Problem 22 - Applied Calculus

Question

The cardiac output, represented by \(c\), is the volume of blood flowing through a person's heart, per unit time. The systemic vascular resistance (SVR), represented by \(s\), is the resistance to blood flowing through veins and arteries. Let \(p\) be a person's blood pressure. Then \(p\) is a function of \(c\) and \(s\), so \(p=f(c, s)\). (a) What does \(\partial p / \partial c\) represent? Suppose now that \(p=k c s\), where \(k\) is a constant. (b) Sketch the level curves of \(p\). What do they represent? Label your axes. (c) For a person with a weak heart, it is desirable to have the heart pumping against less resistance, while maintaining the same blood pressure. Such a person may be given the drug nitroglycerine to decrease the SVR and the drug Dopamine to increase the cardiac output. Represent this on a graph showing level curves. Put a point \(A\) on the graph representing the person's state before drugs are given and a point \(B\) for after. (d) Right after a heart attack, a patient's cardiac output drops, thereby causing the blood pressure to drop. A common mistake made by medical residents is to get the patient's blood pressure back to normal by using drugs to increase the SVR, rather than by increasing the cardiac output. On a graph of the level curves of \(p\), put a point \(D\) representing the patient before the heart attack, a point \(E\) representing the patient right after the heart attack, and a third point \(F\) representing the patient after the resident has given the drugs to increase the SVR.

Step-by-Step Solution

Answer

\( \partial p / \partial c \) measures how blood pressure changes with cardiac output. Level curves represent constant blood pressure values. Graphs illustrate drug effects on cardiac output and SVR.

Step 1: Interpret \( \partial p / \partial c \)

\( \partial p / \partial c \) represents the partial derivative of blood pressure \( p \) with respect to cardiac output \( c \). This measures how blood pressure changes with a small change in cardiac output, holding systemic vascular resistance \( s \) constant.

Step 2: Understanding Level Curves

Level curves are lines on a graph where a function of two variables, in this case \( f(c, s) = p = kcs \), remains constant. Level curves for \( p \) show combinations of \( c \) and \( s \) that yield the same blood pressure value.

Step 3: Sketching Level Curves

To sketch the level curves of \( p = kcs \), plot \( s \) on the vertical axis and \( c \) on the horizontal axis. Since \( p = kcs \) can be rearranged to \( s = \frac{p}{kc} \), each level curve is a hyperbola. These hyperbolas shift as \( p \), the level value, changes.

Step 4: Representing Effects of Drugs for a Weak Heart

On the graph of level curves, illustrate the effect of drugs: moving from point \( A \) to \( B \). Point \( A \) represents the initial state. Post-medication with dopamine (increasing \( c \)) and nitroglycerine (decreasing \( s \)), the resulting state is point \( B \), lying on the same level curve but with decreased \( s \) and increased \( c \).

Step 5: Illustrating Mismanagement After a Heart Attack

On the level curve graph, mark point \( D \) for the patient's state before the heart attack, point \( E \) for immediately after (decreased \( c \) resulting in decreased \( p \)), and point \( F \) for after drugs increase \( s \). \( F \) mistakenly returns to the original blood pressure level by increasing \( s \), rather than correcting \( c \).

Key Concepts

Cardiac Output

Cardiac output is a fundamental concept in understanding how the heart functions as a pump. It refers to the volume of blood that the heart ejects in one minute. This vital measurement is denoted by \(c\) and determines how efficiently your heart circulates blood. The formula for calculating cardiac output is the heart rate multiplied by the stroke volume (the amount of blood pumped with each beat).

• Heart Rate: the number of beats per minute.
• Stroke Volume: the amount of blood pumped out with each beat.

Cardiac output is essential for ensuring the body receives sufficient oxygen and nutrients. If cardiac output is low, organs and tissues may not receive enough blood flow, leading to fatigue and potentially life-threatening conditions. In medical scenarios, medications such as dopamine can be administered to increase cardiac output, especially when the heart is too weak to maintain adequate blood circulation on its own.

Systemic Vascular Resistance

Systemic Vascular Resistance (SVR) is a key concept related to the resistance faced by blood as it moves through the circulatory system. It is represented by \(s\) and quantifies the opposition to blood flow encountered in the blood vessels, mainly the arteries.

• Higher SVR means more resistance; the heart has to work harder to pump blood.
• Lower SVR implies less resistance, easing the heart's workload.

Much like the diameter of a garden hose affects water flow, the diameter of blood vessels impacts SVR. If vessels constrict (become narrower), SVR increases, while dilation (widening) decreases resistance.
Medications such as nitroglycerine are often prescribed to lower SVR, especially beneficial for patients with cardiovascular issues or after heart attacks. This reduction in resistance helps in easing the strain on a weak or compromised heart, ensuring normal blood pressure levels are maintained with less effort from the heart.

Blood Pressure

Blood pressure (BP) is a critical measure of cardiovascular health. It represents the force exerted by blood against the walls of blood vessels.Blood pressure is expressed as two numbers: systolic over diastolic. Systolic pressure is the force when the heart beats, while diastolic is the pressure when the heart rests between beats.

• Systolic: the "top" number, indicating arterial pressure during heartbeats.
• Diastolic: the "bottom" number, indicating pressure when the heart rests.

The partial derivative \( \partial p / \partial c \) in the exercise helps measure how blood pressure changes with varying cardiac output, assuming systemic vascular resistance is constant.
Blood pressure can be calculated using the formula \( p = kcs \), where \( k \) is a constant. This implies that blood pressure depends on both cardiac output and systemic vascular resistance. Careful management of both parameters is crucial, as either can significantly impact blood pressure and overall cardiovascular health. Doctors often use this interplay to guide treatment following heart-related events, aiming to optimize both cardiac output and vascular resistance for stable blood pressure.