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# Calculus in Medicine

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## Allison Heefner

on 8 June 2015

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#### Transcript of Calculus in Medicine

62
ECG
bpm
Thank You!
Tumor Growth
Through Chain Rule
Using the power rule!
Muscle Contractions
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!
The Rate of Muscle Contractions
By: Allison Heefner
Block: 3

Calculus in Medicine
What is a Tumor?
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.
The Math...
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
Medications being Metabolized
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.
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?
Thank You
For Watching!

Example with the Math...
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?
18hr mark
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.
Finding how
fast a Medication
will metabolize
Through Logistic Decay
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.
Time
Milligrams of Drug
62
ECG
bpm
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.
Measuring Cardiac Output

Through Simpson's Rule
Remember Simpson's Rule
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>.
"What Is Cancer?" The Sol Goldman Pancreatic Cancer Research Center. Johns Hopkins, 12 Nov. 2012. Web. 15 May 2015. <http://pathology.jhu.edu/pc/BasicTypes1.php?area=ba>.
Beals, M., L. Gross, and S. Harrell. "Muscle Contraction." Muscle Contractions. 1999. Web. 15 May 2015. <http://www.tiem.utk.edu/~gross/bioed/webmodules/muscles.html>.
Pilachowski, Tim. "Supplemental Sections 11.1–11.2 Logistic Growth." Calculus 131. Web. 15 May 2015. <http://www.math.umd.edu/~tjp/131 11.1-2 supplement logistic growth.pdf>.
"Physiology and Psychology." Performance Benchmarks. Montana State University-Bozeman, 1 Apr. 1998. Web. 19 May 2015. <http://btc.montana.edu/olympics/physiology/pb01.html>.
Circulatory System Diagram Labeled. Digital image. Http://www.google.com/search?q=diagram of Heart Labeled&safe=active&sa=x&es_sm=122&biw=1280&bih=899&tbm=isch&tbo=u&source=univ&ei=qkbuvy2ikuexsatp7ih4ba&ved=0cb0qsaq&surl=1#imgrc=LSOl9ttmVKAGEM%3A;Ax3HaSbEmHB7wM;http%3A%2F%2F2.bp.blogspot.com%. Web.
"Cardiac Output and Blood Pressure." PTDirect. PTDIrect, 2015. Web. 19 May 2015. <http://www.ptdirect.com/training-design/anatomy-and-physiology/cardiac-output-and-blood-pressure>.
Arm Muscles Labeled Clipart. Digital image. Http://clipart-finder.com/clipart/arm-muscles-labeled.html. 1 Jan. 2015. Web.
Mahaffy, Joseph. "Riemann Sums and Numerical Integration." San Diego State University. San Diego State University, 31 Mar. 2004. Web. 19 May 2015. <http://www-rohan.sdsu.edu/~jmahaffy/courses/f00/math122/lectures/riemann_sums/riemanns.html>.
Vandiver, R. "Biological Applications for Calculus." Bryn Mawr College. Web. 28 May 2015. <http://www.brynmawr.edu/math/people/vandiver/BiologicalapplicationstoincorporateintoCalculus.htm>.

When to use Simpson's Rule...
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...
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.
Example...
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
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)
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