Abstract
Relatively little attention has been paid to the venous system and valves from a cardiovascular engineering perspective up to now. 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 as 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.
Introduction
The accurate description of pressure and blood flow in the veins of the human circulation system is relevant to the study of cardiovascular disease and to our understanding of blood pressure response during 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 the change in venous return associated with heart failure or the involvement of venous valves in deep vein thrombosis. Because of 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 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 the particular characteristics of the venous system.1,2
For the purpose of modelling, the venous system has usually been treated in a way similar 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 however does not capture the non-linear 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 waves and can be non-linear in nature, and finally 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
Properties of arteries and veins relevant to modelling
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 pressures in the large veins which make 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 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 Adkins 15 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 venules19–21 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
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 the 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 the intra-abdominal 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 −5 to −7 mmHg, as the chest cavity enlarges. This pressure is transmitted directly to all vessels in the thorax (including the entire pulmonary system, the heart, the thoracic aorta and veins 25 ). The use of the diaphragm might conceivably reduce venous return from 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 for 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 these properties should be addressed.
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 the 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 systems and 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 the 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 the description includes the upright position, the 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.
28
in 1955, 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:
Venous return is approximately proportional to mean circulatory filling pressure minus the right atrial pressure. This is termed as pressure gradient for venous return.
28
The mean circulatory filling pressure (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 The increase in venous return due to an increase in mean circulatory filling pressure 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; ‘Venous return reaches a maximum value when the right atrial pressure falls to −2 to −4 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 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; 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 the references.7,15,28–31
The considerations stated above were incorporated into a model by Snyder and Rideout
32
in 1969 (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 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 drew, 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 return and the response of the circulatory system in astronauts, which experience unusual acceleration conditions during space travel.
The model created and used by Snyder and Rideout
32
to study the venous circulation
Moreno et al. 33 in 1969 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. 34 in 2002 to validate Guyton's analysis of cardiac output and venous return curves. Mukkamala and colleagues 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. 35 in 2003 created a model of venous return for the purpose of simulating vacuum-assisted venous return (VAVR) to provide information about safety and efficiency during cardiopulmonary bypass (CPB). 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 time dependent. 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 dramatically 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 VAVR during CPB, 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 the 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 postoperative state. The main limitation of the proposed model was its application spectrum. It could only be use for studying the healthy paediatric venous return in the total cavo-pulmonary connection. It was not able to describe pressure waves, nor was it able to provide an inside 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
37
in 2005 (Figure 2). 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 anesthetized 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 the venous circulation.
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
Buxton and Clarke 38 in 2006 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 valves dimensions used were not based on anatomical data.
Zervides et al.39,40 in 2008, reported on a 1D 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. While 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 1D mathematical/computational model reported earlier39,40 and to complement the information it provided, in 2010 Zervides
41
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 needs substantial improvements to be considered as a useful research tool.
Three-dimensional model of vein with venous valve
42
Blood velocity profile just before application of gravity as seen if vessel was cut in half along x-axis
42
Blood velocity near the valve region after application of gravity
42
Conclusions
There are lessons that 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 the venous system behaviour and in order to accomplish this we need the input of a variety of sciences from physics 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 in this new field must decide on the sophistication level needed, 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 the 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 the development of valid techniques to improve the 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 the improvement of simulation accuracy. Finally, the potential use of imaging (MRI or computed tomography) 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.
Footnotes
Acknowledgements
The authors acknowledge the financial support of the European Venous Forum under the EVF Pump Priming Grant – 2011/2012.