The Use of Calculus in Medicine

Topics Covered

• Tumor Growth
• Gompertz Function
• Muscle Contraction
• Blood Pressure
• Bacteria Growth
• Drug Sensitivity
• Pharmacology
• Noyes-Whitney Function
• Blood Flow
• Poiseuille's Law

It is no secret that there are nearly an infinite number of applications of Calculus in our world. Today, medical researchers and professionals are using calculus in order to predict and measure bodily functions, as well as abnormalities such as tumor growth. Calculus is essentially allowing doctors to prepare humans for the future.

Patients Reaction

Because doctors prefer to prescribe drugs with a maximum sensitivity, they must also take into account the strength of the patients reaction to the drug. This can be calculated by:

R(x) = csub1 x Msup2 (csub2 −M)

M refers to the total milligrams, and csub1 and csub2 are positive constants.

Medical Applications of Calculus

Drug Sensitivity

When prescribing drugs to their patients, doctors need to know the strength of the drug and how it will affect the patient. The function:

R(M) = 2M√10+0.5M

Where M measures the dosage, or the amount of medicine that is absorbed into the blood. The sensitivity can be found by taking the derivative of R with respect to M and plugging in any positive integer greater than zero for M.

Pharmacology

Final Notes

Many of these functions are only theoretical and are not always used by medical professionals.

A calculus is a term used in the medical field as an alternate name for a stone, or a consecration of material, like a kidney stone.

Calculus is often used in the field of Pharmacology to determine dissolution rates for drugs. The rate of dissolution is very important, as a drug that dissolves too quickly could lead to toxicity while a drug that dissolves too slow could have no effect at all.

Dissolution of Administered Drugs

The Noyes-Whitney equation shows the rate at which a solvent dissolves.

Poiseuille's Law

The flow of fluids through an IV catheter can be calculated by Poiseuillle's Law. This function takes into account the viscosity, which is the internal resistance all flowing liquids have.

Blood Flow

The flow of blood or any liquid flowing through a catheter can be calculated using Poiseuille's Law as shown below. Both the volume flow rate and the resistance to flow can be calculated.

Sources

• http://www-rohan.sdsu.edu/~jmahaffy/courses/s11/math121/beamer_lectures/prod-04.pdf
• http://www.math.ubc.ca/~keshet/M102/M102CourseNotes/M102chap9.pdf
• http://demonstrations.wolfram.com/TheGompertzSigmoidFunctionAndItsDerivative/
• http://mathworld.wolfram.com/SigmoidFunction.html
• https://prezi.com/td2vcitwderg/calculus-and-tumor-growth/
• http://oaji.net/articles/2014/563-1395952206.pdf
• http://www.forbes.com/forbes/2008/1027/074.html
• http://encyclopedia2.thefreedictionary.com/Hill's+Equation+of+Muscle+Contraction
• http://www.brynmawr.edu/math/people/vandiver/documents/Derivative.pdf
• http://www.brynmawr.edu/math/people/vandiver/documents/Optimization.pdf
• http://www.brynmawr.edu/math/people/vandiver/documents/Integration.pdf
• https://bhcc.digication.com/CalculusII_YassineBahbah/Calculus_in_Medicine

Bacteria Growth

As we have frequently throughout the year, the growth or decay of culture of bacteria can be calculated using various types of functions, including exponential, logistic, as well as sine and cosine functions.

Functions

Gompertz Function Model According to Tumor Size

Basic Logistic/Sigmoidal Curve

Gompertz Function

In 1964, Laird showed that tumor growth satisfied the Gompertz function:

G(N)=N(b−aln(N))

Where N is the number of tumor cells, a and b are constants matched to the data and the function is defined for N = 0.

The Gompertz Function is a sigmoidal function, or logistic function, a modeling function for a time series, where growth is slowest at the beginning and end of the time period. This function is often used to calculate tumor growth.

Modeling

By taking the original Gompertz function and substituting W for N, where W is the weight of the tumor in milligrams, we can find the equilibrium weight of the tumor and the maximum growth rate.

G(W)=W (0.5−0.05ln(W))

Tumor Growth

Equilibrium Weight

Equilibrium is the second step in the process of tumor growth. This is the longest of the three processes and can be calculated by simply solving the equation for zero.

G(W)=W(0.5−0.05ln(W))=0

0.5−0.05ln(W)=0

ln(W)=10

W=e<sup>10=22,026mg

Maximum Growth Rate

The maximum growth rate of the tumor can be found by setting the derivative G'(W) equal to zero.

G'(W)=W (−0.05/W) +(0.5−0.05ln(W)) G'(W)=0.45−0.05ln(W)=0

ln(W)=9

W=e<sup>9=8,103mg

The 8,103mg is then plugged back into the original function to yield a rate of 405.2 mg per day.

Tumors are abnormal growths of tissue caused by the rapid multiplication of cells. They grow based on the nutrient supply that is available, and grow outward in the form of rough spherical shapes. Several models have been created to predict tumor growth.

Blood Pressure

Blood pressure is the pressure exerted by the circulating blood on the walls of the arteries. By using the function: p(t) = 90 + 15sin(2.5πt), where t is the number of seconds since the beginning of the cardiac cycle, we can calculate things such as the maximum and minimum pressures.

Use of the Derivative

In 1953 Abbott and Wilkie expanded upon Hill's work and postulated that if contracting muscle has a length l, at time t, then the velocity of its contraction dl/dt is given by:

dl/dt = (Fsub1 – F)b/(F + a)

Where F is the force that overcomes the muscle, Fsub1 is the maximum force of the muscle at which the velocity, and a and b are constants

Muscle Contraction

In 1938, British physiologist A.V. Hill formulated his equation that expresses the change in speed of a muscle contraction as a function of load.

(P + a)(v + b) = b (Psub0 + a)

Where v is the speed of a muscle contraction under a load P, Psub0 is the maximum value of the isometric force during tetanic stimulation of the entire muscle, and a and b are constants.