An important function of the venous circulation is to provide storage capacity within the circulatory loop, typically 60–80 % of the total blood volume is stored within the venous system depending on both posture and activity (Meissner et al. 2007). The storage capacity of the veins arises from the compliance of the vessels, which is related to their geometry, material properties, influence of the surrounding tissue and variation in transmural pressure, defined as the difference between the internal and external pressures acting on the vein.

The vein wall is thinner than that of its companion artery and contains less smooth muscle. Under physiological pressure loading the diameter of the venous lumen is larger than that of neighbouring arteries, particularly in peripheral vessels. The elastic modulus of veins has been shown to vary considerably with vessel location, exhibiting a similar stiffening response to arteries with increase in pressure loading (Wesley et al. 1975). For a particular value of elastic modulus simple linear measures of vessel compliance (Eq. 7.2) predict higher compliance in the venous circulation due to the larger diameter and thinner wall of the veins.
where R is the vessel radius, h the wall thickness and E the Young’s modulus of the vessel wall. The increase in compliance arises from the higher circumferential stress in a larger, thinner vessel at a given pressure.

In addition to the contribution from passive mechanical properties the pressure–area response of the veins is determined by active processes including metabolic control and pressure and flow-mediated changes in venous tone (Monos et al. 1995).

The linear approximation of Eq. 7.2 neglects the large changes in compliance of the veins which arise over the range of transmural pressures experienced by these vessels. When transmural pressure becomes negative, the compliance of the veins is nonlinear as a result of changes in geometry of the vessel cross-section. Under negative transmural pressure the vein does not remain circular in cross-section, becoming elliptical or adopting a dumbbell shape. The form of the vessel cross section is determined by the support conditions with different forms of response observed in vessels surrounded by muscular tissue (e.g. deep calf veins) than in those with less supporting tissue (e.g. vena cava).

To examine the mechanics of venous collapse the vein can be considered a thin-walled elastic cylinder. The initial circular cross-section of the vein will deform to adopt a non-circular cross section under a uniform pressure load, as illustrated in Fig. 7.2. This arises from the elastic instability of the system and the pressure at which this occurs can be obtained through consideration of small deformations of the tube and a linear analytical approach (Fung 1997), which predicts that the vein will adopt a non-circular form at the transmural pressure given by Eq. 7.3.

where p _e is the external pressure, p the internal pressure and ν is the Poisson’s ratio of the vessel wall. Extension of the analysis of this deformation into the post-buckling regime was undertaken by Flaherty et al. (1972) using numerical integration. This analysis captures the self-contact of the cylinder and the rapid changes in cross-sectional area that occur with increase in pressure following the initial buckling response. Figure 7.3 illustrates these effects through the solution of a finite element model of a thin cylinder under uniform pressure load. This analysis recognizes the finite thickness of the cylinder and captures the variation of the stress across the wall.

The response of the finite element model demonstrates significant changes in the geometry of the cross-section as the transmural pressure becomes negative. It should be noted that these deformations arise from nonlinear geometric effects in this analysis and that the strain within the cylinder is relatively low throughout the deformation. These effects are distinct from the nonlinearities observed in both the arterial and venous pressure/area relationship under positive transmural pressure resulting from the nonlinear mechanical response of the vessel to loading due to the vessel wall constituents. For more complex analysis of the deformation of thin-walled elastic tubes, including 3D deformation effects and fluid-structure interaction, the reader is referred to more detailed reports (Grotberg and Jensen 2004).

The results of such analyses provide a description of the nonlinear pressure–area response (compliance) of the venous system, which is generally handled through the definition of a “tube-law” to describe this behaviour. The typical form of an analytical tube-law is shown in Fig. 7.4 along with the response of the vessel computed using the finite element approach for two vessel thicknesses (h/R = 0.1 and 0.05). The equation for the tube-law is given below, with K _p the bending stiffness of the vessel, A the cross-sectional area and A₀ the area at zero pressure:

From Fig. 7.4 it is clear that, as the transmural pressure becomes increasingly negative, small changes in pressure result in large changes in cross-sectional area. This allows the vein to act as a blood reservoir without large increases in venous pressure. With further decrease in transmural pressure the compliance of the vessel reduces, due to the self-contact illustrated in Fig. 7.3e, f which correspond to normalized pressure of −4.86 and −25 respectively.

In addition to the nonlinear form of the pressure–area response, it is also evident that the effective resistance of the vein will change significantly during the collapse phase. As a result, regions of vein which are collapsed present a high resistance to flow, which provides a dynamic mechanism for flow limitation. This effect is clear from Fig. 7.3 where the flow travels through a small gap and viscous losses are high in the collapsed state (d–f) compared to flow within the vessel as the transmural pressure increases (a–c). This effect is also observed in the operation of a Starling resistor, originally used to control flow rates within a heart-lung machine.

More complex behaviour is observed under conditions where the local speed u of the blood within the vein, exceeds the speed c of propagation of waves along the vessel. Due to significant variations in both cross-sectional area and local compliance of the vein, which determines the wavespeed, it is possible to transition between subcritical (u < c) and supercritical (u > c) flows within a single vein. Detailed analysis of the implications of these effects has been examined using a 1D numerical model under steady flow conditions (Shapiro 1977).

As wavespeed decreases with increase in compliance, supercritical flow becomes more likely in conditions when the veins are partially collapsed, this occurs in the jugular vein and the superior vena cava when in the standing position and in the superficial circulation when the limb is raised above heart level, as described in Sect. 7.2.4.

Venous return is defined as the volume of blood returning to the right atrium from the systemic venous circulation. Maintaining venous return is important for obvious reasons to ensure that blood is continuously available to prime the pulmonary circulation thus, in turn, providing oxygenated blood to the left ventricle. The storage capacity of the venous system and the subtle relationship between pressure and volume within the veins allows the venous system to act as a ‘buffer’, regulating levels of venous return under transient changes in cardiac output.

Total venous return is determined by a number of contributing factors including; the pressure gradient between the peripheral veins and the right atrium, the resistance of the venous vascular bed, the influence of transient muscle pump activation (in particular the calf muscle pump) and the effects of the respiratory pump. As discussed above, the resistance of the venous bed can be strongly nonlinear due to the increased resistance of collapsed vascular segments, this is particularly notable in the abdomen during collapse of the vena cava under changes of abdominal pressure (Wallace et al. 2010). The mechanisms underlying the calf muscle pump and respiratory pump are discussed in more detail in the following sections. Both pumping mechanisms rely on the presence of venous valves.

During exercise, increased cardiac output requires an associated increase in venous return. Augmentation of venous return from the legs is achieved through the action of the foot, calf and thigh muscle pumps, allowing venous return to be enhanced beyond the level associated with the pressure gradient between the legs and the heart. The anatomy of the calf muscle pump, which provides the most significant contribution, is shown in Fig. 7.5. The calf pump is formed by the action of the gastrocnemius and soleus muscles which contract and force blood out of the deep veins of the calf (anterior tibial, posterior tibial and peroneal veins) which drain into the popliteal vein at the knee.