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CALCULUS IN CARDIOLOGY

Table of Contents

Cardiac Output Explanation

1. Introduction

2. Cardiac Output Explanation

3. Dye Dilution Method Explanation

4. Cardiac Output Formula

5. Example 1

6. Conclusions

Cardiac Output

Cardiac output is the measurement of the volume of blood pumped out by the heart per minute. Cardiac output can be used to determine certain diseases or disorders of the heart, blood and circulatory system. The simple formula for cardiac output is stroke volume (volume of blood pumped out of ventricle per heart beat) time beats per minute. In addition, according to an article from Boundless.com, "While there are many clinical techniques to measure CO, it is best described as a physiological and mathematical relationship between different variables. When one of the variables change, CO as a whole will change as a result. This can also be used to predict other regulated variables, such as blood pressure and blood volume."

Formula

"Cardiac Output." Wikipedia. Wikimedia Foundation, 01 May 2017. Web. 04 May 2017. <https://en.wikipedia.org/wiki/Cardiac_output>.

Blood Flow

"How Does Blood Flow Through the Heart?" New Health Advisor. N.p., 24 Aug. 2015. Web. 04 May 2017. <https://www.newhealthadvisor.com/Blood-Flow-Through-the-Heart.html>.

The dye dilution method for the calculation of cardiac output involves injecting a specific amount of dye into one of the patient's main veins located near the heart. The dye travels through the heart, into the lungs and back into the heart before passing through the aorta, the main artery in the body.

Dye Dilution Method

A monitor on the aorta measures the concentration of dye in the aorta at specific time intervals. The dye concentration values are then used to calculate the cardiac output of a patient.

Cardiac Output Formula

The following variables are used to create the cardiac output formula:

Formula for amount of dye per time interval:

  • The amount of dye is equal to the concentration of dye times the volume of dye.

  • The volume of blood is equal to the rate of blood being pumped or flow rate of blood multiplied by the time interval. Therefore, the volume of blood in the equation can be replaced by t times Q, cardiac output because cardiac output is the measurement of the flow rate of blood.
  • This formula only calculates the amount of dye for ONE interval. Our goal is to calculate the total amount of dye over multiple time intervals.

Summation notation for total amount of dye over multiple time intervals:

  • The total amount of dye is equal to the sum of the concentration of dye times cardiac output times time interval.

  • To calculate the total amount of dye we let n approach infinity to get the most accurate answer we assume that the time intervals also approach infinity and become infinitely small.
  • The sum notation can now be rewritten as an integral.

  • Cardiac output is assumed to be a constant value and can therefore be moved to the outside of the sum.

Cardiac Output Formula

  • Now the function for the amount of dye can be rewritten as a function for cardiac output.

Problem 1

12 mg of dye is injected into the circulatory system of a patient. From t=0 seconds to t=10 seconds, the dye concentration in the blood of the patient is modeled by the function c(t)=2te^(5t^2)mg/L. Calculate cardiac output over the interval [0, 10].

Example 1

Steps

X coordinates - time in seconds

Y coordinates - c(t)

Desmos Graphing Calculator. N.p., n.d. Web. 02 May 2017. <https://www.desmos.com/calculator>.

Step 1 - Determine variables

Step 2 - Setup function

Step 3 - Evaluate the integral

Step 4 - Input integral into original equation and simplify

Step 5 - Put your answer in the correct units. Cardiac output is usually measured in L/min

Relation to Real World

Importance in real world

Cardiac output is important in medicine because it can help doctors diagnose heart diseases or disorders. Specifically, calculating cardiac output can help doctors find and calculate abnormal heart rate, cardiac muscle strength or stroke volume. Additionally it can help determine heart failure or poor circulation. In general, cardiac output is important in determining the efficiency of the heart in delivering blood to parts of the body.

Normal cardiac output for an adult at rest is about 5 L/min. However, cardiac output varies from person to person because of height, weight, and other factors. Any physical activity or workout increases cardiac output because your heart has to work harder to supply parts of the body with necessary nutrients quicker.

In the second example, where we approximated cardiac output based on data collected from a patient, the cardiac output was equal to about 9 L/min. This means that the patient the data was collected from may have been doing some type of light physical activity at the time of the test or may have been larger in weight or height than a person whose cardiac output at rest is 5 L/min.

THANK YOU

THANK YOU

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