sapheno–femoral or sapheno–popliteal junctions. For example, Labropoulos
and associates studied the site of valvular incompetence in 139 limbs with primary varicose veins and found that in 24% reflux was detected in the main trunk
of the long saphenous or a tributary without junctional incompetence [27].
The main discussion regarding the etiology of varicose veins is whether the
venous dilatation is secondary to a primary problem in the vein wall or a primary
problem in one or more venous valves [20].
The original theory regarding varicose veins etiology suggested that valvular
failure at the sapheno-femoral junction was the primary event with secondary
failure of more distal valves as a result of the increased pressure placed on these
1.4.
Physiology of the venous system
25
Figure 1.18: Effect of breathing on venous return. F, flow in thoracic inferior
vena cava; IAP intra-abdominal pressure; ITP, intra-thoracic pressure (see [7]).
sites (see the red plot in figure 1.17) [32]. If the venous valves of the deep veins of
the calf become incompetent, the muscle pump becomes ineffective. The vertical blood column can no longer be effectively broken up when muscle relaxation
occurs, so the distal veins are subjected to a chronically raised pressure load in
orthostasis. Over time this leads to a permanent dilatation of the veins [28]. The
finding of isolated valvular incompetence at distal sites or tributaries demonstrates that descending incompetence cannot account for all cases of varicose
veins [27].
Another theory claims that a weakening of the vein wall develops initially
as a result of connective tissue problems within the vessel or perhaps loss of
venous tone. The other changes typical of varicose veins, including valvular
incompetence, are secondary changes due to widening of the valve annulus at
the site of vein dilatation (see for example [5]).
Whatever the initiating factor for the venous dilatation, all theories incorporate a common final path of prolonged venous filling, further valvular incompetence and transmitted pressures from the deep to superficial venous systems
[20].
The respiratory pump
Beside muscular pump, the respiratory pump is a protagonist in boosting
the venous flow. Flow in the vena cava increases during inspiration (figure 1.18).
This is because the fall in intra-thoracic pressure expands the intrathoracic veins,
and at the same time the descent of the diaphragm compresses the abdominal
contents, raises abdominal venous pressure and enhances flow from the abdomen
into the thorax increa.sing venous return. Conversely, flow in the vena cava slows
during expiration. The Valsalva manoeuvre is a forced expiration against a
closed or narrowed glotts. Forced expiration is a natural event performed daily
by all of us as a normal accompaniment to activities such as coughing or lifting
heavy weights. This manoeuvre creates high intrathoracic pressure, which evokes
a circulatory response with four phases (figure 1.19).
• Arterial pressure immediately rises (phase I), because the aorta is compressed by the high intrathoracic pressure. Coughing can elevate intrathoracic pressure transiently to 400 mmHg.
• In phase II the mean pressure and especially pulse pressure falls, because
1.4.
Physiology of the venous system
26
Figure 1.19: Blood pressure responses to the Valsalva maneuver in a control
patient (top) and in patients with orthostatic hypotension (bottom). Results
from [19].
the raised intrathoracic pressure compresses the thoracic vena cava. This
impedes venous return and therefore stroke volume. A reflex increase in
sympathetic outflow causes tachycardia and peripheral vasoconstriction,
which arrest the fall in blood pressure in normal subjects (figure 1.19,
top). In patients with orthostatic hypothension, pressure fails to stabilize
during this phase (figure 1.19, bottom).
• Phase III marks the termination of hte Valsalva manoeuvre. Arterial pressure drops abrubtly as intrathoracic pressure falls to normal, decompressing the aorta.
• In phase IV the pulse pressure and mean pressure increase rapidly, because
the normalized intrathoracic pressure allows venous blood to surge into the
thorax, distending the heart and increasing the stroke volume.
Chapter 2
Mathematical model
This chapter details the 1-D mathematical formulation of blood pressure
and velocity. In section 2.1 we present the hypothesis of the model and its
governing equations. In section 2.2 we illustrate the numerical discretization
with the Discontinous Galerkin method.
2.1
Governing equations
Each vein is approximated by a 1-D impermeable tube of length l, centreline
s(x), and cross sectional area normal to s denoted by A(s, t), thickness h(s)
as indicated in figure 2.1 (on left). Our first modelling simplification will be to
assume that the local curvature is everywhere small enough so that the axial
direction can be described by a Cartesian coordinate x as shown in figre 2.1
(on right) so that the problem
can be defined in one-dimension. At each cross
R
dσ
as the area of the cross section S and u(x, t) =
section
we
define
A(x,
t)
=
S
R
R
1
1
u
ˆ
dσ,
p(x,
t)
=
p
ˆ
dσ
as
the
average velocity and internal pressure over
A S
A S
the cross section where u
ˆ(x, σ, t) and pˆ(x, σ, t) denote the values of velocity and
pressure within a constant x-section. We also introduce the dependent variable
Q(x, t) = Au which represents the volume flux at a given section.
We therefore have three independent variables (A, u, p) or equivalently (A,
Q, p). The required three independent equations will be provided by the equations of conservation of mass and momentum and a constitutive equation relating cross sectional area to internal pressure. In what follows, we shall also
assume that the fluid is incompressible and Newtonian and so the density ρ and
dynamic viscosity µ are constant. Our final modelling assumption is that the
structural venous properties are constant at a section [39, 1].
The derivation of the 1-D governing equations in collapsible tubes such as
veins is already well established (see [22]). We therefore assume an unsteady,
incompressible one-dimensional flow along the axis x. The pressure p = p(x, t)
and the velocity u = u(x, t) of the fluid are averaged values of the local variables
over each cross-section (see also [14, 30, 8]).
The governing equations for the fluid flow express the conservation of mass
∂A ∂(uA)
+
= 0.
∂t
∂x
27
(2.1)
2.1.
Governing equations
28
Figure 2.1: Layout of a 1-D compliant vessel. General orientation (left) and 1-D
orientation (right).
Conservation of momentum gives
∂u
∂u 1 ∂p
+ αu
+
+ fv + g sin θ = 0
∂t
∂x ρ ∂x
(2.2)
where ρ is the density of blood, g is the gravitational acceleration and θ is the
angle between the tube axis and the horizontal. The momentum flux correction
factor α accounts for the nonlinearity of the sectional integration [39].
The term fv represents the viscous resistance to the flow (and, in principle,
includes a contribution from the convective inertia terms to account for the fact
that the velocity profile is not flat). It may be obtained by solving the NavierStokes equations for parallel flows in tubes of given cross-section. For laminar
flows with parabolic profile we have (see [1] for more details):
fv =
8πµ
u
ρA
(2.3)
where µ is the viscosity of the blood and A0 = A0 (x) denotes the crosssectional area at the equilibrium state (p, u) = (pe , 0), where pe is the pressure
at the outer surface of the vessel.
Pressure-area relationship
To close the system of equations 2.1 and 2.2 a constitutive law involving the
pressure and the cross-sectional area is required. As we mentioned above, the
tube-law needs to take into account the veins collapse.
We consider a law for the transmural pressure p − pe of the form (see [22,
14, 30, 8, 12, 40])
p − pe = KF (α),
(2.4)
where K is a constant proportional to the bending stiffness, α =
F (α) = αn − αm ,
m ≥ 0, n ≤ 0.
A
A0
and
(2.5)
2.1.
Governing equations
29
With regards to the bending stiffness of the vessel, for a thin wall made
from a homogeneous linearly elastic material we have (see [15] for a detailed
illustration)
3
h
E
,
(2.6)
K=
12(1 − ν 2 ) R0
where E and ν are the Young’s modulus and Poisson’s ratio of the material
and R0 is the radius ratio when the tube is circular and not distended (i.e.
A = A0 ). Undistorted area, wall stiffness and external pressure may all vary with
distance down the vein. Thus A0 , K , and pe are all functions of longitudinal
distance x.
dA
Two closely related concepts are the compliance C = dP
and the distensibil1 dA
ity D = A dP . When dilated (A > A0 ) the tube has circular cross-section and
constant compliance. The tube starts to collapse at A = A0 . As A decreases
the tube becomes increasingly elliptical and compliance steadily declines. As the
collapse intensifies, opposite sides of the wall come into contact and two parallel
conduits are formed. Compliance falls further and the cross-section is reduced
in size. For a thorough description of this post-buckling behaviour refer to [15].
Characteristic variables
Considering equations (2.1) and (2.2) with pressure-area relationship (2.4),
and assuming K and A0 constant along the vessel, we can write the system in
non-conservative form as
∂U
∂U
+ H(U)
= S(U),
∂t
∂x
(2.7)
where
u
A
U=
, H(U) =
1
u
ρDA
A
0
, S(U) =
.
u
−fv − g sin θ
We have a non-linear hyperbolic system with two geninuely nonlinear eigenvalues
λ1,2 (H(U)) = u ± c,
1
is the wave speed for the nonlinear system, so λ1 < 0 and λ2 > 0
where c = √ρD
when the flow is subsonic, i.e. u < c. The matrix of left eigenvectors L, is
c
1
L = ac
−a 1
Finally we can introduce a change of variables such that ∂W
∂U = L, where W =
[W 1, W 2]T is the vector of characteristic variables which transforms equation
(2.7) in a system of decoupled equations that allows us to determine the Riemann
invariants:
W1,2 =
Z
u
u0
du ±
Z
A
A0
c
dA = u − u0 ±
A
Z
A
A0
c
dA,
A
(2.8)
where (u0 , A0 ) is taken as a reference state. We notice that with the tube
law (2.4-2.5) the wave speed value is (see [40])
2.2.
Numerical discretization
c=
s
30
s
A ∂p
=
ρ ∂A
K
(mαm − nαn ),
ρ
(2.9)
which implies
Z
A
A0
c(A)
dA =
A
Z
A
A0
p
c1 Aβ + c2 Aγ dA,
(2.10)
with β = β(m) = m − 2, γ = γ(n) = n − 2, c1 = c1 (K, ρ, A0 , m) =
K m
,
ρ Am
0
n
c2 = c2 (K, ρ, A0 , n) = − K
. This integral cannot be solved analitically, so
ρ An
0
the expression of Riemann invariants will be calculated numerically with an
appropriate quadrature rule.
2.2
Numerical discretization
The non-linear system of 1-D governing equations (2.7) is solved by means
of a discontinuous Galerkin scheme, with a high-order 1-D spectral/hp element
spatial discretization. This scheme is suitable for the 1-D formulation because
it can propagate waves of different frequencies without excessive dispersion and
diffusion errors [23].
We first write the system (2.7) in conservative form:
∂F
∂U
+
= S(U),
∂t
∂U
(2.11)
where
"
#
"
au
A
U=
, F(U) = u2
p , S(U) =
u
−fv − g sin θ −
2 + ρ
1
ρ
0
∂p ∂K
∂K ∂x
+
∂p ∂A0
∂A0 ∂x
The weak form of the system (2.7) is obtained by multiplying it by a vector
of test functions φ and integrating over the domain Ω = (0, l), i.e.
∂U
∂F
,φ
,φ
+
= (S, φ)Ω ,
(2.12)
∂t
∂x
Ω
Ω
where
Z
Ω
u · vdx
is the standard L2 (Ω) inner product.
The domain of each vein is discretized into a mesh of Nel non-overlapping
R
R
L
regions Ωe = [xL
e , xe ], e = 1, . . . , Nel , such that xe = xe+1 for e = 1, . . . , Nel −1,
and ∪e=1÷Nel Ωe = Ω. The integrals of the weak form (2.12) are decomposed
into the elemental regions Ωe , e = 1, . . . , Nel , to obtain
"
# X
Nel
Nel X
∂F
∂U
=
(S, φ)Ωe .
,φ
,φ
(2.13)
+
∂t
∂x
Ωe
Ωe
e=1
e=1
Integrating the second term by parts leads to
"
# N
Nel el
X
X
∂U
dφ
xR
( S , φ)Ωe .
+ [F · φ]xeL =
,φ
− F,
e
∂t
dx Ωe
Ωe
e=1
e=1
(2.14)
#
.
2.2.
Numerical discretization
31
To get the discrete form of our problem, we choose U to be in the finite
space of L2 (Ω) functions which are polynomial of degree P on each element. We
indicate an element of such a space using the superscript δ. We also note that
Uδ may be discontinuous across inter-element boundaries. To attain a global
solution in the domain we need to allow information to propagate between the
elemental regions. Information is propagated between elements by upwinding
the boundary flux in the third term of the right hand side of equation (2.14).
Denoting the upwinded flux as Fu , we can now write the discrete weak formulation as
Nel
X
e=1
"
∂Uδ δ
,φ
∂t
!
Ωe
# N
el δ
X
R
δ dφ
u
δ xe
.
S(Uδ ), φδ
− F(U ),
+ [F · φ ]xL =
e
dx Ωe
Ωe
e=1
(2.15)
Following the traditional Galerkin approach, we choose the test function
within each element to be in the same discrete space as the numerical solution
Uδ . Finally we select our expansion bases to be a polynomial space of order P
and expand our solution on each element e in terms of Legendre polynomials.
We note that the choice of discontinuous discrete solution and test functions allow us to decouple the problem on each element, the only link coming
through the upwinded boundary fluxes. Legendre polynomials are particularly
convenient because the basis is orthogonal with respect to the L2 (Ωe ) inner
product. To complete the discretisation, we require a time-integration scheme.
In the current scheme we have adopted a second-order Adams-Bashforth scheme
in accordance with the code developed in [39].
The calculation of the upwind flux Fu is an essential component of the discontinuous Galerkin formulation. Through the evaluation of the upwind flux we
are able to enforce information propagation between elemental boundaries either
within a single vessel or at a junction. It also allows us to impose both inflow
and outflow boundary conditions in a weak sense. For an elemental boundary
within a single vessel the upwinded flux is evaluated by determining the upwinded characteristic variables at the elemental interface. For the subcritical
system we are considering (i.e., u < c) this involves determining W1 from the
backward boundary and W2 from the forward boundary. The upwinded variables can then be determined and subsequently the upwinded flux Fu is then
evaluated. For a more detailed discussion on this numerical implementation see
references [1, 3].
Chapter 3
Numerical Results
In this chapter we validate our model and our code with significant test
cases. In section 3.1, we extend our code for arteries into a general code working
with general tube laws. In section 3.2 we perform two specific physiological tests
that allow us to verfy the behaviour of the code when the exact solution is a
rarefaction wave and a shock wave respectively. Section 3.3 presents the results
of a physiological situation in which muscular pressure occurs. We will first
neglect gravity and then we will add it and analyze its function. Finally section
3.4 consideres a particular case of a swollen and inclined lower limb, in which
the emptying process is led by gravity only (according to the inclination from
the horizontal) and not by external muscular pressure.
3.1
Verification of the code
We first want to make a verification of our code. More specifically our aim
here is to verify that the code used in [39] for the simulation of the arterial flow
works even with different tube laws. In order to do that we need to modify all
the parts of the code that change whenever a different tube law is applied.
These tests do not have a valid physiological relevance. We just want to
perform numerical tests. After these verifications we will be able to run physiological tests.
As usual we consider tube laws of the form
p − pe = K(αm − αn )
m > 0, n 6 0,
√
and we obtain the arterial law when K = β A0 , m = 12 and n = 0. Therefore
we have different laws depending on the role played by the exponents m and n.
In particular negative values of n take collapse into account.
A changing in the tube law modifies the wave speed (see equation 2.9) and the
characteristic variables, whose expression is given by an integral (see formula
2.10) which has to be calculated numerically. For this purpose we apply the
Simpson quadrature rule.
The multiplicative parameter Kp is not a key factor in this analysis as it is a
coefficient that summarizes some characteristics of the tube. We notice that it
is proportional to the wave speed. Then we just need a minimum value of Kp to
ensure the flow is subsonic. When m = 21 and n = 0, we may use K = 1360 P a
32
Verification of the code
3.1.
33
18
1
Exact integration
Simpson formula
Velocity (m/s)
Pressure (KPa)
16
Exact integration
Simpson formula
14
12
0.5
0
10
8
6
6.2
6.4
6.6
Time (s)
6.8
7
−0.5
6
6.2
6.4
6.6
Time (s)
6.8
7
Figure 3.1: Comparison of pressure (on left) and velocity (on right) when calculating the exact integral for the characteristic variables or using the Simpson
formula.
which means β ≃ 76729.79 P a/m. Given the same m, the choice of n different
from 0 will increase the wave speed. Then we run all the tests with these values.
We make use of parameters that have been used previously for the simulation
of the flow in the aorta with the original code. As we are not interested in the
physiology at the moment, these tests do not intend to be realistic and may be
run with any tube.
We may consider a period of 1 second in our analysis (assuming a sinuisoidal
flow at the inlet). Initially we test the Simpson rule in the arterial case since in
this case we may exactly calculate the integral expression for the characteristic
variables. We see from figure 3.1 that the Simpson formula is very reliable.
Starting from the existing code, we have implemented a version with more
general tube law, wave speed and charactic variables. We want to verify if we
obtain the same results when we bring it back to the original case. We set
m = 21 and n = 0. This test can be used as a further verification that the
modifications are correct. The plots of pressure and velocity in figure 3.2 show
that the previous version is exactly a particular case of the new one.
Now we slightly modify the law by considering different values of m and n.
We aim to analytically identify a region where the modified law is still similar
to the arterial one. Then we could check if the numerical results are in agreement with the theoretical analysis of the curves. The arterial tube law does not
consider the possibility of collapse so we look for a region when α > 1.
Given the function ψ = αm − αn we have that ∀α > 0
ψ ′′ (α) < 0
∀ 0 < m < 1, n 6 0
whereas we obtain a point of inflection if m > 1, ∀n < 0. The trends of the
concavity can be seen in figure 3.3. On the left we keep the value m = 21 and we
consider different negative values n = −0.05, −0.10, −0.50. As n becomes more
negative the transmural pressure increases continuously compared to the arterial
one in the region where α > 1. In any case, with not too much negative values
Verification of the code
3.1.
34
18
1
Arterial
m=0.5, n=0
Velocity (m/s)
Pressure (KPa)
16
Arterial
m=0.5, n=0
14
12
0.5
0
10
8
6
6.2
6.4
6.6
Time (s)
6.8
−0.5
6
7
6.2
6.4
6.6
Time (s)
6.8
7
Figure 3.2: Comparison of pressure (on left) and velocity (on right) when using
the arterial law from the original code or from the new general version.
1.5
2
1
1
0.5
n
α −α
m
0
α
1/2
−α
n
3
−1
−0.5
n=0
n=−0.05
n=−0.50
−2
−3
0
0
2
α
4
6
−1
−1.5
0
Figure 3.3: Analytical tube laws. Exponent m =
(left); m > 1 and n = −0.01 (right).
m=0.5, n=0
m=1.05, n=−0.01
0.5
1
2
1
α
1.5
2
and different negative n
3.1.
Verification of the code
35
18
m=0.5, n=0
m=0.5, n=−0.05
Pressure (KPa)
16
14
12
10
8
6
6.2
6.4
6.6
Time (s)
Figure 3.4: Changes in pressure when m =
6.8
1
2
7
and n = 0, −0.05.
of n (like n = −0.05, −0.10) we expect to obtain values of pressure similar to
the arterial case in a very wide range of values of α. On the contrary the curve
with n = −0.50 is very different from the arterial one immediately after the
intersection α = 1.
On the right we consider m > 1 which means that the concavity of the curves
becomes positive for high values of α. In this case the curves have a complete
different structure from the arterial one and the difference of concavity after the
intersection α = 1 makes the research of a similarity region quite complicated
even when m = 1.05, 1.10 and n = −0.01.
We can combine the three cases on the left of figure 3.3 by putting together
the evolution of pressure as in figure 3.4. As we expected, the evolution of
pressure is stable with respect to little changing in the exponent n. Besides
the differences are clearer at the end of the simulation, when α increases. In
particular, the highest values of pressure correspond to the wave associated
with the most negative exponent. These considerations are in agreement with
the analytical laws.
As further verification we compare the analitical tube law with the numerical
results. This is to check that pressure and area are properly related in the code.
This will also allow to test the numerical behaviour of pressure and area when
the exponent m is big. In fact we consider the law that is widely acknowledged
to fit the behaviour of collapsible tubes such as veins. We take m = 10 and
n = − 23 . We obtain a very comforting result in figure 3.5 where we plot the
analytical and the numerical values of the non-dimensional transmural pressure
(divided by K) over α. We find out that the numerical values fit the analytical
curve. We use this as a verification that all the code modifications have given
birth to the expected results. We can now move on to physiological tests.
3.2.
Riemann solutions
36
15
Analytical
Numerical
(p−pe)/K
p
10
5
0
1
1.1
α
1.2
1.3
Figure 3.5: Comparison of analytical and numerical results of scaled transmural
pressure over the ratio α = AA0 when m = 10 and n = − 23 .
L = 0.5m
T = 10s
m = 10
pe = 0 Pa
K = 2 KPa
µ = 0 Pa · s
L
m
∆x = 25
∆T = 10−4 s
n = − 23
R0 = 3 ∗ 10−3 m
A0 = 9π ∗ 10−6 m2
α=1
Table 3.1: Venous physiological parameters. Testing numerical solutions with
exact ones in [40].
3.2
Riemann solutions
In this section we will perform our first two physiological venous tests. We
refer to [40] where the exact solutions of several Riemann problems for collapsible tubes can be found. Our idea here is to implement similar test cases in order
to check if our numerical solutions agree with the exact ones. In particular our
goal here is to check if the numerical method is able to capture rarefaction and
shock waves properly. We will run two test cases in order to achieve this. In
both cases we use the parameters listed in table 3.1:
where L is the length of the vein, ∆x and ∆T are the space and time steps,
T is the duration of the simulation, m and n are the exponents of the tube-law,
R0 is the vein radius at equilibrium (and obviously A0 = πR02 ), K is the bending
stiffness and pe is the external pressure. We may also neglect dinamic viscosity
µ and assume a flat profile, hence α = 1. As we already stated, blood is treated
as incompressible, with constant density ρ = 1050Kg/m3.
3.3.
Calf muscle pump
37
Rarefaction waves
In this test we aim to identify rarefaction waves with our numerical scheme.
We consider the parameters in table 3.1 and we add to it the initial conditions:
A(0) = 1.14 A0 , u(0) = 0 m/s,
and the boundary conditions
Q = 0 m3 /s,
A = A0 ,
x = 0,
x = L.
(3.1)
These conditions describe a problem of a tube initially loaded (as A(0) > A0 )
and which is forced to drain towards the right hand side (flow is forbidden at
the inlet) until it reaches the equilibrium (the section on the right is set equal
to A0 ).
In figure 3.6 we can see the numerical results of pressure, α, velocity and
speed index with boundary conditions (3.1) at time t = 0.02s
These plots agree with those in [40]. As we expected, pressure drops moving along the tube and the cross sectional area follows accordingly. Hence the
velocity increases with a rarefaction wave process.
Shock waves
In this test we aim to identify shock waves with our numerical scheme. We
consider the parameters in table 3.1 and we add to it the initial conditions:
A(0) = 1.14 A0 , u(0) = 0 m/s,
and the boundary conditions
Q = 0.2 m/sA0 ,
A = 1.25 A0 ,
x = 0,
x = L.
(3.2)
These conditions describe a problem of a tube initially loaded as in the
previous test but the boundary conditions are considerably different. A flow
at the inlet is imposed but the cross sectional area at the outlet is fixed at a
bigger value than the general inizial condition. Thus we expect a drop in velocity
moving alongside the tube as the waves on the left hand side are meant to meet
the ones from the right hand side, thus causing a shock wave.
In figure 3.7 we can see the numerical results of pressure, α, velocity and
speed index with boundary conditions (3.1) at time t = 0.02s.
These plots agree again with the analytical solution in [40]. As we expected,
the pressure wave slightly decreases close to the left hand side of the vessel
(because of the boundary condition at the inlet) and then rapidly increases
moving towards the right hand side of the vessel as the right hand side condition
becomes more and more significant. The drop in velocity represent the elastic
jump due to the shock wave.
3.3
Calf muscle pump
The results collected in the previous section allow us to follow with a more
complex case. We start considering a vein subjected to external pressure. In
Calf muscle pump
3.3.
38
6
1.15
1.1
4
α=A/A0
Pressure (KPa)
5
3
1.05
2
1
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
0
1
0.8
0.2
0.6
0.15
Speed index S=u/c
Velocity (m/s)
0
0
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
0.1
0.05
0
0
Figure 3.6: Numerical results of pressure, α, velocity and speed index with
boundary conditions (3.1) at time t = 0.02s.
Calf muscle pump
39
25
1.3
20
1.25
α=A/A0
Pressure (KPa)
3.3.
15
10
1.2
1.15
5
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1.1
0
1
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
0.5
Speed index S=u/c
Velocity (m/s)
0.05
0
−0.5
0
−0.05
−1
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
−0.1
0
Figure 3.7: Numerical results of pressure, α, velocity and speed index with
boundary conditions (3.2) at time t = 0.02s.
3.3.
Calf muscle pump
40
30
External pressure (KPa)
25
20
15
10
5
0
0
2
4
6
8
10
Time (s)
Figure 3.8: Calf muscle pump. Pressure evolution.
L = 0.39m
T = 10s
m = 30
pe (t) = 3 ∗ 104 tanh
K = 200 Pa
µ = 0 Pa · s
t
2.2
− 0.363 H(t − 0.8) Pa
L
∆x = 20
m
∆T = 10−4 s
n = − 32
g=0
A0 = 1.15 ∗ 10−4 m2
α=1
Table 3.2: Physiological parameters for the vein of the lower limb subjected to
the calf muscle pump.
particular we make use of the external pressure used in [30, 12] to model the
calf muscle pump during walking. As we can see in figure (3.8), the action of
the muscle becomes effectie after 0.5 seconds and then raises rapidly up to 30
KPa.
We refer mainly to [30] in this part of the analysis and we have the parameters
in table 3.2.
At first we don’t take gravity into account. We have these initial conditions:
A(0) = 1.09 A0 , u(0) = 0 m/s,
and the boundary conditions
Q = 0 m3 /s,
R = 7.3 ∗ 107 mP3a/s ,
x = 0,
x = L,
(3.3)
(3.4)
where R is the resistance of the venous flow according to the Windkessel
model described in [41]. See also [34, 3] for a detailed description of the Windkessel models and the coupling of 1-D models with 0-D models governed by
ordinary differential equations that relate pressure to the flow at the outflow of
1-D terminal vessels. Hence we have an analogy with circuits: pressure acts as
potential and flow as electric current. Figure 3.9 illustrates the scaled transmural
pressure, area and speed index along the vessel at several time steps.
Calf muscle pump
3.3.
41
15
1.2
t=0.02
t=0.05
t=1
1
α=A/A0
(p−pe)/K
10
5
0
−5
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
0.2
0
1
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
−0.5
0.4
−1
−1.5
0.2
0
0
t=0.02
t=0.05
t=1
t=1.13
0
t=0.02
t=0.05
t=1
t=1.13
(p−pe)/K
Speed index S=u/c
0.6
0.6
0.4
1
0.8
0.8
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
−2
0.5
0.6
0.7
0.8
α=A/A
0.9
1
0
Figure 3.9: Calf muscle pump (no gravity). Numerical results of scaled transmural pressure (top left), α (top right), and speed index (bottom left) in the
external pressure test at several time steps. Tube law at time t = 1s (bottom
right).
Calf muscle pump
3.3.
42
1.2
1.5
Velocity (m/s)
α=A/A0
1
0.8
0.6
0.4
0.2
0
Left
Centre
Right
0.2
0.4
1
Left
Centre
Right
0.5
0
0.6 0.8
Time (s)
1
0
0.2
0.4
0.6 0.8
Time (s)
1
Figure 3.10: Time evolution of α and velocity at at the left hand side, at the
centre and at the right hand side of the vessel. External pressure test with no
gravity.
The results are conforting. The simulated flow becomes supersonic after 1.14
seconds as can be seen in the way speed index raises rapidly. This is because
the action of the muscle becomes remarkable after 0.5 seconds and then it contributes to empty the vessel very quickly. The draining process can clearly be
seen in the plot of pressure and area in particularly. Transmural pressure is
negative in every point of the tube in just 1 second, that means that the area is
smaller than the section at equilibrium, i.e. the vein is collapsed as we expected.
On the bottom right we consider also the plot of scaled transmural pressure
over the scaled cross sectional area at t = 1s as a further confirmation that the
external pressure works properly in our code.
Furthermore we show the time evolution of area and velocity, at the inlet
(left) at the centre and at the outlet (right) of the vessel (figure 3.10).
The section at the outlet collapses abruptely after about 0.8 seconds before
marking the transition from subcritical to supercritical flow.
The effect of gravity
We can now make the test more realistic by adding gravity to our code. We
consider the parameters of table 3.2 with the modification of g = 9.81m/s2. The
initial conditions and the boundary conditions are the same as the previous test,
i.e. (3.3) and (3.4) respectively.
It should be noticed tha the condition of zero flow at the inlet is much clearer
now that we take gravity into account. It acts as the bottom venous valve that
closes when pressure above is higher in order to prevent venous backflow. This
is a simple wave to model a physiological competent venous valve.
Figure 3.11 illustrates the scaled transmural pressure, area and speed index
along the vessel at several time steps.
We notice that the emptying process is still relatively fast but we can also
see that gravity contributes in partly slowing down the draining. For instance,
Calf muscle pump
3.3.
43
30
1.5
t=0.02
t=0.05
t=1
20
α=A/A0
(p−pe)/K
1
10
0.5
0
−10
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
0
0
1
0.6
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
20
t=0.02
t=0.05
t=1
t=1.13
15
(p−pe)/K
Speed index S=u/c
0.8
t=0.02
t=0.05
t=1
t=1.13
0.4
10
5
0.2
0
0
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
−5
0.2
0.4
0.6
0.8
α=A/A
1
1.2
0
Figure 3.11: Calf muscle pump with gravity considered. Numerical results of
scaled transmural pressure (top left), α (top right), and speed index (bottom
left) in the external pressure test at several time steps. Tube law at time t = 1s
(bottom right).
Inclined limb
44
1.2
40
1
30
0.8
20
(p−pe)/K
α=A/A0
3.4.
0.6
0
0
10
0
0.4
0.2
Bottom
Centre
Top
Bottom
Centre
Top
−10
0.5
Time (s)
1
−20
0
0.2
0.4
0.6 0.8
Time (s)
1
Figure 3.12: Time evolution of α and scaled transmural pressure at at the left
hand side, at the centre and at the right hand side of the vessel. External pressure
test with gravity considered.
at t = 1s the transmural pressure is negative only close to the top section of
the tube whereas at that time the vein would be already completely collapsed
without gravity. At t = 1.13 speed index is around 0.7 instead of 1.
As we did above, on the bottom right we consider also the plot of scaled
transmural pressure over the scaled cross sectional area at t = 1s as a further
confirmation that the external pressure works properly in our code. Due to the
effect of gravity there are lower minimum values of α (around 0.4) compared to
the previous case (around 0.6) at the same exact instant.
We show the time evolution of area and pressure, at the inlet (bottom) at
the centre and at the outlet (top) of the vessel (figure 3.12).
The evolution of transmural pressure in particular shows the influence of
gravity: the three points appear clearly translated. This shows the action of
gravity in modification of hydrostatic pressure with height and indicates that
our code considers gravity properly.
These tests have also been performed adding viscous forces and considered a
parabolic velocity profile (α = 34 ) and no significant differences have been found.
This is also in accordance with theory.
3.4
Inclined limb
It is now legitimate to analyze the effect of gravity in an inclined vessel,
to check if the code has been properly extended to take into account not only
horizontal or vertical vessels, but also tubes at any angle θ with respect to the
horizontal axis, so that the gravity term would be g sin θ. This is important
because it can be the building block of a network of more than one single vein,
which extends in a complex way and with several angles.
We consider the vein of the calf, with no external pressure and inclined of
θ = π8 from the horizontal. This describes the situation of a person lying with
Inclined limb
3.4.
45
10
1.2
1
α=A/A0
(p−pe)/K
5
0.8
t=0.02
t=0.05
t=1
t=1.5
t=3
0.6
0
t=0.02
t=0.05
t=1
−5
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
0.4
1
0.2
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
Speed index S=u/c
0.1
0.08
0.06
t=0.02
t=0.05
t=1
t=1.5
t=3
0.04
0.02
0
0
0.2
0.4
0.6
0.8
Distance along the tube ξ=x/L
1
Figure 3.13: Emptying process of inclined lower limb (θ = π8 ). Numerical resultsof transmural pressure, α and speed index at several time steps.
his leg lifted slightly upward. Thus in this case gravity accelerates the flow.
The initial conditions and the boundary conditions are still the same as
the previous test, i.e. (3.3) and (3.4) respectively. The difference now is that
the emptying process is caused only by gravity. Our idea comes from the tests
performed in [8, 30], in particular the ones about the emptying process of the
jugular vein of a giraffe upon tilting upright. These tests are highly supercritical
and more importantly prove that the schemes can capture elastic jumps. Hence
we perform a similar process with subcritical conditions: a person with swollen
veins of the calf (for example after a long run if he is not very fit) lifts his legs
to reallow the venous blood to move upward more easily.
Figure 3.13 illustrates transmural pressure, area and speed index along the
vessel at several time steps.
Observe that in this test, the section associated with the non-dimensional
distance along the tube ξ = 0 corrisponds to the section close to the ankle
and pressure increases properly moving toward the knee, in contrast with the
3.4.
Inclined limb
46
opposite situation of the test above. The speed index plot shows clearly that
our numerical scheme is able to capture the elastic jumps that we expected. In
fact, at a given time t, speed increases moving toward the knee and then drops.
Besides, as time goes on velocity decreases and the peak of the jump moves
towards the knee until the situation becomes steady.
Chapter 4
Conclusions and future
work
In this work we have presented a model for collapsible tubes. Starting from
a code developed and conceived for arteries, we have been able to extend it into
a general code working with general tube laws.
The Simpson formula has played a crucial role in succesfully approximating
the integral of Riemann invariants. This problem does not occur with arteries
as those integrals can be calculated exactly. After we have done that, we have
been able to perform several physiological tests.
We have seen that the Discontinuous Galerkin method with second order
Adam-Bashforth time discretization has been appropriate to reaching this goal.
The expansion with Legendre polinomial makes calculations much faster. In
particular these scheme captures elastic jumps as we have seen in several situations.
The tests have also proved that the external pressure performed by the calf
muscle pump and gravity have been extended correctly to the code, also with
different inclinations of the vessel. The scheme has produced comforting results.
These tests need to take into account many variables, so lots of future work
can be done especially because venous flow has much less been studied so far
compared to arterial one. As gravity and external pressure are crucial, an extension of the code that allows to analyze supersonic flow would be extremely
helpful. An accurate model of valves of the lower limb could also be important
in improving the accuracy of simulations.
Turbulent flow could also be considered, as it has been observed that this
occurs during forced expiration [30]. More complicated windkessel models for
boundary conditions (for example including venous compliance) can be done as
well.
This code has also been projected to develop a network of more than one
vessel, so junctions and bifurcations could also be analyzed.
47
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
Stylized arrangement of the circulation . . . . . . . . . . . .
Wall-lumen ratio in different blood vessels . . . . . . . . . .
Pressure and blood velocity in systemic circulation . . . . .
Distribution of blood volume in a resting man . . . . . . . .
Arterial and venous histology . . . . . . . . . . . . . . . . .
Histology of medium and large vein . . . . . . . . . . . . . .
Valve of a vein from the orbital sinus . . . . . . . . . . . . .
Pulmonary vessels . . . . . . . . . . . . . . . . . . . . . . .
Anatomy of the superficial and perforating veins of the foot.
Superficial veins of the lower limb . . . . . . . . . . . . . . .
Superficial and deep veins of the lower limb . . . . . . . . .
Perforating veins of the lower limb . . . . . . . . . . . . . .
Pressure and flow in human superior vena cava . . . . . . .
Effect of gravity on pressure in a standing human. . . . . .
Displacement of venous blood volume on changing position
Calf muscle pump . . . . . . . . . . . . . . . . . . . . . . .
Pressure of the dorsal vein of the foot during orthostasis . .
Effect of breathing on venous return . . . . . . . . . . . . .
Blood pressure responses to the Valsalva maneuver . . . . .
.
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7
8
8
9
11
12
13
14
15
16
16
17
19
20
21
22
23
25
26
2.1
Layout of a 1-D compliant vessel . . . . . . . . . . . . . . . . . .
28
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
Simpson formula applied to arterial law . . . . . . . . . . . . . .
General tube law . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analytical tube laws . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure evolution with respect to changes in the tube law . . . .
Venous tube law . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rarefaction waves: pressure, area, velocity and speed index . . .
Shock waves: pressure, area, velocity and speed index . . . . . . .
Calf muscle pump. Pressure evolution . . . . . . . . . . . . . . .
External pressure test (no gravity): pressure, area and speed index
External pressure test (no gravity): time evolution . . . . . . . .
External pressure test (gravity): pressure, area and speed index .
External pressure test (gravity): time evolution . . . . . . . . . .
Inclined limb test: pressure, area and speed index . . . . . . . . .
33
34
34
35
36
38
39
40
41
42
43
44
45
48
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.
List of Tables
3.1
3.2
Parameters for testing rarefaction and shock waves solutions . . .
Parameters for testing the calf muscle pump . . . . . . . . . . . .
49
36
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