Illuminations: Cardiac Output, Rates of Change, and Accumulation

# Cardiac Output, Rates of Change, and Accumulation

This activity explores the measurement of the amount of blood being pumped by a heart. The goal of this investigation is a rich exploration of rates of change and accumulation in context. As presented here, the measurement is done from a discrete perspective, but the generalization to a continuous perspective and use of calculus (integrals) can be done easily. The measurement is based upon two common problems: (1) a flow rate and accumulation problem and (2) a concentration problem. Given a flow rate, a concentration of ice water and a duration, how much ice water is pumped over the time interval? How can you determine the concentration of ice water by measuring the temperature of the mixture? The activity is separated into four parts. These four segments are available at the bottom of every page.

### Math Content

The goal of this investigation is a rich exploration of rates of change and accumulation in context. As presented here, the measurement is done from a discrete perspective, but the generalization to a continuous perspective and use of calculus (integrals) can be done easily.

### Individual Lessons

Lesson 1 - Make a Conjecture

Doctors and veterinarians are interested in a quantity called the "cardiac output" which is a measure of the rate of blood flow being pumped by the heart. To measure the cardiac output, a catheter (tube) is inserted through the heart. The catheter measures the temperature of the surrounding blood near the tip of the probe. The catheter has a small balloon at the tip which is inflated to help the probe move through the heart and then deflated. Once the catheter is inserted, ice water is injected through the catheter and emerges from a small hole approximately 12 inches before the end of the catheter.

Lesson 2 - Gather Data

A simple experiment which can be used to model the cardiac output measurement using a catheter can be constructed using:

• an aquarium pump
• a temperature probe
• a bucket
• tube with a port for injecting ice water
• syringe and valve
• data collection device such as a CBL.

The pump and tubing is available at most aquarium stores and many home improvement stores. The syringe and valve port are available through science supply stores. The temperature probe that comes with the CBL is sufficient.

Lesson 3 - Analyze the Data

Students will analyze and graph the data taken.

Lesson 4 - Reflecting on Your Work

The methods explored in the measuring of cardiac output can be applied to other situations. Two of these situations are described here. The first examines the sediments flowing from the Des Moines River near Saylorville, Iowa. The second situation investigates the measurement of blood flow through the brain.

### NCTM Resources

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