Arterial Hemodynamics

Pulsatile Pressure and Flow Waves

Figure 8.1 illustrates pressure waveforms down the arterial system, from the aorta to the capillaries, and then on to the venous side of the systemic circulation. The depicted behavior corresponds to healthy conditions: arterial occlusive disease may significantly alter the patterns of pressure and flow.

Mean arterial pressure (MAP) is relatively constant in the aorta and elastic and muscular arteries but experiences a sharp decline at the level of arterioles. Pulsatile behavior is apparent in the aorta, elastic arteries and muscular arteries. Pulsatility also decays greatly in the arterioles and is absent thereafter. Next, an overview of several physical parameters key to understanding pulsatile flow in elastic arteries is provided.

Important Physical Concepts in Blood Flow

Resistance : Resistance refers to the ratio of drop in pressure to flow in a vascular territory:

It is apparent that the arterioles constitute the vascular territory with the largest resistance, as they experience the largest drop in pressure. These vessels are known as resistive arteries. The resistance is determined by several factors, including vessel length and diameter. Poiseuille flow is a useful concept from fluid mechanics to understand the relationship between flow, pressure, and resistance (see Fig. 8.2 ). Here, a pressure gradient ( $\Delta \text{P}$ ) drives flow through the vessel, which moves from a point of higher pressure (Pressure _proximal ) to a point of lower pressure (Pressure _distal ). The flow has a parabolic shape, with maximum velocity (Velocity _max ) at the center of the lumen, and zero velocity at the interface with the endothelial surface.

In Poiseuille’s flow, the relationship between flow (Q), pressure drop $\Delta \text{P}$ , and Resistance are:

where $\mu$ is the blood viscosity , L is the vessel length over which the given pressure drop $\Delta \text{P}$ takes place, and R is the vessel radius. It is thus apparent that the vessel radius plays a much larger role on vascular resistance than the vessel length, due to its power of 4 exponent. This explains why relatively small changes in vascular tone significantly alter vascular resistance (e.g., a 10% vasoconstriction results in an increase of over 50% in vascular resistance).

Another important biomechanical concept that can be easily illustrated with Poiseuille’s flow is wall shear stress ( $\tau$ ), the tangential stress of the flowing blood on the endothelial surface of the blood vessel ( Fig. 8.2 ). Stress is force per unit area. Therefore, wall shear stress and pressure have both the same units. Their orientation and magnitude are drastically different though: while typical values of wall shear stress in the arterial system are around 10–100 dynes/cm ² acting tangentially to the endothelial surface, pressure is over a thousand times larger (100 mm Hg = 133,322 dynes/cm ² ) and acts perpendicularly to the vessel wall. ^, Figure 8.3 shows a computational fluid dynamics (CFD) analysis of wall shear stress and pressure values acting on a thoracic aortic endograft, highlighting the difference in magnitude and orientation between these quantities.

Using Poiseuille’s solution, the wall shear stress ( $\tau$ ) can be estimated by:

For a given vessel radius, larger velocities will lead to larger wall shear stress. Larger viscosity also results in larger wall shear stress. Both pressure and wall shear stress are important drivers of vascular mechanobiology, as discussed later.

Hoop wall stress : the hoop wall stress ( ${\sigma }_{\text{hoop}}$ ) refers to the stress induced by the blood pressure inside the vessel wall. This stress acts circumferentially within the wall and is a key biomechanical driver of vessel function. The stress can be understood by another fundamental equilibrium equation in biomechanics, Laplace’s law ( Fig. 8.4 , left). Laplace’s law states that the hoop stress ${\sigma }_{\text{hoop}}$ is directly proportional to the product of the pressure P and the vessel radius R and inversely proportional to the wall thickness h:

It is important not to confuse the hoop stress with the wall shear stress. The hoop stress acts within the vessel wall. Assuming a ratio R/h = 10, and a mean pressure to 100 mm Hg, typical magnitudes of the hoop stress are 1,333,333 dynes/cm ² , which is approximately 30,000 times larger than the magnitude of the wall shear stress.

Axial wall stress : the axial wall stress ( ${\sigma }_{\text{axial}}$ ) is due to the tethering of the blood vessel in the axial direction, which imposes a force F _axial on the vessel ( Fig. 8.4 , right). The axial stress is the result of dividing the axial force by the cross-sectional area of the vessel:

Vascular compliance and stiffness : the pulsatility of the pressure waveforms is greatly influenced by the stiffness of the arterial wall. ^{, ,} Vascular stiffness (E) is determined by the ratio of elastin to collagen fibers (the more collagen, the stiffer) and by the thickness (h) of the vessel wall (the thicker the wall, the stiffer). The intrinsic stiffness of the vessel wall is known as material stiffness . The product of the thickness and material stiffness is known as structural stiffness E _structural = E⋅h . Vascular stiffness is widely accepted as a metric of vascular health, as it plays a critical role in dampening the pulse of pressure as it travels down the arterial system. ^, Often, terms such as compliance, distensibility, and stiffness are loosely used to refer to the elastic properties of arteries. For the reader’s clarity, simple definitions are given below:

Compliance : Compliance (C) is the change in volume ( $\Delta \text{V}$ ) imposed on the vessel by a given change in pressure ( $\Delta \text{P}$ ) such as the pulse pressure (PP) between systole and diastole, i.e.:

The compliance of an artery depends on the pressure that it is subjected to. The larger the pressure, the smaller the compliance (see left panel in Fig. 8.5 ).

Distensibility: distensibility (D) refers to the ratio of changes in luminal area between systole and diastole, divided by the pressure pulse (PP) between systole and diastole, i.e.:

Distensibility is therefore a similar metric to compliance, obtained via changes in luminal area of the vessel rather than via changes in volume.

Stiffness : Stiffness (E) is the ratio between increments in stress ( $\sigma$ ) and strain (ε):

In general, both stress and strain are multi-axial quantities with components in the circumferential and axial directions of the vessels. Therefore, they are more general than the simple pressure, area, and volume quantities used in the definitions of compliance and distensibility. This is the reason engineering analysis often relies on complex, multi-axial stiffness characterization of blood vessels.

Blood vessels exhibit a nonlinear relationship between stress and strain. At low strains and pressures, the burden of bearing the stress is carried by the elastin matrix, which has low values of stiffness. At higher stress, the collagen fibers become engaged and confer the vessel a much stiffer behavior. Note that the shape of the compliance and stiffness curves for vascular tissue is different, due to the axes of stress/pressure and strain/volume being switched ( Fig. 8.5 , right).

Blood viscosity: viscosity ( $\mu$ ) is another important biophysical concept in blood flow. It refers to the friction between layers of fluid as they slide relative to each other. Figure 8.6 (left) shows a classic experiment used to determine viscosity known as Couette flow. Here, a fluid is contained between two parallel plates. The bottom plate is fixed, and the top plate is subject to a force, F, parallel to the bottom plate, which results in the plate moving at a constant velocity v . Therefore, the fluid has a velocity ranging from zero at the interface with the bottom plate to v at the interface with the top plate. Here, the shear rate ( $\dot{\gamma }$ ) is the ratio of change in velocity over a certain distance, $\gamma =\nu /h$ , where h is the gap between the plates. In Poiseuille’s flow, the shear rate can be approximated by the maximum centerline velocity of blood divided by the vessel radius, see Fig. 8.2 , Eq. 8.4 .

The bottom plate experiences a wall shear stress ( $\tau$ ), given by the product of the fluid viscosity and the shear rate. Given that in this experiment the shear stress ( $\tau$ ) can be obtained by dividing the force exerted on the top plate F by its area A , the experiment provides a formula to measure the viscosity of the fluid given by:

As we saw earlier, viscosity is an important contributor to resistance to flow ( Eq. 8.3 ). In simple fluids like plasma, the viscosity is constant property of the fluid. However, whole blood has a complex viscosity behavior, due to being a mixture of plasma, cells, and proteins. ^, This results in viscosity being higher at lower values of shear rate, typically occurring on the venous circulation and on regions of slow re-circulating flow ( Fig. 8.6 , right). Furthermore, hematocrit levels greatly affect viscosity as well. The right panel of Figure 8.6 shows experimental data for the relationship between viscosity and shear rate for human plasma (straight lines) and increasing levels of hematocrit (from 12.6 to 67.4). Higher hematocrit levels greatly increase the viscosity for all shear rates, with more pronounced effects in the lower shear rate range.