Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Poiseuille's law states that the resistance \(R\), measured in dynes, of blood flowing in a blood vessel of length \(l\) and radius \(r\) (both in centimeters) is given by $$R=f(l, r)=\frac{k l}{r^{4}}$$ where \(k\) is the viscosity of blood (in dyne-sec/cm \(^{2}\) ). What is the resistance, in terms of \(k\), of blood flowing through an arteriole \(4 \mathrm{~cm}\) long and of radius \(0.1 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

The resistance of blood flowing through an arteriole 4 cm long and of radius 0.1 cm is \(R = 40,000k\), where k is the viscosity of the blood in dyne-sec/cm².

Step 1: Write down the given information

We are given: - Length of arteriole: \(l = 4\, cm\) - Radius of arteriole: \(r = 0.1\, cm\) - Poiseuille's law: \[R = \frac{k l}{r^4}\]

Step 2: Plug in values and calculate resistance

Now, we substitute the given values for \(l\) and \(r\) into Poiseuille's law to find the resistance: \[R = \frac{k \cdot 4}{(0.1)^4}\]

Step 3: Simplify and write the final answer

Now, we simplify the expression to get the final answer: \[R = \frac{4k}{0.0001}\] \[R = 40,000k\] So, the resistance of blood flowing through an arteriole 4 cm long and of radius 0.1 cm is \(R = 40,000k\), where k is the viscosity of the blood in dyne-sec/cm².

Key Concepts

Fluid Dynamics

Fluid dynamics is a branch of physics concerned with the movement of fluids (liquids and gases) and the forces acting on them. Understanding fluid dynamics is crucial in the field of medical science, especially when studying blood flow through the circulatory system.

In our case, Poiseuille's Law provides a mathematical framework to describe how blood flows through vessels. It considers blood a fluid with specific properties, such as viscosity, and vessel characteristics, such as length and radius. By doing so, we can calculate the resistance fluid encounters due to these properties, which is a critical factor in understanding conditions such as hypertension (high blood pressure) and atherosclerosis (narrowing of the arteries).

The law particularly applies to laminar flow, which is a type of flow where the fluid moves in parallel layers with minimal disruption between them, important for blood flow in smaller vessels like arterioles and capillaries. Deviations from laminar flow can indicate medical conditions or the presence of obstructions within blood vessels.

Vascular Resistance

Vascular resistance is the resistance that blood vessels impose on the circulatory flow of blood. It's a key concept in cardiovascular physiology, as it contributes significantly to blood pressure and flow within the body.

Using Poiseuille's Law, we understand that resistance depends on both the physical properties of blood, such as viscosity (denoted by the symbol 'k'), and the dimensions of the blood vessel - including its length (l) and radius (r). An interesting aspect of this law is the profound effect of the vessel's radius on resistance, as it's raised to the fourth power in the equation. This means even small changes in the radius can majorly impact the resistance and thereby the blood flow.

In the exercise, calculating the resistance for a particular arteriole could be crucial for understanding how alterations in vessel size or blood properties could affect overall vascular resistance and patient health. This insight can help medical professionals manage and treat cardiovascular diseases.

Applied Mathematics

Applied mathematics engages with mathematical methods to solve practical problems in science, engineering, business, and other areas. Here, it combines with fluid dynamics to provide solutions related to blood flow and vascular resistance.

The step-by-step solution demonstrates how applied mathematics can simplify complex real-world phenomena into an equation that can be strategically manipulated to find a solution. By substituting the realistic values, such as vessel length and radius, into Poiseuille's Law, we can mathematically derive a value for the resistance to blood flow, expressed in terms that relate directly to physiological conditions.

This approach is an excellent example of how mathematical modeling is applied in healthcare, helping us understand biological systems quantitatively. Accurate calculation is essential, as it affects diagnostics and treatment strategies in cardiology and other medical fields.