Subfield: Thermodynamics

Physics of Live Systems

Data: Normal atmospheric pressure, $P_{0}=1.013 \times 10^{5} \mathrm{~Pa}=760 \mathrm{mmHg}$

Part A.The physics of blood flow

In this part you will analyse two simplified models of blood flow in vessels.

Blood vessels are approximately cylindrical in shape, and it is known that for a steady, non turbulent flow of an incompressible fluid in a rigid cylinder, the difference in pressure of the fluid at the two ends of the cylinder is given by

$$
\Delta P=\frac{8 \ell \eta}{\pi r^{4}} Q
\tag{1}
$$

where $\ell$ and $r$ are the length and radius of the cylinder, $\eta$ is the fluid viscosity and $Q$ is the volumetric flow rate, i.e. the fluid volume that passes the cylinder cross section per unit time. This expression is often able to provide the correct order of magnitude for the pressure difference in a vessel, even without taking into account the pulsatile flow, the vessel's compressibility and irregular shape, and the fact that blood is not a simple fluid but a mixture of cells and plasmA.Moreover, this expression has the same form as Ohm's law, with the volumetric flow rate being interpreted as a current, the difference in pressure as a voltage, and the factor $R=\frac{8 \ell \eta}{\pi r^{4}}$ as a resistance.

Consider for example the symmetrical network of arterioles (small arteries) depicted in Figure 1 that delivers blood to the capillary bed of a tissue. In this network, at each bifurcation a vessel is divided in two identical vessels. However, the vessels of higher levels are thinner and shorter: consider that the radii and lengths of vessels in two consecutive levels, $i$ and $i+1$, are related by $r_{i+1}=r_{i} / 2^{1 / 3}$ and by $\ell_{i+1}=\ell_{i} / 2^{1 / 3}$.

<img_4505>

Level $\quad \begin{array}{lllllll}0 & 1 & 2 & 3 & \cdots & N-1\end{array}$

Figure 1. Network of arterioles.
Context question:
A.1 Obtain an expression for the volumetric flow rate, $Q_{i}$, in a vessel at any level $i$, as a function of the total number of levels $N$, of the viscosity $\eta$, of the radius $r_{0}$ and length $\ell_{0}$ of the first vessel, and of the difference $\Delta P=P_{0}-P_{\text {cap }}$ between the pressure at the arteriole at level $0, P_{0}$, and the pressure at the capillary bed, $P_{\text {cap }}$.
Context answer:
\boxed{$Q_{i}=\Delta P \frac{\pi r_{0}^{4}}{2^{i+3} N \ell_{0} \eta}$}


Question: A.2 Calculate the numerical value of the volumetric flow rate $Q_{0}$ of the arteriole at level 0 , if its radius is $6.0 \times 10^{-5} \mathrm{~m}$ and its length is $2.0 \times 10^{-3} \mathrm{~m}$. Consider that the pressure at the arteriole inlet is $55 \mathrm{mmHg}$ and the vessel network has $N=6$ levels linking this arteriole to the capillary bed at the pressure $30 \mathrm{mmHg}$. Consider that the blood viscosity is $\eta=3.5 \times 10^{-3} \mathrm{~kg} \mathrm{~m}^{-1} \mathrm{~s}^{-1}$. Express your result in $\mathrm{ml} / \mathrm{h}$.

Solution:
Replace values in the formula and change units appropriately

$$
\begin{aligned}
Q_{0} & =\frac{\Delta P \pi r_{0}^{4}}{8 N \ell_{0} \eta}= \\
& =\frac{(55-30) \times 1.013 \times 10^{5} \times 3.1415 \times\left(6.0 \times 10^{-5}\right)^{4}}{760 \times 48 \times 2.0 \times 10^{-3} \times 3.5 \times 10^{-3}}=4.0 \times 10^{-10} \mathrm{~m}^{3} / \mathrm{s}
\end{aligned}
$$

Final answer: $1.5$ $\mathrm{~m} \ell / \mathrm{h}$