Calculus in Medicine

Sources

• "The Chain Rule, Related Rates, and Implicit Differentiation." Chapter 7. 4 Sept. 2009. Web. 15 May 2015. <http://www.ugrad.math.ubc.ca/coursedoc/math102/2011/keshet.notes/Chapter7.pdf>.
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The Rate of Muscle Contractions

ECG

Muscle Contractions

62

bpm

• This equation uses load and velocity to determine muscles contractions.
• (P + a)(V + b) = c
• a, b, c = constants
• P = Force generated by a muscle
• V = Speed of muscle contractions
• Therefore, to find the rate at which a muscle contracts you must use calculus to find the derivative!

Calculus in Medicine

Using the power rule!

The Math...

(P + a)(V + b) = c

PV + Pb + aV + ab = c

(PV' + VP') + (Pb' + bP') + (aV'+ Va') + (ab' + ba') = rate of

contractions

a, b, c = constants

P = Force generated by a muscle

V = Speed of muscle contractions

By: Allison Heefner

Block: 3

Measuring Cardiac Output

Through Simpson's Rule

ECG

When to use Simpson's Rule...

Medications being Metabolized

62

bpm

Tumor Growth

Cardiac output can be measured as a function, table, or graph. When the data is presented in a table or graph we can use Simpson's Rule, when the data is presented as a function we use the following formula...

Through Chain Rule

Example...

Once a medication enters the blood stream the body begins to metabolize it (break it down). Doctors can use logistic decay to estimate the amount of medication left in a patient's body after a set amount of time. This ensures the patient does not receive too much of the medication.

Patient X is injected with 4 milligrams of dye. The following table represents the time and the amount of dye in milligrams left in the patients heart at that time. Using this information and the equation on the previous slide calculate the cardiac output.

2/3[0 + 4(4.6) + 2(5) + 4(6.3) + 2(7.2) + 4(6.1) + 2(4.9) + 4(2.8) + 1.2]

2/3[0 + 18.4 + 10 + 25.2 + 14.4 + 24.4 + 9.8 + 11.2 + 1.2]

2/3[114.6]

75.636

This number now can be plugged into the flow equation for the integral portion.

Flow (cardiac output) = 4/75.636

F = 0.05288487 milligrams per unit of time

The Math...

• Example: A 45 year old female has a spherical, benign tumor on her left kidney. To determine if surgery is needed you must find the rate at which the tumor's volume is expanding. If the tumors current radius is 5 centimeters and the radius increases 0.000025 centimeters every day at what rate is the volume of the tumor expanding?

The human heart is divided into four chambers, the bottom two chambers are called ventricles. Cardiac output is how much blood the ventricles can pump through the body per unit of time. When measuring cardiac output doctors focus on the left ventricle because this is the ventricle that pumps blood to our muscles and organs.

What Does this Mean?

This means that over the course of a year (365 days) the tumor's volume will grow another 2.86671 cubic centimeters. Therefore, a doctor could determine that this is an exceptionally large tumor and should be removed.

What is a Tumor?

Example with the Math...

Remember Simpson's Rule

Patient X was brought to the ER with third degree burns presenting over 30% of their body. Patient X needs 80mg of Oxycontin to help control the pain. The amount of Oxycontin in a patient's blood stream decreases 34% every hour. Let "y" equal the total amount of medication in the patients blood stream. In how many hours will the medication be fully metabolized?

A tumor is a mass that can be benign or malignant. Tumors can also be either spherical or more irregularly shaped. When tumors grow their radius expands at a rate; therefore, so long as the tumor is spherical, to determine the growth rate of a patient's tumor doctors can use the chain rule.

Flow = Cardiac output (volume of blood per unit of time)

A = Dye initially injected (usually in milligrams)

Integral = Dye readings (from table, graph, or equation)

Milligrams of Drug

This graph shows that by roughly 18 hours the patient's body will have completely metabolized the Oxycontin and it would be safe for another dose to be administered.

18hr mark

Time

It is important to remember that Simpson's Rule only accounts for the integral portion of the previous formula (the denominator). In the flow equation "A" (the numerator) is the total amount of dye injected into the patient.

Finding how

fast a Medication

will metabolize

Through Logistic Decay

Thank You!

Thank You

For Watching!