Review article
Computational phlebology: reviewing computer
models of the venous system
C Zervides* and A D Giannoukas†
*European University of Cyprus, School of Sciences, Department of Health Sciences, Nicosia, Cyprus; †Department
of Vascular Surgery, Faculty of Medicine, University of Thessalia, Larissa, Greece
Abstract
Relatively little attention has been paid to the venous system and valves from a
cardiovascular engineering perspective till date. Given the involvement of venous valve
haemodynamics in the development of deep vein thrombosis this is an area that needs
more detailed investigation and close collaboration between clinicians and cardiovascular
engineers. The purpose of this review article is to provide an indication of the
physiological conditions that need to be included in any computational model of the
venous system, based on recommendations from clinicians, and to summarize published
computational models of the venous system by trying to explore their limitations and
application range. A MEDLINE search was carried out on the relevant literature from 1940
until today. Several models have been developed with a specific purpose in mind to
coincide with the aim of each individual study. The model complexity and laws used in
each model vary significantly. There are more simplistic computational models based on
electric circuit analogies, termed lumped parameter models, which can be used to provide
boundary conditions to one-dimensional (1D) and three-dimensional (3D) domain models,
followed by 1D continuous models based on analytical equations, which allow the
description of pressure wave and can be non-linear in nature. Finally, there are the more
advanced 3D models, which are based on the principles of haemodynamics, and consider
the compliance of the venous system and the effect that venous valves have on the
cardiovascular system. In conclusion, it appears that computer modelling of the venous
system can contribute greatly to our understanding of venous physiology and allow us to
evaluate the haemodynamic interactions that occur in the venous system under different
physiological conditions.
Keywords: venous flow; venous pressure; venous valves; venous simulations; venous
modelling
Introduction
The accurate description of pressure and blood
flow in the veins of human circulation is relevant
to the study of cardiovascular disease and to our
understanding of blood pressure response during
Correspondence: Dr C Zervides, PhD, CSci, European
University of Cyprus School of Sciences, Department of
Health Sciences, 6, Diogenes Str., Engomi, P.O. BOX 22006,
1516, Nicosia, Cyprus. Email: [email protected]
Accepted 18 February 2012
activities such as running or walking. Blood flow
in the systemic veins is of interest to researchers
and clinicians concerned with a wide range of
clinical problems such as change in venous return
associated with heart failure or the involvement
of venous valves in deep vein thrombosis. Owing
to the level of complexity involved, and the difficulty of achieving closed-form analytical solutions,
the venous system lends itself to analysis by computer simulation in order to study the above problems. Mathematical models of the cardiovascular
system have contributed greatly to the quantitative
understanding of its behaviour. The majority of
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Review article
C Zervides and A D Giannoukas. Computational phlebology
Table 1 Properties of arteries and veins relevant to modelling
Arteries
Transmural
pressure
(mmHg)
Relative blood
volume
Vessel collapse
Importance of
muscle tone
Existence of
valves resisting
retrograde flow
Effect of
intrathoracic
pressure
Flow
characteristic
Veins
Large
Aorta
Small
i.e.
Arterioles
Large
Vena cava
Small
i.e.
Venules
80 –100
30– 50
4– 8
10–20
6%
12%
15%
65%
No
None
No
Some
Yes
None
No
Significant
No
No
Yes
Yes
Little
Little
Significant
Some
Pulsatile
Pulsatile
Not
pulsatile
Not
pulsatile
we have the more advanced 3D models which are
based on the principles of haemodynamics, and
consider the compliance of the venous system and
the effect that venous valves have on the cardiovascular system.
Methods
MEDLINE was searched up to 1 December 2011 for
studies evaluating the use of computer models to
study the venous system. Search terms used were
‘venous flow’, ‘venous pressure’, ‘venous valves’,
‘venous simulations’, ‘venous models’ and ‘cardiovascular modelling’ in various combinations. The
reference lists of the gathered reports were manually read. This produced additional items, which
were also considered.
Key parameters for modelling the venous system
these were developed with the aim of investigating
the relationship between pressure and flow in the
arterial system and in those situations where the
entire circulation has been modelled, the focus is
on the arterial system or the heart. Models created
to represent the arterial tree have adequately
explained the key phenomena, but many responses
are primarily determined by particular characteristics of the venous system.1,2
For the purpose of modelling, the venous system
has usually been treated in a similar way to the
arterial system using different parameter values3 – 5
due to the extreme complexity of modelling the
venous system and the enormous computational
power needed to accomplish such a task. This
methodology though does not capture the nonlinear behaviour of the venous system (expansion
and collapse) and disregards the presence of
venous valves.
This manuscript highlights the most important
computer models created to represent the human
venous system. Each model was developed with a
specific purpose in mind to coincide with the aim
of each individual study. The model complexity
and laws used in each model vary significantly.
There are more simplistic computational models
based on electric circuit analogies, termed lumped
parameter models, which can be used to provide
boundary conditions to one-dimensional (1D) and
three-dimensional (3D) domain models, followed
by 1D continuous models based on analytical
equations, which allow the description of pressure
wave and can be non-linear in nature and finally
210
In order to develop a representative model of the
venous system, it is important to have an understanding of the properties of the veins and the
way in which these differ from those of the arteries.
The main differences of veins and arteries of systemic circulation in this context are summarized
in Table 1.1,6 – 9
The main differences include: (1) the fact that
large veins are thin-walled, able to collapse and
hold a large volume of blood compared with
arteries, (2) low pressure in the large veins which
makes them very susceptible to changes in extravascular pressure and particularly to the possibility
of negative transmural pressures, (3) a very small
percentage change in the volume of blood in the
venules due to increased venous tone, which is
able to shift a considerable volume of blood to
other parts of the circulation, (4) valves resisting retrograde flow which are present in the venous
system but are absent in the arterial system and
(5) the fact that arterial flow is pulsatile while
venous flow is not.
The ability and tendency of veins to collapse is
a major difference between veins and arteries.
Work on vessel collapse started with Holt10,11 and
has been continued by others.12 – 14 Holt measured
pressures in the right atrium and femoral vein in
10 anaesthetized dogs. Simultaneously, he measured the mean right atrial pressure using a saline
manometer connected to a cannula, which passed
into the right atrium via the external jugular vein.
He reported that as atrial pressure was raised
above atmospheric pressure, there was a concomitant increase in femoral pressure, but as atrial
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C Zervides and A D Giannoukas. Computational phlebology
pressure was lowered, the femoral pressure could
not be reduced below 8 mmHg. This allowed him
to conclude that this has occurred because a
region located somewhere in the inferior vena
cava had collapsed. He then performed experiments using thin-walled rubber tubes to simulate
the effects of collapse. Effectively, the collapsed
region presents a near-zero pressure boundary condition to the distal venous vasculature, a point
of extreme importance for modelling the venous
system.
Guyton and Adkins15 performed experiments on
living and cadaveric dogs to determine the relationship between thoracic vena cava pressure and
femoral venous pressure for different values of
intra-abdominal pressure. They showed that
femoral venous pressure could not drop below the
intra-abdominal pressure irrespective of how low
the pressure was in the thoracic vena cava. Finally,
Brecher et al. 16 concluded that the vena cava does
not collapse at a specific point at its entry into the
thorax but, rather, a transition zone is formed
between collapsed and non-collapsed segments.
This is a conclusion of importance when a 3D
model of the venous system is created since this
transition zone needs to be included.
The next important factor is the effect of gravity.
Gravity has a significant influence on vessel collapse. Knowledge of the response of the cardiovascular system when gravity and changing
gravitational forces are acting is important.
Gravity causes a shift in blood volume and it is
well known that individuals with orthostatic hypotension faint upon standing due to a sudden
pooling of blood in the lower parts of the body,
and a consequential drop in cardiac output.8
The presence of valves in veins is well known but
data on their number and distribution are limited.
Valves occur in great numbers in the long
intermediate-size veins of the legs (the long saphenous vein has 3 major and 20 minor valves, the
short saphenous has 1 major and 6 –12 minor
valves while the common femoral has only 1
valve1,17,18), but are entirely absent in the great
veins of the abdomen and thorax.1,6,9 Currently,
there is conflicting information about the existence
of valves in very small veins and venules,19 – 21 but
as medical imaging equipment become more
advanced the resolution they offer increases and
more valves can be seen.
The competency of venous valves ranges from
perfect to offering a slight resistance to backflow.
Venous valves aid venous return in that they
direct blood towards the heart and prevent retrograde flow.1,6
Review article
In the lower limbs, venous valves serve the
important function of directing flow from superficial to deep veins and act as venous flow modulators as indicated by Lurie et al. 22 Moreover, the
deep veins in the legs are surrounded by large
muscle groups that compress the deep veins when
the muscles contract. Venous compression increases
the pressure within the veins, closes upstream
valves and opens downstream valves, thereby providing a pumping mechanism.8 Compression may
also be applied by body weight acting on the underside of the foot (when standing) or by manually
squeezing the calf. Thus, changes in posture can
enhance venous flow. A ‘venous pump unit’ comprises a vein with an adequate lumen, one or two
effective valves and intermittent compression of
the vein provided by contraction of surrounding
muscles. A large number of such units are spread
throughout the lower limbs. Together, these are
referred to as the skeletal muscle pump.1,6,8 If
venous valves are damaged or absent, the skeletal
muscle pump does not function properly.8 Finally,
venous valves, unlike cardiac valves, have an
activity that has no regular action associated with
cardiac pulsatility.
Abdominal muscle contractions affect intraabdominal pressure, which in turn affects the
pressure in the abdominal vessels.23,24 Other important extraluminal pressure variations are due to the
intrathoracic and intra-abdominal pressures that
accompany breathing. On inspiration, the intrathoracic pressure decreases by a few mmHg, reaching
values of 25 to 27 mmHg, as the chest cavity
enlarges. This pressure is transmitted directly to
all the vessels in the thorax (including the entire
pulmonary system, the heart, the thoracic aorta
and veins25). The use of the diaphragm might conceivably reduce venous return from the legs
during inspiration because of constriction of the
inferior vena cava.
Finally, in a recent study published in
2011, Pierce et al.26 measured venous flow with
the use of magnetic resonance imaging (MRI).
Except from stating that venous blood velocity
in the extremities is generally, slow, i.e. less
than 0.1 m/s, they indicated the importance of
breathing in venous flow, since maximum flow
was observed during expiration and minimum
during inspiration. In addition, the difference in
pulsation between arterial and venous flow was
evident.
Based on the physiological differences discussed
above, it is clear that, when building a computational model, all of these properties should be
addressed.
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System models
Systems’ modelling refers to the interdisciplinary
study of the use of models to conceptualize and
construct systems. It helps us to obtain a deeper
understanding of the functionality of the system
under investigation. This can be accomplished by
examining an idealized version of a system, i.e. an
idealized version of the venous system, or by examining how a developed therapeutic approach affects
a system, i.e. use of Heparin as anticoagulation
therapy and its effects on the venous system, or
by examining more patient-specific systems, i.e.
based on MRI images which have been processed
to create a patient-specific computer model of the
venous system.
Different models can be used to represent the
same system from different perspectives and viewpoints. The external perspective shows the system’s
context or environment and indicates the more
important external factors that interact with the
system at hand. The behavioural perspective
shows the way that the system will behave under
different stimulations and functional conditions.
Finally, the structural perspective shows the
system architecture, allowing different parts of it
to be modelled. For such a system model to be
created, it is important to have essential fundamental knowledge of how it will behave. All
models have to be criticized regarding their sufficiency and also for fidelity to nature, completeness
and their relationship to larger models of which
they are a part.27
Venous models
There are comparatively few system physiology
models describing flow in the venous system.
Analogue models of the venous system require at
least three elements: a resistor, a capacitor and an
inductor, with the latter being of more importance
in the venous than in the arterial system. A resistor
represents viscous terms, a capacitor represents
elastic compliance of the vessel and an inductor
represents mass inertia. A potential difference
source can be used for considering gravity effects
and a rectifier (diode) can be used to represent
venous valves. Non-linearities have to be considered in order to have a complete understanding
of the system. Thus, consideration of pressure/flow
relationships in the small venules during venous
collapse, or low flow conditions is imperative.
Venous capacitance is also non-linear. The models
have to include time varying pressure sources
created by respiration and skeletal muscles, and if
212
the description includes the upright position,
partly unidirectional flow through the venous
valves has to be considered. Typically, branching
circuit models describe the anatomy of the venous
system with varying degrees of complexity.
Guyton et al. in 1955,28 with the aid of a mathematical circuit analysis, identified factors important in the control of venous return and tested
these experimentally in dogs. They found that:
(1) Venous return is approximately proportional
to mean circulatory filling pressure (MCFP)
minus the right atrial pressure. This is termed
pressure gradient for venous return.28 The
MCFP was defined as: ‘When heart pumping
is stopped by shocking the heart with electricity to cause ventricular fibrillation or is
stopped in any other way, the flow of blood
everywhere in the circulation stops a few
seconds later. Without blood flow, the pressures everywhere in the circulation become
equal after a minute or so. This equilibrium
pressure level is called the MCFP’;7,29,30
(2) The increase in venous return due to an
increase in MCFP is not proportional to the
above pressure gradient. This is because
while MCFP is a force that tends to push
blood to the right atrium it also increases the
diameters of the blood vessels, thus decreasing
the impedance to venous return;
(3) ‘Venous return reaches a maximum value
when the right atrial pressure falls to 22 to
24 mmHg and remains at this maximum
value down to infinitely low negative pressures. As the right atrial pressure rises to positive values, venous return falls and reaches
zero when the right atrial pressure has risen
to equal the mean circulatory pressure’;7,29,30
(4) Change in resistance to blood flow in different
regions of the peripheral circulatory system
has less and less effect on venous return as
distance from the right atrium increases;
(5) Venous return is proportional to arterial
pressure only when all peripheral resistances
and capacitances remain constant.
For a more comprehensive description of Guyton’s
analysis, the reader is referred to.7,15,28 – 31
The considerations stated above were incorporated into a model by Snyder and Rideout in
196932 (Figure 1). They created a physical analogue
computer model of the human cardiovascular
system, with detailed attention to the representation
of pressure and flow events in the veins. They also
included the effects of gravity on the venous and
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Review article
Figure 1 The model created and used by Snyder and Rideout32 to study venous circulation
arterial trees, the effect of venous collapse, the effect
of breathing and the action of venous valves. Their
model included a control loop for heart rate and
was validated against human venous pressure
waveforms and against the response of humans to
tilt-table experiments. From the comparisons the
authors of the paper performed, they found that
their model was only valid for the study of postulated venous tone control characteristics. This was
useful when studying the mechanisms of venous
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Figure 2 Simple electric circuit analogue representation of the
cardiovascular system. Capacitors represent the distensibility of the
arterial (Ca), capillary (Cc) and venous (Cv) wall, resistances
represent the resistance to flow through the arterial (Ra), capillary (Rc)
and venous (Rv) compartment. Resistance R0 was included so that
flow could be established between the input pressure P0 and the
arterial pressure P1. P2 is the capillary pressure, P3 is the venous
pressure and P4 is the right atrial pressure. Q0 is the input flow, Q1 is
the arterial flow, Q2 is the capillary flow and Q3 is the venous return
return and the response of the circulatory system in
astronauts, who experience unusual acceleration
conditions during space travel.
Moreno et al. in 196933 introduced more general
models of venous return and stressed the necessity
for a better classification of the parameters of the
system. Furthermore, they introduced the definition
of the venous return system into three primary subsystems. This study demonstrated respiratory
changes between the regional contributions of the
systemic inferior caval and splanchnic subsystems.
A very simple model was created by Mukkamala
et al. in 200234 to validate Guyton’s analysis
of cardiac output and venous return curves.
Mukkamala et al. used a non-linear computational
model of the pulsatile heart and circulation. They
developed two sets of open circulation models
capable of creating cardiac output and venous
return curves, by varying the average right atrial
pressure. They showed that their models support
the validity of Guyton’s analysis.28,30,31
Brown et al. in 200335 created a model of venous
return for the purpose of simulating vacuumassisted venous return to provide information
about safety and efficiency during cardiopulmonary
bypass. This model was developed using the
Bernoulli equation and assumed that blood is a
Newtonian fluid, an erroneous assumption since
blood is a non-Newtonian fluid. This assumption
though, did not alter the results significantly and
was a valid one to make, since the simulation
running time was short. This is because in
Newtonian fluids, the coefficient of viscosity is constant while in non-Newtonian fluids it can be timedependent. Thus, if the simulation run time is
short, a constant coefficient of viscosity can be used
for non-Newtonian fluids, since the coefficient of viscosity does not change significantly during a very
short run time. Unfortunately, the main limitation
of the proposed model was its application spectrum,
since it was designed to investigate only vacuumassisted venous return during cardiopulmonary
214
bypass, with no clear way of examining venous
return when it was not vacuum assisted or when
the patient was not under anaesthesia.
In 2003, Pittaccio et al. 36 described a lumped parameter model for the study of venous return in the
total cavo-pulmonary connection. A model of
healthy paediatric blood circulation was created
based on a full range of characteristic constants
that describe the way that the pulmonary, systemic
circulation, the heart and in particular venous
return, behave. The blocks describing major veins
include terms for resistance, compliance, inertance,
venous valves and take into consideration the effect
of venous collapse, even though this is not clearly
defined. Furthermore, their model considers the
effects of respiration and of the moving diaphragm.
Finally, it accurately predicted the tracings and
absolute values of the time variables (velocities in
vena cava, aortic and intrathoracic pressures)
in both a healthy and a postoperative state. The
main limitation of the proposed model was its
application spectrum. It could only be used for
studying healthy paediatric venous return in total
cavo-pulmonary connection. It was not able to
describe pressure waves, nor was it able to
provide an insight into local haemodynamic characteristics. Finally, as the authors themselves state,
there is a need to include oxygen consumption
and active short-term baroflex regulation.
A very simple model which reports derivative
observations to the ones reported by Mukkamala
et al.34 was created by Zervides and Hose in 2005
(Figure 2).37 This computer model of the human
cardiovascular system was based on Guyton’s
closed circuit analysis of the heart and the peripheral circulatory system. This model was compared
with Guyton’s experiments performed in anaesthetized dogs regarding the normal venous return
curve, the effect of MCFP and the importance of
arterial, venous and capillary resistance. The comparisons indicated that the simple model was
valid for the study of Guyton’s experimental work
and could form the basis of a more complex
model of the cardiovascular system with specific
attention to venous circulation.
Buxton and Clarke in 200638 presented a 3D
computer simulation focused on the dynamics of
a venous valve. They showed that they could
capture the unidirectional nature of blood flow in
venous valves. In addition, they investigated the
dynamics of the valve opening area and the blood
flow rate through the valve. Even though this is
the first reported 3D model of venous valves, it is
far from ideal. This is because Buxton and Clarke
simulated the veins as rigid tubes and the venous
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Review article
Figure 3 Three-dimesional model of vein with venous valve (blue shading ¼ vein wall, yellow shading ¼ venous valve)42
valves dimensions used were not based on anatomical data.
Zervides et al. 39,40 in 2008 reported on a onedimensional mathematical/computational model
of a collapsible tube with the facility to introduce
valves at any position. The model was exercised to compute transient pressure and flow
distributions along the vein under the action of an
imposed gravity field (standing up). A quantitative
evaluation of the effect of a valve, or valves, on the
shielding of the vein from peak transient pressure
effects was undertaken. They reported that with
the help of their model they showed that a valve
decreased the dynamic pressures applied to a vein
when gravity is applied by a considerable amount,
over 40 mmHg, and they concluded that the
model has the potential to increase understanding
of dynamic physical effects in venous physiology.
Although the model was very versatile, it provided
a limited understanding of the haemodynamic and
structural characteristics of venous valves in normal
operation, particularly the detail local to the valve
and the dynamic closure and opening characteristics associated with changes of posture under
the action of gravity.
In order to resolve the issues seen by the onedimensional mathematical/computational model
reported earlier39,40 and to complement the information it provided, in 2010 Zervides41 reported on
a 3D model (Figure 3). The 3D model allowed studying the haemodynamics of the opening and closing
phases on the venous valve (Figures 4 and 5). The
model results were used to develop a method of
measuring blood ‘washout’ from behind the valve
leaflets. A methodology was created and adopted
in order to accomplish that and showed that regarding ‘washout’ of blood particles, the application of
gravity helps to remove blood from the locations
where flow-stasis occurs. This was in agreement
with the findings of Lurie et al.,22 who indicated
that ‘the vertical stream behind the valve cusps,
prevents stasis inside the valve pocket’ and that
‘the central jet possibly facilitates outflow’. Unfortunately, the created model was not physiologically
correct since the sinuses of the valves were not
included (they play an important role in the functionality of the valve) and need substantial improvements to be considered as a useful research tool.
Conclusions
There are lessons, which we need to learn from
the history of venous physiology. We are at a
point in time that an update to the definition of
venous physiology is needed in order to include
computational modelling as an integral part of
the process. There is an imperative need for a
specialist understanding of venous system behaviour and in order to accomplish this we need
the input of a variety of sciences from physics
Figure 4 Blood velocity profile immediately before application of gravity as seen if the vessel was cut in half along the x-axis42
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Figure 5 Blood velocity near the valve region after application of gravity42
and mathematics to pharmacology and biology.
Improving fundamental understanding is a goal
for clinicians and scientists alike, which unfortunately is rarely achieved. This is where systems
biology and modelling come into play since they
allow us to put our integrated knowledge within a
specific context. This fast-growing area requires
computational phlebology scientists to take on a
role that focuses on modelling, simulation and
analysis of venous disease in collaboration with
specialized clinicians. This synergy will facilitate
the translation of basic research into clinical
strategies for care of patients with venous related
disorders.
This manuscript examines published computer
models of the venous system, and provides an
overview of their development and applications.
Venous model configurations are becoming increasingly sophisticated and advanced, and various
developed models have seen wide and successful
applications in the study of venous physiology
and more.
Models are developed to achieve specific purposes laid down for each specific study, and for
that reason, the individual complexity of the
models must fit the purposes of the studies that it
was created for. An over-simplified model will
produce inadequate accuracy but this does not
mean that a more complex model will always
produce results that are more accurate. There is no
universally optimal model that suits every application. Scientists involved with this new field
must decide on the sophistication level needed,
216
which will be more suitable for their needed
outcome.
The venous system does not work in isolation.
It has close interaction with other systems like the
arterial system, the respiratory and the nervous
system even with the endocrine system. Studying
for example the response of the venous system
under neuro-regulation and hormone control, or
the simulation of coupled venous dynamics and
transportation of nutrients/metabolic remaining,
will bring venous modelling to a higher level, and
such results will improve our understanding of
venous physiology.
The future of the field needs scientists and
researchers to work on developing sophisticated
models for the venous system, covering major
venous branches in vessel anatomy. Assigning
realistic diameter, length, thickness and elasticity
values to individual venous segments will allow
for successful model simulations. Future effort
may be made in this area for development of
valid techniques to improve parameter settings in
venous modelling.
With the development of computer hardware
and numerical analysis techniques, haemodynamic
analysis using computational fluid dynamics in 3D
can be performed. To address the requirement of
high accuracy and ability to simulate the interaction
between for example venous valves and blood particles concurrently, it is necessary to build multidimensional models, something that is currently
possible but still needs further improvement.
Advancements in this field will be very helpful to
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the improvement of simulation accuracy. Finally, the
potential use of imaging (MRI or CT) in computational modelling not only for the venous but
also for the cardiovascular system as a whole is a
new and interesting field that will eventually lead
to patient-specific models, which is the ultimate goal.
18
19
20
Acknowledgements
21
The authors acknowledge the financial support of
the European Venous Forum under the EVF Pump
Priming Grant – 2011/2012.
23
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