Problem 70 - Physics

Question

In the human body, blood vessels can dilate, or increase their radii, in response to various stimuli, so that the volume flow rate of the blood increases. Assume that the pressure at either end of a blood vessel, the length of the vessel, and the viscosity of the blood remain the same, and determine the factor \(R_{\text {dilated }} / R_{\text {normal }}\) by which the radius of a vessel must change in order to double the volume flow rate of the blood through the vessel.

Step-by-Step Solution

Answer

The radius must increase by a factor of \(\sqrt[4]{2} \approx 1.1892\).

Step 1: Understanding the Relationship

The volume flow rate of blood flowing through a vessel is determined by Poiseuille's Law, which is given by the formula: \[Q = \frac{\pi \, \Delta P \, R^4}{8 \, \eta \, L}\]where \(Q\) is the volume flow rate, \(\Delta P\) is the pressure difference, \(R\) is the radius of the vessel, \(\eta\) is the viscosity of the blood, and \(L\) is the length of the vessel. Since \(\Delta P\), \(\eta\), and \(L\) are constant, the flow rate \(Q\) is proportional to the fourth power of the radius \(R\). This implies \(Q \propto R^4\).

Step 2: Solving for the New Radius

We want the volume flow rate \(Q\) to double, i.e., \(Q_{\text{new}} = 2Q_{\text{normal}}\). Using the proportionality \(Q \propto R^4\), we can set up the following equation for the normal and dilated radii: \[R_{\text{dilated}}^4 = 2 \, R_{\text{normal}}^4\]

Step 3: Calculating the Radius Ratio

To find the factor by which the radius changes, take the fourth root of both sides of the equation:\[R_{\text{dilated}} = \sqrt[4]{2} \, R_{\text{normal}}\]Thus, the ratio of the dilated radius to the normal radius is:\[\frac{R_{\text{dilated}}}{R_{\text{normal}}} = \sqrt[4]{2}\]

Step 4: Conclusion

The factor by which the radius of the vessel must increase to double the volume flow rate is \(\sqrt[4]{2}\). Numerically, \(\sqrt[4]{2} \approx 1.1892\), indicating a roughly 18.92% increase in the radius.

Key Concepts

Blood Flow

Blood flow is a critical concept in the study of the cardiovascular system. It refers to the movement of blood through the vessels, which is essential for delivering nutrients and oxygen to, and removing waste products from, tissues throughout the body. The efficiency of blood circulation is influenced by several factors, especially the diameter or radius of the blood vessels. According to Poiseuille's Law, the volume flow rate of blood can be expressed as:

• \( Q = \frac{\pi \, \Delta P \, R^4}{8 \, \eta \, L} \)

Here, \( Q \) is the volume flow rate, \( \Delta P \) represents the pressure difference, \( R \) is the radius of the vessel, \( \eta \) is the blood viscosity, and \( L \) is the length of the vessel.

This equation highlights that blood flow is highly sensitive to changes in the radius of the vessel. A small increase in radius leads to a markedly greater blood flow due to the fourth power relationship. This means that if a vessel's radius is increased, blood flow increases significantly, which is vital for regulating how much blood reaches different parts of the body.

Vessel Dilation

Vessel dilation is the process by which blood vessels enlarge, increasing their radius. This physiological adjustment can occur in response to various stimuli, such as exercise, heat, or changes in metabolic demand. Within the context of Poiseuille's Law, vessel dilation plays a crucial role in enhancing blood flow. If other factors like pressure difference and blood viscosity are held constant, increasing the vessel's radius is the most effective way to boost blood flow without increasing heart workload.

• When a vessel dilates, its radius \( R \) is multiplied by a factor, which in turn increases the blood flow without a proportional increase in pressure.
• This process is particularly important during physical exertion when muscles require more oxygen.

The equation given in the exercise, \( \frac{R_{\text{dilated}}}{R_{\text{normal}}} = \sqrt[4]{2} \), shows that a small increase of approximately 18.92% in vessel radius can double blood flow. This illustrates the body's efficient response to increased demand for oxygen and nutrients.

Viscosity

Viscosity refers to the thickness or internal resistance to flow within the blood. It is a key factor in determining how easily blood can move through the circulatory system. Higher viscosity implies greater resistance to flow, making it harder for the heart to pump blood efficiently. Conversely, lower viscosity means blood moves more readily, facilitating improved flow dynamics.

In Poiseuille's Law:

• Viscosity is represented by \( \eta \), directly influencing the denominator of the formula \( Q = \frac{\pi \, \Delta P \, R^4}{8 \, \eta \, L} \).
• This means that reductions in blood viscosity can enhance flow by decreasing the resistance blood experiences as it navigates through vessels.

While viscosity was held constant in the original exercise, understanding its role is crucial when considering broader physiological contexts. Modifications in blood viscosity, through changes in temperature or hydration levels, for example, can also significantly affect blood flow efficiency.