6.1 Pressure and flow in the cardiovascular system
6.1.1 Arterial anatomy
Inspired by the presentation in [5], an overview of the of the arterial anatomy is given in this section. The aorta is the main artery with a proximal diameter of 2-3 cm in the human adult (see (54)). Originating from the left ventricle, the aorta is separated from the ventricle lumen during diastole,whereas there is direct communication between the left ventricle and the aorta during systole.
Figure 51: The systemic arteries
The aorta tapers and the distal aortic diameter is about 1 cm. Normally, three main branches are found on the top of the aortic arch, bifurcating into the arteries towards the head and the upper limbs. The descending aorta is divided into the thoracic and the abdominal part as the aorta crosses the diaphragm. In the abdomen, important aortic branches provide blood flow to the kidneys (constituting about 15\% of the blood volume) and the gastro-intestinal organs. At the distal end, the aorta bifurcates into two iliac arteries, providing blood supply to the lower extremities.
Towards the head, the most important branches are the left and right common carotid arteries, which bifurcate further into the internal and external carotid arteries. These arteries are superficial, and are therefore readily accessible for non-invasive measurements of pressure and flow. The pressure pulsation can easily be felt in then neck. Blow supply to the upper limbs are provided by the subclavian artery, which changes name to the brachial artery in the upper arm, and bifurcates at the elbow to the radial and the ulnar artery. These arteries are the most commonly used sites for non-invasive pressure recordings: the brachial for sphygmomanometric measurements; the radial and subclavian arteries for tonometric measurements.
In the abdomen, the most important branches are those which supply the gastro-intestinal organs with blood; hepatic artery to the liver; gastric artery to the stomach, splenic artery to the spleen, the superior and inferior mesenteric arteries to the intestines, and the renal arteries to the kidneys. The renal arteries captures up to 15\% of the total stroke volume, whereas all abdominal arteries receives about 40\% of the total stroke volume.
Figure 52: Propagation of pressure and flow waves in the systemic circulation of a dog (adopted from [6])
During systole, the ventricle ejects blood into the arteries, giving rise to pressure and flow waves traveling from the heart towards the periphery (see figure 52). The pressure peak is delayed with increasing distance from the heart, indicating wave propagation along the aorta with a finite pulse wave velocity or wave speed. Further, the pressure wave is characterized by a progressively increasing amplitude (systolic pressure), and a steepening of the front [7]. Only a moderate fall in the mean pressure with increasing distance is observed, indicating minor effects of viscous friction. The viscous effects are limited to only a few mmHg in the large and mid-sized arteries (e.g., radial artery).
The rise in the pressure amplitude on the other hand, is determined by the varying local characteristic impedance (i.e., elastic properties of the arteries and the vessel dimensions), as well as heart rate and stroke volume. As we will learn in more detail in the chapter 7 Blood flow in compliant vessels; any change in the characteristic impedance, which depends on the local pulse wave velocity (or compliance) and the cross-sectional area of the vessels, will give rise to wave reflections.
For now we settle for the explanation that the observed wave phenomenon in the arterial tree is a direct consequence of the locally varying distensibility or compliance (see the section 6.1.2 Compliance and distensibility) of the arterial wall, allowing for a partial and temporary storage of the blood ejected from the heart.
In accordance with the increase in pressure amplitude, the flow wave is damped progressively due to the reflections, and to the buffering capacity (or compliance) of the arteries towards the periphery. In total, the pulsatile flow in the aorta is transformed to a quasi stationary flow at the arteriolar and capillary level.
The peak value of the pressure, measured at any location, is called the systolic pressure \( p_s \), the minimal pressure is denoted the diastolic pressure \( p_d \). These values are frequently measured with a sphygmomanometer at the upper arm (brachial artery).
However, it is clear from the previous paragraphs, that these values are not constant over the arterial tree. The mean arterial pressure \( p_m \) may be estimated from \( p_s \) and \( p_d \) at the brachial artery from the formula: $$ \begin{equation} p_m = p_d + \frac{p_s - p_d}{3} = p_d + \frac{pp}{3} \tag{6.1} \end{equation} $$ where the pulse pressure \( pp=p_s-p_d \), has been introduced.
The blood pressure is traditionally expressed in mmHg (1 mmHg = 133.3 Pa).
Figure 53: Spatial variation of pressure, mean velocity and cross sectional area for the systemic and pulmonary circulatory system (adopted from [8]).
The mean arterial pressure is mainly determined by the peripheral resistance in the smaller arteries/arterioles/capillaries, whereas the systolic and diastolic pressures result from the characteristic impedance (i.e., elastic and geometric properties) of the large arteries (wave reflection), heart rate and stroke volume.
Spatial pressure, mean velocity and cross-sectional variations in the systemic and pulmonary circulatory systems are illustrated in figure 53. Only a slight pressure decrease from the aorta (at approximately 100 mmHg) to the smaller arteries. The major part of the pressure drop takes part at the arterioles and the capillaries, yielding a venous pressure of about 5 mmHg. The arterioles and capillaries are therefore commonly referred to as the resistance vessels, whereas the large arteries are referred to as the conduit vessels. The total peripheral resistance of all systemic capillaries, \( R \) is calculated from the mean arterial pressure \( p_m \), the venous pressure \( p_v \) and the mean flow rate \( Q \) (or cardiac output \( CO \)): $$ \begin{equation} R = \frac{p_m - p_v}{Q_m} \approx 60 \frac{p_m}{CO} \tag{6.2} \end{equation} $$ As an example: a typical mean pressure of \( p_m \) = 100 mmHg and \( CO \) = 6 l/min, correspond to a total peripheral resistance of R = 1 mmHg/(ml/s). Note that the right heart operates at a much lower pressure than the left heart. Consequently the wall thickness of the left ventricle will be larger than that of the right ventricle, to maintain the wall stresses in the myocardium at approximately the same levels.
Despite the bifurcating anatomy of the aorta and the progressive decrease of the volumetric flow rate at greater distance from the heart, the mean blood velocity is more or less constant, due to the geometric tapering of the aorta (see figure 54)
Figure 54: Illustration of the aortic tapering (adopted from [9]) and the progressive increase in pulse wave velocity (adopted from [10])
Apart from the geometric tapering of the aorta, the aorta also exhibits a progressive increase in the wall stiffness and accordingly, an increase in the pulse wave velocity (see figure 54). This property is a consequence of an increased radius/wall thickness ration and an intrinsic increase in the aortic elasticity modulus.
6.1.2 Compliance and distensibility
Figure 55: Distensibility \( D \) and compliance \( C \) in a population of normotensive (open circles) and hypertensive (filled circles). Adopted from [11].
The compliance and distensibility of a blood vessel is a local property depending on the local transmural pressure and the elastic properties of the vessel wall. The relation between the cross-sectional area \( A \) and the transmural pressure \( p \) is nonlinear and can be rather complicated. Moreover, it varies from one vessel to another. Important quantities used to describe this relation are the compliance which is defined as: $$ \begin{equation} C = \partd{A}{p} \tag{6.3} \end{equation} $$ and the distensibility \( D \) given by: $$ \begin{equation} D = \frac{1}{A} \, \partd{A}{p} = \frac{C}{A} \tag{6.4} \end{equation} $$
In the sequel these quantities will be related to the material properties of the arterial wall. For thin walled vessels, with radius % \( a \), cross sectional area \( A \), and wall thickness \( h \) one may show that (see equation (7.107)): $$ \begin{equation} C = 2 \frac{a}{h} \frac{A}{\eta} \qquad \text{and} \qquad D = 2 \frac{a}{h} \frac{1}{\eta} \tag{6.5} \end{equation} $$ where \( \eta \) denotes the Youngs modulus of the vessel wall. Thus, we see that appart from the material properties of the vessel represented by \( \eta \), the geometric properties \( (a,h) \) also play an important role. The value of the the radius/wall thickness ratio \( a/h \) varies strongly along the arterial tree.
From equation (6.5) we see that the compliance \( C \) is a local property of the vessel wall, which depends both on the geometry (\( \frac{a}{h} \) and \( A \)) and the mechanical property of the vessel \( \eta \).
Subject to the assumption of a linear material, the distensibility \( D \) seem to be somewhat more of a mechanical property from equation % (6.5), as \( D \) is not explicitly depending of cross-sectional area \( A \). But even for this simple material law, \( D \) still depends on the radius/wall thickness ratio, and may thus not be regarded as a strict mechanical property.
Arteries are know to exhibit a nonlinear pressure-area relationship, leading to nonlinear compliance and distensibility. This should be kept in mind for the comparison of normotensive and hypertensive patients, as illustrated in figure 55, adopted from [11]. With respect to their own mean pressures, hypertensives (mean 90 mmHg) have lower compliance \( C \) and distensibility \( D \) than normotensives.
Figure 56: Pressure and flow in the aorta based on the data given in the table below.
6.1.3 Mathematical representation of periodic pressure and flow
The flow is driven by the pressure gradient and thus locally determined by the propagation of the pressure wave. Due to the rhythmic contractions of the heart, the pressure will have a periodic character. To describe the flow phenomena we distinguish between the steady and the unsteady part of the pulses generated by the heart. Often it is assumed that the unsteady part may be described by linear theory, in order to introduce pressure and flow waves as superpositions of several harmonics: $$ \begin{equation} p = \sum_{n=0}^N p_n \, e^{j\omega_n t} \quad \text{and} \quad q = \sum_{n=0}^N q_n \, e^{j\omega_n t} \tag{6.6} \end{equation} $$ where \( p_n \) and \( q_n \) are complex Fourier coefficients, and \( \omega_n = n \, \omega_1 \) and \( \omega_1 \) represents the angular frequency of the fundamental harmonic of the functions (signals) \( p \) and \( q \). Note that the functions are implicitly assumed to be periodic with the representation in equation (6.6), and that the steady parts of the pressure and flow are represented by their average values \( p_0 \) and \( q_0 \), respectively. Further, a complex notation had been used in equation (6.6) in which: $$ \begin{equation} e^{j\omega t} = \cos (\omega t) + j \sin (\omega t) \tag{6.7} \end{equation} $$ with \( j = \sqrt{-1} \).
| \( n \) | \( \vert p\vert \) | \( \angle(p) \) | \( \vert Q\vert \) | \( \angle(Q) \) |
| 0 | 110 | 0 | 85 | 0 |
| 1 | 202 | -0.78 | 18.6 | -1.67 |
| 2 | 156 | -1.50 | 8.6 | -2.25 |
| 3 | 103 | -2.11 | 5.1 | -2.61 |
| 4 | 62 | -2.46 | 2.9 | -3.12 |
| 5 | 47 | -2.59 | 1.3 | -2.91 |
| 6 | 42 | -2.91 | 1.4 | -2.81 |
| 7 | 31 | 2.92 | 1.2 | 2.93 |
| 8 | 19 | 2.66 | 0.4 | -2.54 |
| 9 | 15 | 2.73 | 0.6 | -2.87 |
| 10 | 15 | 2.43 | 0.6 | 2.87 |
Table: The first 10 harmonics of the pressure and flow in the aorta (adopted from [7]).
The actual pressure and flow may be obtained by taking the real part of these complex functions. Representing a function of time as the real part of a complex or exponential expression simplifies the calculations to such an extent that it is widely used in hemodynamics, though it must be remembered that it can be applied only to sinusoidal (periodic) components obtained by frequency analysis, not to natural pulsations [7]. Vascular impedance at a given frequency, for example, is defined as the ratio of complex pressure to complex flow (see the section 6.1.4 Vascular impedance), the latter being complex exponential expressions whose real part are harmonics of the observed pressure and flow. In many situations from 6 to 10 harmonics are sufficient to represent most of the features of the pressure and flow waves in the cardiovascular system. The table above is adopted from [7] [3] and represents the modulus and phase for the first 10 harmonics of pressure and flow in the aorta. The corresponding pressure and flow are given in figure 56.
6.1.4 Vascular impedance
Figure 57: A typical \( Z_i \) for a young healthy adult (from [6]).
Based on the assumption that the relation between pressure and flow can be represented by a linear theory, as outlined in the section 6.1.3 Mathematical representation of periodic pressure and flow,vascular impedance is introduced to represent the relation between corresponding harmonics of pressure and flow.
Thus, one may roughly think of impedance as some frequency dependent resistance. The concept of impedance is of great importance in biofluid dynamics and has many different areas of application. Therefore, four different definitions of impedance have been introduced [6].
Characteristic impedance:
The characteristic impedance \( Z_c \), and the wave speed are two important parameters characterizing the wave transmission and reflection properties in a blood vessel [12]. These two quantities can be derived from the linearized mass and momentum equations for compliant blood vessel flow (see the section 7.3.7 Characteristic impedance). These linearized and simplified equations are similar to those for so-called transmission-line theory, and thus the involved physics has analogies to what happens in telegraph lines, or antenna cables, for the transmission of electromagnetic waves. The characteristic impedance is defined as the ratio of the forward propagating pressure and the forward propagating flow and has (Under certain simplifying assumptions, see the section 7.3.7 Characteristic impedance) the representation: $$ \begin{equation} Z_c = \frac{\rho c}{A} \tag{6.8} \end{equation} $$
where \( c \) represents the wave speed, \( \rho \) the fluid density, and \( A \) the vessel cross-sectional area. Note that this quantity is not influenced by wave reflections, and is thus a local characteristic property of the vessel wall.
Input impedance \( Z_i \) and terminal impedance: \( Z_T \)
The input impedance \( Z_i \) is defined in a very similar manner as the characteristic impedance, namely as the ratio of the pulsatile components of pressure and flow: % $$ \begin{equation} Z_i(\omega_n) = \frac{p_n(\omega_n)}{q_n(\omega_n)} \tag{6.9} \end{equation} $$ Importantly, the input impedance is not restricted to unidirectional waves, i.e., reflected wave components are included. Thus, \( Z_i \) is a global quantity that characterize the properties distal (downstream) to the point of measurement. The cumulative effect of all distal contributions is incorporated in the input impedance. In the aorta \( Z_i \) represents the afterload on the heart.
In Figure 57 the input impedance of a young healthy subject is depicted. For high frequencies the phase angles are close to zero, as high frequency components are more damped and reflections tend to cancel out. The negative phase angle for the low-frequency components correspond to that flow components lead pressure, i.e., the aorta first sees flow and the pressure.
The terminal impedance \( Z_T \) is defined in the same way as the input impedance, the only difference being that the input impedance normally alludes to the aorta, whereas the terminal impedance may be measured to represent the load anywhere in the vascular tree.
Longitudinal impedance: \( Z_l \)
It is the pressure gradient which drives the flow, and the relation of the two is expressed by the longitudinal impedance: $$ \begin{equation} Z_l = \partd{\hat{p}}{z}/\hat{q} \tag{6.10} \end{equation} $$ The longitudinal impedance is a complex number defined by the complex pressure gradient and complex flows. It can be calculated by frequency analysis of the pressure gradient and the flow that have been measured simultaneously. \( Z_l \) plays an important role in the characterization of vascular segments. In the chapter 5.6.3 Longitudinal impedance for pulsatile flow in straight tubes, \( Z_l \) is derived analytically for pulsatile rigid pipe flow using Womersley theory. For small Womersley numbers \( \alpha \) the we get that \( Z_l \approx Z_p = 8 \mu/\pi a^4 \), i.e., an Poiseuille resistance, whereas for large \( \alpha \) the \( Z_l \approx j \alpha^2/8 \).
Transversal impedance: \( Z_t \)
The compliance of an arterial segment is characterized by the transverse impedance defined by: $$ \begin{equation} Z_t = \hat{p}/\partd{\hat{q}}{z} \approx -\hat{p}/j\omega \hat{A} \tag{6.11} \end{equation} $$ The transverse impedance expresses the flow drop due to the storage of the vessel resulting from the radial motion of the wall [3], which in turn is caused by the pressure \( \hat{p} \). From the mass conservation equation (7.14) for compliant vessels we have: $$ \begin{equation} \partd{A}{t} = -\partd{q}{z} \tag{6.12} \end{equation} $$ and therefore from Fourier transformation: $$ \begin{equation} j \omega \hat{A} = - \partd{\hat{q}}{z} \tag{6.13} \end{equation} $$ which may be substituted into the definition of \( Z_t \) in equation (6.11) to provide the given approximation.
4: i.e. pressure difference over the vessel wall.