**Applications of Calculus in Medicine**

**Ericka Randazzo**

AP Calculus BC

Period 1

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