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# Applications of Calculus in Medicine

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#### Transcript of Applications of Calculus in Medicine

Applications of Calculus in Medicine
Ericka Randazzo
AP Calculus BC
Period 1

Drug Sensitivity
Doctors must understand the characteristics of the drugs they prescribe to patients
Strength of drug is given by
R(M
) where
M
measures the dosage
The sensitivity of the patient's body to the drug is the derivative of
R
with respect to
M
Sensitivity corresponds to approximately how much change to expect in
R
as the result of a unit change in the dosage
Generally, doctors prefer to prescribe dosages of maximum sensitivity
The strength of a patient's reaction to a dose of
M
milligrams of a certain drug is represented by

R(x) =
c(1)
M^2(
c(2
)-M)

c(1) and c(2) are positive constants
Doctors try to determine:
For what value of
M
the strength is at a maximum
What the exact maximum value is
For what value of M is the sensitivity a maximum
What is the maximum sensitivity
Tumor Growth
Calculus can be used to determine how fast a tumor is growing or shrinking, the size when the tumor will stop growing, when certain treatments should be given, the volume of a tumor, and how many cells make up the tumor
Gompertz Function
: assumes that growth rate declines over time due to a prolonged cell cycle, decreased oxygen/nutrient availability, and increased rate of cell death
Differential equation where
V: volume at a certain time
N: number of cells
(Note: V and N can be interchangeable)
a: growth constant
b: constant for growth retardation
Blood Flow
Calculus can be used to determine the blood flow in an artery or a vein at a given point in time
Cardiac Output
Calculus can be used to find the amount of blood pumped through the heart per unit time
Cardiac Output: the volume of blood that the heart pumps out per unit time (aka rate of blood flow into aorta)
Dye dilution method can be used to measure cardiac output using dye placed in right atrium and a probe that's placed in the aorta
Probe checks concentration of dye leaving the aorta at equal time intervals until dye runs out
Blood flows from right atrium through pulmonary veins to pulmonary arteries and into left atrium where the aorta sends blood around the body (see diagram)
Origination of Calculus
Mathematicians from all over the world contributed to the development of calculus but Issac Newton and Gottfried Wilhelm Leibniz are credited as the main discoverers
Debate over which man truly deserves credit continues to this day
Both were instrumental to the creation of calculus, but they thought of the main concepts in different ways
Thought in terms of graphs rather than functions
Issac Newton
Gottfried Wilhelm Leibniz
Considered variables changing with time
Used quantities x' and y' , which were finite velocities to compute the tangent
Used a different notation for differentiation
Dot notation: dot placed over a function name to denote the time derivative of that function; called this a fluxion
Studied calculus from more of a geometrical viewpoint
Thought of variables x and y as ranging over sequences of infinitely close values
Introduced dx and dy as differences between successive values of these sequences
Knew that dy/dx gives the tangent but didn't use it as a defining property
Studied calculus from more of an analytical standpoint
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