(II) Assuming a constant pressure gradient, if blood flow is reduced by 65%, by what factor is the radius of a blood vessel decreased?

When a fluid flows through a circular tube, then the pressure difference is determined using Hagen-Poiseuille’s Law.

The expression for the pressure change is given by,

\(\Delta P = \frac{{8\eta QL}}{{\pi {r^4}}}\)

Here, \(\Delta P\) is the pressure difference, \(\eta \) is the viscosity of the blood and \(L\) is the length of the blood vessel.

The initial volume blood flow is \({Q_1}\).

The final volume blood flow is \({Q_2}\).

The reduction in the volume blood flow is, \(\frac{{{Q_2} - {Q_1}}}{{{Q_1}}} = - 0.65\).

The initial radius of the blood vessel is \({r_1}\).

The final radius of the blood vessel is \({r_2}\).

Using the Poiseuille’s law, the expression for the reduction in the volume flow of the blood is given by,

\(\begin{array}{c}\frac{{{Q_2} - {Q_1}}}{{{Q_1}}} = - 0.65\\{Q_2} = - 0.65{Q_1} + {Q_1}\\{Q_2} = 0.35{Q_1}\\\frac{{\Delta P\pi {r_2}^4}}{{8\eta L}} = 0.35 \times \frac{{\Delta P\pi {r_1}^4}}{{8\eta L}}\end{array}\)

Solving the above expression,

\(\begin{array}{c}{r_2}^4 = 0.35{r_1}^4\\{r_2} = 0.7692{r_1}\\\frac{{{r_2}}}{{{r_1}}} - 1 = 0.7692 - 1\\\frac{{{r_2} - {r_1}}}{{{r_1}}} = - 0.231\end{array}\)

The negative sign represents the decrease in the radius of the blood vessel and it can be written as,

\(\frac{{{r_2} - {r_1}}}{{{r_1}}} = - 23.1\% \)

Thus, the radius of the blood vessel decreases by a factor of 23.1%.