This chapter mainly deals with the physiology of the arterial system and the hemodynamic analysis of pressure and flow in the arterial tree. Although most of the topics covered are generic and applicable to the systemic and pulmonary circulation, there is no doubt that the main focus is on the systemic circulation because it is the most studied, and also the most accessible, certainly for noninvasive measurement technology.

Before tackling arterial system physiology, we first provide the reader with a concise overview of some elementary laws of flow dynamics that are frequently used and cited in clinical literature, but without going in detail on the mathematical derivation.

The chapter first provides the “classic” analysis of hemodynamic data in the frequency domain, which is based on the parallelism between flow dynamics and electric system analysis. Input and characteristic impedance are defined and discussed, and it is demonstrated how arterial wave reflection can be assessed and quantified from pressure and flow. In addition to this classic view, we also consider wave intensity analysis, which is an alternative method of analyzing arterial hemodynamic data and wave reflection, but in what is perhaps a more intuitive way in the time domain.

A subsequent section is dedicated to the analysis of “arterial function” and “arterial stiffness,” which is a new emerging domain of clinical research. Different methods are described, ranging from local measurement of arterial properties to parameters describing the complete arterial tree, most of them emerging from an electrical analog representation of the arterial tree. The clinical applicability of the methods, their strong and weak points, and the relevance of measuring arterial stiffness in general are discussed.

Blood flow in the arterial tree is of a relatively complex nature, for many different reasons. First of all, blood in itself is a complex liquid.^1,2 As a suspension of biochemically active cells in plasma, it is a non-Newtonian liquid, which implies that it has a nonconstant viscosity. When not in motion, red blood cells coagulate and rouleaux formation occurs, increasing the viscosity of blood. Under the action of shear forces within the blood (see later in the chapter), these rouleaux break up when blood flows and the viscosity of the blood decreases. Blood is therefore known as a “shear thinning liquid,” with viscosity at high shear rates in the order of 3 to 4 mPa s (or cPoise). Blood can be generally considered as a homogenous liquid, except at the level of the arterioles and capillaries, where the size of the red blood cells (8 μm) becomes relatively large with respect to the vessel diameter (100 μm and less). Here, more complex rheological models should be considered, or blood should be treated as a multiphase liquid.

To define “shear rate,” we take the example of flow of a liquid in a long, straight cylindrical tube. When the liquid starts to flow, a velocity profile will be installed, with maximal flow at the center. Near the tube wall the velocity is zero, which means that over the cross section of the tube the velocity gradients exist. The slope to the velocity profile, that is, the derivative of velocity with respect to the radial coordinate, ∂u/∂r, is the shear rate. One can visualize the flow as consisting of concentric cylindrical shells gliding over each other (thus exerting a shear force), with the outer shell standing still and the inner core having the highest velocity (Figure 7-1).

Steady Laminar Flow: Poiseuille Flow

It has been shown by Poiseuille³ that for a Newtonian liquid in fully developed steady flow with “no-slip” boundary conditions (zero velocity near the wall) and up to a certain flow level, the velocity profile is parabolic, with the maximum velocity, U_max, at the center of the tube (Figure 7-1). With r the radial distance from the center and R_a the radius of the tube, the velocity at a given distance r is

The mean velocity, U_mean, equals U_max/2. The absolute value of the shear rate (making abstraction of the sign of γ) is

where γ is zero at the center of the blood vessel and maximal near the wall, where it becomes 2U_max/R_a. The shear stress, τ, that is exerted on the wall by the blood flow is the product of dynamic blood viscosity (μ) and γ, and thus becomes 2μU_max/R_a. Equilibrium of forces requires that the resultant of the shear stresses along the vessel lumen over a segment L is balanced by the pressure difference (Δ P) that exists over that segment, and this results in

where Q is the flow through the tube. The resistance of a blood vessel, Δ P/Q, is thus proportional to the dynamic viscosity of the blood, the length of the vessel, and inversely proportional to the vessel radius to the fourth power! As such, it is mainly the small diameter vessels that generate the fluid dynamic resistance in the arterial tree.

The above relations apply only to highly structured, organized (laminar) steady flow in a straight tube. The dimensionless parameter that expresses whether a flow is laminar or not is the Reynolds (Re) number (with ρ the density of blood and D the vessel diameter)⁴:

Flow is laminar for Reynolds numbers up to 2000. For higher values, the fluid loses its structure and turbulences will occur, introducing extra energy (pressure) losses and flattening the velocity profile. Note that the presence of a vortex does not necessarily imply that the flow is turbulent. In bends and branches, steady vortices may occur without losing the structured character of the flow. In that case, the flow is still laminar.

Oscillatory Laminar Flow: The Womersley Number

In the arterial tree, and then especially at the level of the large arteries, blood flow is not steady but pulsatile. Flow can still be perfectly laminar in these conditions. Oscillatory (sinusoidal), laminar flow in a tube has been studied by Womersley around 1940–1950; he developed a mathematical framework that allows to analytically calculate velocity profiles in these conditions.^5,6 Full elaboration of the Womersley theory is beyond the scope of this chapter, and can be found elsewhere. It suffices here to define the so-called Womersley number, often termed α:

where f is the frequency of the oscillation. The Womersley parameter is, as the Reynolds number, a dimensionless parameter and it expresses the ratio of the inertial effects over the viscous friction effects. The higher the α, the more the inertia will play a role.

The effect of α on the velocity profiles is illustrated in Figure 7-2. For small α, the velocity profile is close to a parabola, with an amplitude that changes with time. The higher the α becomes, the more the inertia will become important, especially for the core flow at the center of the blood vessel. Near the wall the friction remains important and the flow is in phase with the pressure gradient. In the core, phase lags will occur, with peak flow being reached 90 degrees out of phase with the pressure gradient that drives the blood flow. The velocity profile also becomes flatter.

In the systemic arterial tree, the highest Womersley numbers occur in the aorta (values in the order of 20) with flattened velocity profiles. In the small arteries, viscous effects dominate and velocity profiles will rather be parabolic.

Bernoulli’s Law: Conservation of Energy

Finally, to conclude this section, it is worth mentioning Bernoulli’s law,⁷ which expresses the conservation of energy along a streamline in an incompressible, steady, isothermal flow, ignoring the effects of viscosity, the heat transfer, and the work done.⁴ Along a stream line, it applies that

where p is the static pressure, g the gravitational constant, h the height of the point above a reference level, v the velocity in that point, and ρ the density. p + ρgh represents the potential energy of the liquid, with p the static pressure and ρgh the hydrostatic pressure. ρv²/2 represents the kinetic energy (dynamic pressure).

The Bernoulli’s law can be extended with an extra term Δ F, which stands for extra losses that occur between points 1 and 2 (e.g., because of turbulences occurring as a consequence of a sudden expansion or narrowing).

The Bernoulli equation, in a simplified form, is often used to calculate the pressure drop over a stenosis (Figure 7-3). In that case, neglecting Δ F, the blood is accelerated in the stenosis, with conversion of potential energy into kinetic energy. Assuming point 2 to be in the stenosis and v₁ negligible compared to v₂, Δ P over the stenosis approximates ρv₂²/2. With a blood density of 1050 kg/m³ and accounting for a conversion factor 133.3 to convert pressure in Pa to pressure in mm Hg (1 mm Hg = 133.3 Pa), this equation is in the clinical setting often approximated as

where v is the velocity in the stenosis, expressed in meters per second and Δ P is the pressure drop, expressed in millimeters of mercury. This expression is known as the “simplified Bernoulli equation” and generally overestimates the true pressure drop because v₁ is not totally zero and, more important, there is commonly a partial recovery of the pressure distal to the stenosis, with restoration of part of the kinetic energy into potential energy.

The blood pressure in the large arteries results from the interaction between the cyclic ejection of blood from the ventricle into the systemic or pulmonary circulation. It is forcing an open door when we state that the blood pressure is determined by both the function of the heart and by the properties of the arterial system.^8,9 In clinical terms, “blood pressure” is commonly interpreted as the upper (systolic; SBP) and lower (diastolic; DBP) value of pressure measured at the level of the brachial artery. Epidemiologic studies, the most famous one being the Framingham Heart Study, have demonstrated that normal ageing is associated with a gradual monotonic increase in SBP, while DBP rises up to the age of 60 and declines beyond that age¹⁰ (Figure 7-4). The same “uncoupling” between systolic and diastolic blood pressure is also found in patients with so-called isolated systolic hypertension, i.e., those who have an elevated systolic blood pressure but normal-to-low diastolic blood pressure. Studies have demonstrated that an increased pulse pressure (PP), i.e., the difference between systolic and diastolic blood pressure, implies an increased cardiovascular risk and that PP has a higher prognostic power than SBP and/or DBP by themselves.¹¹

The determining factors of arterial (pulse) pressure become immediately clear when considering the most simple mechanical equivalent of the cardiovascular system, that is, that of an antique fire hose (Figure 7-5), where the pulsatile action of the pump is attenuated by an air chamber, which transforms the pulsatile flow into an almost steady outflow at the level of the nozzle.^8,12 The pump obviously stands for the heart, while the air chamber stands for the buffering function of the large elastic arteries (the “total arterial compliance,” C). The nozzle represents the total vascular resistance (R), generated mainly in the arterioles and capillaries. This model was first suggested by reverend Stephen Hales (1733).¹³ In 1899, Otto Frank translated this into a mathematical formulation,^14,15 and it was Frank who introduced the terminology “windkessel models,” windkessel being the German word for air chamber, after the buffer chamber used in the antique fire hose. Assuming venous pressure to be negligible compared to arterial pressure, the arteriovenous pressure difference approximates mean arterial pressure (MAP), which can be deduced using the Ohm’s law as the product of cardiac output (CO) and total vascular resistance: MAP = CO × R. The PP is (mainly) determined by the capacity of the air chamber and the stroke volume⁸ (SV): PP ≈ SV/C. It is clear that a loss of arterial compliance gives rise to an increase in PP.

The windkessel model is, however, too simple an approximation of the arterial system. In physiological terms, it is much more accurate to consider it as a (complex topological) network of elastic arteries. When the left or right ventricle ejects its stroke volume into the systemic or pulmonary circulation, it generates pressure and flow waves that travel within the respective arterial networks. Measurements of pressure and flow in mammals and humans along the aorta and arterial tree in the second half of 20th century evidenced the existence of these waves. From the ascending to abdominal aorta, for instance, there is a measurable time delay between the onset of rise of pressure (or flow), and the wavefront (rising slope) of the pressure is steeper at the abdominal aorta. Furthermore, it was found that the peak of the pressure wave (systolic blood pressure) is higher at the abdominal than at the ascending aorta.¹⁶ These observations can only be explained by wave reflection since in the absence of wave reflection the dissipation would lead to less steep wavefronts, attenuation of maximal pressure, and one would also expect similarity of pressure and flow wave morphology, which is certainly not the case. The existence of wave reflection is, of course, not surprising given the complex anatomical structure of the arterial tree, with geometric and elastic tapering (the further away from the heart, the stiffer the vessel), its numerous bifurcations, and the arterioles and capillaries making the distal terminations.

As such, the blood pressure waveform measured at any location within the arterial tree is the result of the superposition of forward and backward running pressure waves. Besides the magnitude of the reflected waves, the timing of their arrival in the central aorta is a major determinant of the morphology and amplitude of the pressure wave (Figure 7-6). In young subjects, arteries are most elastic (deformable) and the pulse wave velocity (PWV—the speed with which a pressure or flow wave propagates through an artery) is much lower than in older subjects, or in patients with hypertension or diabetes. In the young, it takes more time for the forward wave to travel to the reflection site and for the reflected wave to arrive at the ascending aorta. Both interact only in late systole (late systolic inflection point), causing little extra load on the heart (this waveform was called a “C-type” waveform by Murgo et al.). In older subjects, on the other hand, PWV is much higher, and the inflection point shifts to early systole. The early arrival of the reflected wave literally boosts systolic pressure, causing an augmentation of systolic and PP.¹⁷ Murgo et al. classified this waveform type as an “A-type” waveform.

Since the pulse wave velocity is directly related to the stiffness of the arterial tree, the overall conclusion is that, irrespective of the way the arterial tree is being studied (as a windkessel model or as an elastic tube system), an increase in arterial stiffness results in an increase in systolic blood pressure and PP. Given their association with an increased cardiovascular risk, there has been an increased interest in measuring arterial stiffness and wave reflection in the past two to three decades and also because an increase in arterial stiffness and/or wave reflection may be an early, intermediate phenotype of atherosclerosis.¹⁸

Pressure and Flow Waves as a “Series of Harmonics”

In the pioneering studies on arterial hemodynamics, the analysis of the arterial system and the (invasively measured) pressure and flow waveforms was based on parallelisms between hemodynamics and the electrical network and electrical transmission line theory. This also explains why models of the arterial system are often represented as an “electrical analog” model, consisting of resistors, inductors, and/or capacitance elements. The electrical analog of the basic windkessel model, for instance, is a parallel connection of a resistance (total vascular resistance, R) and a capacitance (total arterial compliance, C).

In these models, blood pressure is the analog of electrical voltage, and flow is the analog of electrical current. An electrical signal is generally considered as the superposition of a steady, direct current (the DC signal) and an alternating, oscillatory current (the AC signal), which is a pure sinusoidal wave (the mean value is zero) with a given frequency. Although arterial pressure and flow waves are approximately periodic (assuming a regularly beating heart), the flow is pulsatile and certainly not oscillatory.

Luckily, the Fourier theorem resolves this problem. Using this theorem, the pulsatile pressure and flow waves can be transformed into a “Fourier series,” consisting of a steady component and sinusoidal waves with an incremental frequency (“harmonics”) that, superimposed, yield the original pulsatile wave. In general, 10 to 15 harmonics are sufficient to describe hemodynamic variables such as pressure or flow. The frequency of the first (fundamental) harmonic is the cardiac frequency, and the higher harmonics have a frequency that is a natural multiple of the fundamental frequency. Assuming that the superposition principle is valid (“the sum of the effects of all individual harmonics is the same as the effect of the superimposed harmonics”), the hemodynamic analysis can be performed for a given harmonic with a given frequency. Thus, instead of considering the measured pressure and flow wave, individual harmonics are studied, and the properties of the arterial tree are displayed as a function of frequency.

The sinusoidal waves or harmonics can be described by their amplitude and their “phase angle,” which can mathematically be represented using a complex formalism (Figure 7-7). For the nth harmonic, the pressure (P) and flow (Q) component can be written as

where |P_n| and |Q_n| are the amplitudes (or modulus) of the pressure and flow sine waves, having phase angles Φ_Pn and Φ_Qn (to allow for a phase lag in the harmonic), respectively. Time is indicated by t and ω is the fundamental angular frequency, given by 2π/T with T the duration of a heart cycle (RR interval). For a heart rate of 75 beats/min, T is 0.8 seconds. The fundamental frequency is 1.25 Hz, and ω becomes 7.85 rad/s.

The Impedance of the Arterial Tree

Impedance is a general measure of the “resistance to flow” and is defined as the ratio of pressure and flow: Z = P/Q. As it also accounts for pulsatile flow, it surpasses the definition of systemic vascular resistance, which only accounts for mean pressure and flow. Z has the dimensions of mm Hg mL⁻¹ s (kg m⁻⁴ s⁻¹ in SI units; dyn cm⁻⁵ s in older units) if pressure is expressed in mm Hg and flow in mL s⁻¹. Z is calculated for each individual harmonic, and displayed as a function of frequency (Figure 7-8). Since both P and Q are complex numbers, Z is complex as well and also has a modulus and a phase angle, except for the steady (“DC”) component at 0 Hz, which is nothing but the ratio of mean pressure and mean flow, that is, the value of total vascular resistance. For all higher harmonics, the modulus of the nth harmonic is given as |Z_n| = |P_n| / |Q_n|, and its phase Φ_Zn is given as Φ_Pn − Φ_Qn.

Arterial impedance requires simultaneous measurement of pressure and flow at the same location. Although it can be meaningfully calculated at any location along the arterial tree, it is most commonly measured at the entrance of the systemic or pulmonary circulation and is called “input” impedance, often denoted as Z_in. Z_in fully captures the relation between pressure and flow, and is determined by all downstream factors influencing this relation (arterial network topology, branching patterns, stiffness of the vessels, vasomotor tone, …). It provides a powerful description of the arterial circulation, since it captures all effects, but this is at the same time its greatest weakness, as Z_in is not very sensitive to local changes in arterial system properties, such as focal atherosclerotic lesions.

It is not always easy to give an interpretation to an impedance pattern. This is, however, facilitated when one considers the impedance of basic electrical or mechanical “building blocks”^19,20,21:

• 
  

  When a system behaves strictly resistive, there is a linear relation between the pressure (difference) and flow. Pressure and flow are always in phase, Z_in is a real number, and Φ_Z is zero.

  
  
• 
  

  In case there is only inertia in a system, pressure is ahead of flow; for sine waves, the phase difference between both is a quarter of a wavelength, or +90 degrees in terms of phase angle.

  
  
• 
  

  In the case that the system behaves like a capacitor, flow is leading pressure, again 90 degrees out of phase, so that Φ_Z is –90 degrees.

Figure 7-8 displays the input impedance calculated from a measured aorta pressure and flow waveform. The impedance modulus drops from the value of total systemic vascular resistance at 0 Hz to much lower values at higher frequencies. The phase angle is negative up until the fourth harmonic, showing that capacitive effects dominate at these low frequencies, although the phase angle never reaches –90 degrees, indicating that inertial effects are present as well. For higher harmonics, the phase angle is close to zero, or at least oscillating around the zero value. At the same time, the modulus of Z_in is nearly constant. For these high frequencies, the system seems to act as a pure resistance, and the impedance value, averaged over the higher harmonics (typically from the third to the 10th harmonic), has been termed the characteristic impedance (Z₀).

Input Impedance Explained from the Viewpoint of Wave Transmission and Reflection

The above-mentioned characteristic impedance, Z₀, is an important property of the arterial tree that deserves some further attention. In the hypothetical case that the arterial tree were an infinitely long and uniform tube, wave reflection would be totally absent. Under these conditions, the relation between pressure and flow harmonics would be solely dependent on the geometric and mechanical properties of the tube itself. The ratio of pressure and flow harmonics, Z, would be a constant—the characteristic impedance Z₀, which can be approximated as

where ρ is the density of blood (1050 kg/m³), PWV the pulse wave velocity of the tube, and A the cross-sectional area of the tube. In that hypothetical case, the input impedance thus equals the characteristic impedance. Later in the chapter we address how Z₀ can be assessed in vivo.

The arterial system, however, is not an infinitely long, uniform tube free of reflections, which is the reason the input impedance is more complex than a simple real constant. Despite the fact that it is difficult (and perhaps impossible) to fully pinpoint wave reflection in the arterial tree, the problem is often approached in a pragmatic and simple way, with the arterial tree being considered as a tube with an “effective length,”^22,23 and a single reflection site at the end. The input impedance of such a system can be calculated, as demonstrated in Figure 7-9. In that example, a uniform tube of diameter 1.5 cm and PWV of 6.2 m/s is assumed, yielding a value for Z₀ of 0.275 mm Hg mL⁻¹ s. It is further assumed that the tube has a length of 50 cm, and is ended by a (linear) resistance of 1 mm Hg mL⁻¹ s. The impedance mismatch between the tube and its terminal resistance gives rise to wave reflections and oscillations in the input impedance pattern, and many features, observed in vivo,²⁴ can be explained on the basis of this model.

Assume there is a sinusoidal wave running in the system with a wavelength λ being four times the length of the tube. This means that the phase angle of the reflected wave, when arriving back at the entrance at the tube, will be 180 degrees out of phase with respect to the forward wave. The sum of the incident and reflected wave will be minimal, since they interfere in a maximally destructive way. If this wave is a pressure wave, measured pressure at the entrance of the tube will be minimal for waves with this particular wavelength. There is a relation between PWV, λ, and frequency f, that is, λ = PWV/f. Thus, in an input impedance spectrum, the frequency f_min where input impedance is minimal corresponds to a wave with a wavelength that is equal to four times the distance to the reflection site, L: 4L = PWV/f_min, or L = PWV/4f_min and f_min = PWV/4L. Applied to the example of the tube, f_min is expected at 6.2/2 = 3.1 Hz. This equation is known as the “quarter wavelength” formula, and is used to estimate the effective length of the arterial system.