# The role of venous valves in pressure shielding

## Abstract

### Background

It is widely accepted that venous valves play an important role in reducing the pressure applied to the veins under dynamic load conditions, such as the act of standing up. This understanding is, however, qualitative and not quantitative. The purpose of this paper is to quantify the pressure shielding effect and its variation with a number of system parameters.

### Methods

A one-dimensional mathematical model of a collapsible tube, with the facility to introduce valves at any position, was used. The model has been exercised to compute transient pressure and flow distributions along the vein under the action of an imposed gravity field (standing up).

### Results

A quantitative evaluation of the effect of a valve, or valves, on the shielding of the vein from peak transient pressure effects was undertaken. The model used reported that a valve decreased the dynamic pressures applied to a vein when gravity is applied by a considerable amount.

### Conclusion

The model has the potential to increase understanding of dynamic physical effects in venous physiology, and ultimately might be used as part of an interventional planning tool.

## Keywords

Peak Pressure Reverse Flow Transient Pressure Venous Valve Chronic Venous Deficiency## Background

The motivation behind this study was a desire to understand the physiological effects of compression cuff therapy for prevention of deep vein thrombosis. It is generally accepted [1, 2, 3, 4], that deep vein thrombosis is associated with flow stasis, particularly in and around the venous valves and their sinuses. From a survey of the literature, it rapidly became apparent that the role and quantitative performance of venous valves, even in the normal physiological state is poorly understood. Texts on venous physiology always identify the role of the valves as the control of reverse flow [4, 5, 6, 7]; most often in the context of muscle pump action to maintain flow in the direction of the heart and sometimes in the context of postural changes and of exercise. The purpose of this paper is to explore the effects of gravity on the pressure and flow distribution in a simple representation of a vein in the leg, and in particular to quantify the role of the valves in pressure shielding under the action of standing. The effects of a range of parameters on the shielding performance of the valves are examined. It is demonstrated that the effects depend not only on the distribution, location and performance of the valves themselves, but also on the geometric and mechanical characteristics of the veins. It is anticipated that this information will be of direct interest to the vascular surgeon because it provides an indication of the likely effect of interventions, including removal or repair of valves as well as insertion of bypass grafts, on peak pressure distributions in the peripheral vasculature.

A person who stands, inactive, for a period of time will be subjected to the full hydrostatic pressure gradient in the venous system and the pressure in the veins in the foot will reach something of the order of 100 mmHg [8, 9, 7]. This is confirmed by Pollack [10] and by Arnoldi [11]. The presence of a valve or valves cannot shield against this static pressure – which will be associated with the physiological phenomenon of blood pooling, and related to oedema through the Starling equation [7], but it can alleviate the transient maximum that will occur as posture is changed. In the absence of valves, a simple analysis would suggest that the transient pressure peak experienced in the vein might be double the final standing pressure. Neglecting inertial and viscoelastic effects in the vessel wall, the stress in the wall of the vein is proportional to the instantaneous applied pressure, and it is postulated that incompetent valves do not provide adequate transient pressure shielding and thus might be strongly implicated in the formation of varicosities [9, 12, 13]. A primary quantitative measure of valve, or rather system, performance is given by its effectiveness in reducing the pressure peaks associated with the transient response. A second measure might be one of the effectiveness of a postural change, and the associated action of gravity on valve opening and closure characteristics and thus on the 'wash out' of the sinuses and displacement of stationary pockets of blood. Although a full three-dimensional analysis is required to address this question in detail, the one-dimensional model presented in this paper is used to examine whether the properties of the system are such that there might be sufficient backflow to close the valve for realistic geometries.

## Materials and Methods

A series of ordinary differential equations are written to represent the electrical system. The nonlinear elastic properties of the vein, including those associated with collapse, are represented by a tube law [14, 15, 16, 17]. The main purpose of the tube law is to capture the vein's flexibility at small negative pressures as collapse is initiated, whilst maintaining the properties of a stiffening response for higher negative or positive pressures. A penalty of this formulation is that it does not reduce to the standard linear approximation at small positive pressures, and for the current work, a modification has been implemented to remedy this deficiency. A number of numerical techniques are available for solution of the derived equations [18]. The one adopted for the current study is a Lax Wendroff formulation, which is accurate to second order in time and space. For completeness, the governing equations and the numerical discretisation are listed in Appendix 1. This formulation has been adopted in other studies of the cardiovascular system [19], although Brook [16, 20], has identified conditions under which numerical instabilities might be manifest. Numerical testing has indicated that the system is stable under the pertinent conditions for the current study.

One of the important properties of the system that will have significant influence on the results is the boundary conditions applied at the proximal and distal ends of the vein segment. For the purposes of the current study, a constant atmospheric pressure boundary condition has been applied at the proximal end and a constant flow boundary condition at the distal end. It is recognised that the prescribed boundary conditions might represent a gross simplification of physiological flow in the venous segments of interest. The important feature of the proximal pressure boundary condition is that it allows unimpeded reverse flow into the vein segment as gravity acts. It would be possible to apply a negative pressure representative of that in the thoracic cavity, but as a constant offset this would not significantly affect the results. A transient thoracic pressure representative of respiration could also be applied, but primary focus in this paper is on relatively short term events associated with a near-instantaneous application of gravity. The relatively low frequency respiratory cycle would not significantly modify the results. It has further been assumed that there is a constant flow into the 'bottom' (distal end) of the vein, based on average steady state drainage into the femoral vein. It is unlikely that there will be significant backflow through the distal end during gravity application, due to the higher resistance of the smaller vessels. More sophisticated descriptions of transient flow waveforms measured under a range of conditions can be found in the literature. Of most direct interest is the study reported by Raju S et al [21], who describe flow conditions under ambulatory conditions but not under first application of the gravity field, whilst Neglen and Raju [22] also focus on the measurement of ambulatory pressures in individuals with signs of chronic venous deficiency. Willeput R et al [23] and Abu-Yousef M [24] focus on rest and respiratory conditions. Again, it is suggested that the frequencies associated with these temporal variations are relatively low compared with those associated with the phenomenon addressed in this paper. Furthermore, the starting condition for the analysis is a steady flow through the system (equal to the distal end flow), with no gravity applied.

This paper focuses on the transient pressure and flow distributions in a vein segment, with and without valves, under a near-instantaneous application of gravity. The system is considered passive, and effects of the muscle pump are not included: similarly other relatively low frequency external load factors are neglected. A body force is applied, in the opposite direction to flow, representing the action of gravity under a change of posture from horizontal to vertical: this force is sigmoidal in time, so that there is smooth transition from zero to the full gravity force, which is then held constant.

## Results and Discussion

### Baseline condition, no valve

A series of numerical tests were performed, to ensure that the model performed properly and returned accurate results for simple conditions, including for example using a linear tube law, for which analytical comparisons were available, and for other conditions for which numerical results have been published [17, 19]. Once these tests were passed, a first baseline analysis was performed using the following parameters: vein diameter 1.19 cm [25], vein thickness-to-diameter ratio 0.2 [26], vein length 1 m, wall stiffness 1 MPa [27], blood viscosity 0.004 Pa.s, blood density 1000 kg/m^{3}, distal (inlet) flow 15.1 ml/s [25], proximal (outlet) pressure 0 mmHg (0 Pa), near instantaneous body-force application (gravity increased from zero to 9.8 m/s^{2} over 0.01 milliseconds). These values are given in convenient units: all analyses were performed in SI units. The initial condition, prior to the application of gravity, was a steady flow in the opposite direction to that in which gravity would be applied (i.e from distal to proximal end of the tube).

Baseline condition, valve performance and gravity application time results

Parameters | No valve | Perfect valve | "Real" valve | Leaky valve | Gravity test |
---|---|---|---|---|---|

| 1.2 | 1.2 | 1.2 | 1.2 | 1.2 |

| 1 | 1 | 1 | 1 | 1 |

| 1000 | 1000 | 1000 | 1000 | 1000 |

| 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

| 4 | 4 | 4 | 4 | 4 |

| 1000 | 1000 | 1000 | 1000 | 1000 |

| Near instantaneous | Near instantaneous | Near instantaneous | Near instantaneous | 0.1 |

| No valve | One | One | One | One |

| No valve | 0.5 | 0.5 | 0.5 | 0.5 |

| No valve | Perfect | "Real" | Leaky | Perfect |

| 136.9 (18.2 kPa) | 60.2 (8.01 kPa) | 60.3 (8.02 kPa) | 65.4 (8.69 kPa) | 50.3 (6.69 kPa) |

| 127.1 (16.9 kPa) | 81.9 (10.9 kPa) | 81.9 (10.9 kPa) | 115.8 (15.4 kPa) | 72.6 (9.66 kPa) |

| 136.9 (18.2 kPa) | 93.3 (12.4 kPa) | 93.9 (12.4 kPa) | 115.8 (15.4 kPa) | 99.3 (13.2 kPa) |

| 1.83 | 1.25 | 1.25 | 1.51 | 1.33 |

| No valve | 0.17 | 0.17 | 0.11 | 0.16 |

| 0.02 | 20.2 | 20.4 | 3.57 | 8.17 |

| 15.1 | 9.89 | 9.89 | 12.5 | 10.6 |

The analysis of the baseline condition gives some confidence in the operation of the model, and also provides quantitative information on the peak pressure that can be expected at the distal end of the vein in the absence of protection from venous valves. Further confidence has been developed by comparison of the results with those from three dimensional models using a commercial finite element code, but the reporting of these results is beyond the scope of this paper.

### Effect of perfect, real and incompetent valves

The model was next used to evaluate and to provide quantitative information about a hypothesis often expressed in text book descriptions of venous physiology, e.g. Browse [13]:

'The venous valves normally protect the wall of the vein below each valve from the pressure in the vein above it.'

In a final test on the reference configuration with a perfect valve, the time over which gravity was applied was increased from near instantaneous to 100 ms (more consistent with the likely time taken to stand up). As expected, the first and second pressure peaks were lower (by the order of 10%) but perhaps surprisingly, the absolute peak was a little higher. This was due to different interactions of the wave reflections in the system, but it does not affect the overall shape of the response, nor indeed the conclusions.

### Parameter studies

Parameter variation test results

Parameters | Perfect valve | Diameter test | Stiffness test | Distribution test | Length test | Location test | ||||
---|---|---|---|---|---|---|---|---|---|---|

| 1.2 | 0.8 | 1.6 | 1.2 | 1.2 | 1.2 | 1.2 | |||

| 1 | 1 | 1 | 1 | 0.50 | 0.75 | 1 | |||

| 1000 | 1000 | 500 | 2000 | 1000 | 1000 | 1000 | |||

| 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | ||||

| 4 | 4 | 4 | 4 | 4 | 4 | ||||

| 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | ||||

| Near instantaneous | Near instantaneous | Near instantaneous | Near instantaneous | Near instantaneous | Near instantaneous | ||||

| One | One | One | Two | One | One | ||||

| 0.5 | 0.5 | 0.5 | 0.25 and 0.75 | 0.5 | 0.25 | 0.75 | |||

| Perfect | Perfect | Perfect | Perfect | Perfect | Perfect | ||||

| 60.2 (8.01 kPa) | 66.3 (8.81 kPa) | 56.8 (7.55 kPa) | 56.3 (7.48 kPa) | 63.2 (8.4 kPa) | 32.6 (4.34 kPa) | 33.2 (4.41 kPa) | 46.8 (6.23 kPa) | 32.6 (4.33 kPa) | 87.2 (11.6 kPa) |

| 81.9 (10.9 kPa) | 114.3 (15.2 kPa) | 74.9 (9.97 kPa) | 74.6 (9.92 kPa) | 95.5 (12.7 kPa) | 55.6 (7.39 kPa) | 55.9 (7.44 kPa) | 57 (7.58 kPa) | 99.3 (13.2 kPa) | 90.2 (12 kPa) |

| 93.3 (12.4 kPa) | 115.8 (15.4 kPa) | 89.5 (11.9 kPa) | 90.9 (12.1 kPa) | 104.5 (13.9 kPa) | 92.9 (12.36 kPa) | 57.2 (7.61 kPa) | 77.5 (10.31 kPa) | 110.5 (14.7 kPa) | 105.3 (14 kPa) |

| 1.25 | 1.47 | 1.22 | 1.21 | 1.41 | 1.25 | 1.53 | 1.39 | 1.49 | 1.41 |

| 0.17 | 0.11 | 0.19 | 0.26 | 0.11 | 0.09 and 0.17 | 0.06 | 0.11 | 0.16 | 0.15 |

| 20.2 | 3.64 | 24.5 | 48.2 | 6.06 | 35.6 | 3.58 | 10.6 | 3.25 | 42.3 |

| 9.89 | 12.1 | 9.87 | 21.2 | 5.41 | 9.89 | 5.92 | 7.19 | 11.9 | 11.5 |

Changing the overall length of the system (whilst maintaining the position of the valve at 0.5 m from the inlet), or changing the position of the valve along the length of the 1 m vein, increased the dynamic pressure ratio, suggesting that the optimal position for a valve in a vein with the imposed boundary conditions is near to the midpoint. Finally, a test with two valves, one at one-quarter length and one at three-quarters length produced a pressure shielding of the same magnitude as that obtained with a single valve halfway along the vein.

## Conclusion

The one-dimensional model reported in this paper permits the quantitative evaluation of the effects of venous valves on the loads and geometrical changes induced by the action of gravity. It is an important first step in a longer-term study of venous valves, venous diseases and their prevention. With refinements to the venous valve description, applied tube law and boundary conditions, a more physiologically realistic model can be created in an equivalent form to the Westerhof arterial model [29]. This will enable the model to be validated against physiological data. For the purposes of this paper though the greatest interest is in the dynamic pressure ratio, which provides a measure of the increase of the peak local pressure in the system (due to dynamic effects) over the corresponding hydrostatic pressure. It is demonstrated that, for a configuration typical of the femoral vein, the dynamic pressure ratio without a valve is 1.83, and that with a perfect valve located halfway along the vein is 1.25. The absolute pressure reduction is over 40 mmHg (5320 Pa). The model has been used to investigate the quantitative influence of variation of a number of parameters. Following extensive *in vitro* and *in vivo* validation, this model might be used to evaluate the effects of valve incompetence on venous pressure distributions and could have implications for the understanding of the progression of disease in the context of varicose veins. It might also be used as part of an interventional planning tool.

The reported study is entirely theoretical. Validation against other reported numerical studies has been performed to give confidence in the numerical implementation, but validation against experimental data is an important next step. Some experimental data does exist for collapsible tubes with gravity effects, for example of the filling under gravity of an initially collapsed tube [30], but none of direct relevance to the current study. A preliminary experimental model that can be used directly to validate the current model has been reported by Potter [31] and Burnett [32] but this is not yet sufficiently mature for detailed comparative evaluation. Validation against physiological data, such as that presented in the works referenced in the section on boundary conditions, will require first the construction of an improved numerical model with a more complex network representation of the venous circulation in the lower limb. In the longer term a detailed three dimensional (3D) model is required to compute the haemodynamic characteristics in the region of the valve, and to evaluate the effects of local geometric and material variations. The 1D model described in this paper provides important mutual validation data for such a 3D model, as well as the potential to provide local boundary conditions for it in the region of the valve.

## Appendix: Equations and discretisation

### A1: Governing equations

The variables used in equations 1–7 were: A_{0} – undeformed vessel area-, A^{n}_{i} – vessel area at point i along the vessel length at time n-, μ – fluid viscosity-, ρ – fluid density-, E – vessel wall Young's modulus, σ-vessel Poisson's ratio, h – vessel wall thickness and finally r – vessel radius-. All variables used were described in SI units.

### A2: Lax-Wendroff discretisation

## Notes

### Acknowledgements

This study was funded by the British Heart Foundation under BHF PhD studentship FS/05/086/19462. Many useful synergies arose between this and a parallel study on deep vein thrombosis, EPSRC GR/S86464/01, funded by the Engineering and Physical Sciences Research Council.

## Supplementary material

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