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figure a

Description

With laminar and steady flow through a uniform tube of radius r i , the velocity profile over the cross-section is a parabola. The formula that describes the velocity (v) as a function of the radius, r is:

$$ {v}_{r}=\frac{\Delta P \cdot \text ({r}_{i}-{r}^{2})}{4 \cdot \eta \cdot \text l}={v}_{\mathrm{max}}(1-{r}^{2}/{r}_{i})$$

ΔP is the pressure drop over the tube of length (l), and η is blood viscosity. At the axis (r = 0), velocity is maximal, v max , with v max = ΔPr i 2/4ηl, while at the wall (r = r i ) the velocity is assumed to be zero. Mean velocity is:

$$ {v}_{mean}=\Delta P \cdot \text {r}_{i}^{2}/8\text h \cdot \text l=v_{\mathrm{max}}\text /2=Q/\pi {r}_{i}^{2}$$

and is found at r ≈ 0.7 r i .

Blood flow (Q) is mean velocity, v mean , times the cross-sectional area of the tube, πr i 2, giving:

$$ Q=\Delta P \cdot \pi \cdot \text {r}_{i}^{4}/8 \cdot \text h \cdot \text l$$

This is Poiseuille’s law relating the pressure difference, ΔP, and the steady flow, Q, through a uniform (constant radius) and stiff blood vessel. Hagen, in 1860, theoretically derived the law and therefore it is sometimes called the law of Hagen-Poiseuille. The law can be derived from very basic physics (Newton’s law) or the general Navier-Stokes equations.

The major assumptions for Poiseuille’s law to hold are:

  • The tube is stiff, straight, and uniform

  • Blood is Newtonian, i.e., viscosity is constant

  • The flow is laminar and steady, not pulsatile, and the velocity at the wall is zero (no slip at the wall)

In curved vessels and distal to branching points the velocity profile is not parabolic and the parabolic flow profile needs some length of straight tube to develop, this length is called inlet length (Fig. 2.1). The inlet length, l inlet , depends on the Reynolds number (Re, see Chap. 4) as:

$$ {l}_{inlet}/D\approx 0.06\: Re$$

with D vessel diameter. For the aorta mean blood flow is about 6 l/min, and the diameter 3 cm, so that the mean velocity is ∼15 cm/s. The Reynolds number is therefore ∼1,350 (Chap. 4). This means that l inlet /D is ∼80, and the inlet length ∼240 cm, which is much longer than the length of the entire aorta. In the common iliac artery the Reynolds number is about 500 and diameter ∼0.6 cm, giving an inlet length of ∼18 cm. In other, more peripheral arteries the inlet length is much shorter but their length is shorter as well. Clearly, a parabolic flow profile is not even approximated in the arterial system. Nevertheless, the law of Poiseuille can be used as a concept relating pressure drop to flow.

Fig. 2.1
figure 1

Inlet length. Flow entering a side branch results in skewed profile. It takes a certain inlet length before the velocity develops into a parabolic profile again

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A less detailed and thus more general form of Poiseuille’s law is Q =ΔP/R with resistance R being:

$$ R=8 \cdot \text h \cdot \text l/\pi {r}_{i}^{4}$$

This law is used in analogy to Ohm’s law of electricity, where resistance equals voltage drop/current. The analogy is that voltage difference is compared to pressure drop and current to volume flow. In hemodynamics we also call it Ohm’s law. Thus:

$$ \Delta P/Q=R$$

This means that resistance can be calculated from pressure and flow measurements.

Calculation of Wall Shear Stress

The wall shear rate can be calculated from the slope of the velocity profile near the wall (angle θ in Fig. 2.2), which relates to the velocity gradient, tan θ = dv/dr, near the wall (see Chap. 1). The derivative of the velocity profile gives the shear rate γ =(ΔP/l) · r/2η. Shear stress is shear rate times viscosity τ = (ΔP/l) · r/2. The shear rate at the vessel axis, r = 0, is zero, and at the wall, r = r i , it is τ = (ΔP/l) · r i /2, so the blood cells encounter a range of shear stresses and shear rates over the vessel’s cross-section.

Fig. 2.2
figure 2

The shear rate at the wall of a blood vessel can be calculated from the ‘rate of change of velocity’ at the wall, as indicated by angle θ. Relations between shear rate and flow or pressure gradient are given in the text

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The shear stress at the wall can also be calculated from basic principles (Fig. 2.3). For an arterial segment of length l, the force resulting from the pressure difference (P 1 − P 2 )=ΔP, times the cross-sectional area, πr i 2, should equal the opposing force generated by friction. This frictional force on the wall equals the shear stress, τ, times the lateral surface, 2πr i · l. Equating these forces gives, ΔP · πr i 2 = τ · 2πr i · l, and

Fig. 2.3
figure 3

Shear stress at the wall can also be calculated directly by the balance of pressure, ΔP = P 1 − P 2, and frictional forces: τ = (ΔP/l)·(r i /2)

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$$ t=(\Delta P/l) \cdot \text ({r}_{i}/2)$$

This formulation shows that with constant perfusion pressure an increase in viscosity does not affect wall shear stress.

The wall shear stress may also be expressed as a function of volume flow using Poiseuille’s law

$$ t=4 \cdot \text h \cdot \text Q/\pi {r}_{i}^{3}$$

this is a more useful formula for estimating shear stress because flow and radius can be measured noninvasively using ultrasound or MRI, whereas pressure gradient cannot.

Example of the Use of Poiseuille’s Law to Obtain Viscosity

A relatively simple way to obtain viscosity is to use a reservoir that empties through a capillary (Fig. 2.4). Knowing the dimensions of the capillary and using Poiseuille’s law viscosity can be calculated. Even simpler is the determination of viscosity relative to that of water. In that case only a beaker and stopwatch are required. The amounts of blood and water obtained for a chosen time are inversely proportional to their viscosities. The practical design based on this principle is the Ostwald viscometer.

Fig. 2.4
figure 4

A wide bore reservoir, maintaining constant pressure, provides the blood flow through a capillary. The application of Poiseuille’s law, or comparison with water, gives absolute or relative viscosity, respectively

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Murray’s Law

Murray’s law (1926) was originally proposed by Hess in 1913 and assumes that the energy required for blood flow and the energy needed to maintain the vasculature is assumed minimal [1]. The first term equals pressure times flow and, using Poiseuille’s law, this is P · Q = Q 2 · 8 · η · l/πr i 4. The second term is proportional to vessel volume and thus equals b·π r i 2 l, with b proportionality constant. The total energy, E m , is:

$$ {E}_{m}={Q}^{2} \cdot \text 8h \cdot \text l/\pi {r}_{i}^{4}+\text {b} \cdot \pi {r}_{i}^{2}l$$

The minimal value is found for dE m /dr = 0 and this leads to:

$$ Q=(\pi /4l) \cdot \text {(\text {b}/h)}^{0.5} \cdot \text {r}_{i}^{3}=\text {k} \cdot \text {r}_{i}^{3}$$

For a bifurcation it holds that

$$ {Q}_{mother}={Q}_{daughter1}+{Q}_{daughter2}$$

and thus

$$ {\text {k}}_{\text {m}}{r}_{mother}^{3}={\text {k}}_{\text {d,1}}{r}_{daughter1}^{3}+{\text {k}}_{\text {d,2}}{r}_{daughter2}^{3}$$

with two equal daughters, same radii and lengths, thus equal k’s, it holds that:

$$ {r}_{mother}^{3}=2 \cdot \text {r}_{daughter}^{3}$$

and we find that

$$ {r}_{daughter}={({\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.})}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{r}_{mother}\approx 0.79{r}_{mother}$$

The area of both daughters together is 2·0.792 ≈ 1.25 the area of the mother vessel. This area ratio is close to the area ratio predicted by Womersley on the basis of the oscillatory flow theory (Chap. 8), to obtain minimal reflection of waves at a bifurcation, namely between 1.15 and 1.33 [2]. Thus Murray’s law suggests a minimal size of blood vessels and an optimum bifurcation [1].

Physiological and Clinical Relevance

The more general form of Poiseuille’s law given above, i.e., Q =ΔP/R allows us to derive resistance, R, from mean pressure and mean flow measurement.

The wall shear stress, i.e., the shear force on the endothelial cells plays an important role in the short term, seconds to minutes, and the long term, weeks, months or years (Chap. 27). Short-term effects are vasomotor tone and flow mediated dilatation (FMD). Long-term effects are vascular remodeling, endothelial damage, changes in barrier function, and atherosclerosis (Chap. 27).

Shear stress plays a role in the embryonic development of the cardiovascular system. On the one hand shear stress, through gene expression, affects (ab)normal cardiovascular growth [3], and on the other hand it activates blood-forming stem cells [4]. It is of interest to mention that shear stress depends on vessel size and is different between similar vessels (e.g., aorta) of mammals [5].

It is still not possible to directly measure wall shear stress or shear rate in vivo. Shear rate is therefore derived from the velocity profile and blood viscosity at the wall. Velocity profiles can be measured with MRI and Ultrasound Doppler. However, the calculations to obtain shear rate require extrapolation of a flow velocity profile, because very near the wall velocity measurements are not possible. From the velocity profile, based on either Poiseuille or Womersley’s oscillatory flow theory (Chap. 8), the velocity gradient, dv/dr = γ, at the wall is then calculated. To calculate wall shear stress, τ = η γ, the blood viscosity near the wall has to be known as well, but viscosity close to the wall is not known because of plasma skimming. Plasma skimming refers to the relative absence of erythrocytes in the region near the wall. Also the diameter variation over the heartbeat is almost impossible to account for. All in all wall shear stress cannot be estimated accurately.

Wall shear stresses are ∼10–20 dynes/cm2, which is about 10,000 times less than the hoop stress (Chap. 9). Despite this enormous difference in magnitude, both stresses are equally important in the functional wall behavior in physiological and pathological conditions (see Chaps. 27 and 28).