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# Calculus in Medicine

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## Claudia Choi

on 29 May 2014

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#### Transcript of Calculus in Medicine

One of the practices in the medicinal career field is analyzing the patients’ sensitivity to drugs. The concept of derivative is useful in obtaining the degree of sensitivity to a particular drug. For example, the strength of a certain drug is given by R(M)= 2M√(10+0.5M) where M is the dosage. The sensitivity of the body to the drug is measured by taking the derivative of R with respect to M. R’(M) corresponds to the change in R as a result of a change in the dosage.
integration
Pharmacologists as well apply calculus when they test drugs. To test the characteristics of newly designed drugs, they use the principle of integration. For instance, when they have the rate at which an experimental drug changes the temperature of the patient’s body, they can calculate the total change in temperature resulting from a certain amount of the drug using the rules of integration.
optimization
Doctors need to prescribe drugs at a dosage with maximum strength. To know how strong a patient’s reaction to the dose of a drug will be, doctors use the equation R(x) = c_1 M^2 (c_2 - M) where c_1 and c_2 are constants and R(x) measures the strength of a dose. In order to maximize the strength of the drug, doctors find the value of M for which the sensitivity is a maximum by following the steps of optimization.
Modeling the growth of tumors is helpful in the study of a tumor’s progression in the patient’s body. Using calculus, doctors can gather important information about the spread of tumors and schedule the best type of therapy. Tumor growths usually follow an exponential growth model, V= V_0 e^at , where a is a constant of exponential growth, and V and V_0are the volumes of the tumor at times t and 0. From the equation above, doctors derive the specific growth rate (SGR) of tumor: SGR=1/V dV/dt where SGR stands for the relative change in tumor volume per unit of time. If a tumor has a high value of SGR, doctors can interpret it as being a rapidly growing tumor and make decisions about the apt form of therapy to cure the tumor.
Today, digital blood pressure devices accurately measure blood pressure. However, doctors can also model blood pressure by using differential calculus. In order to calculate blood pressure, doctors need to know the rate of flow entering the aorta and the rate flow leaving the aorta. The difference between the two rates of flow is the same as the change in the arterial volume. By knowing the underlying calculus in obtaining blood pressure, doctors can better understand the values that machines give.
The same concept can be applied to calculate cardiac output, the rate of blood flow produced by the heart and the pulse, or heart rate.
Drug sensitivity
Tumor growth
Blood pressure
Calculus in Medicine
Derivative
Ehrenberg equation
The Ehrenberg equation gives the relationship between the weight in kilograms and the height in meters of children. Tying this concept with differential calculus, doctors can estimate the rate of change of height with respect to weight. By taking the derivative in terms of weight, they can find the relationship between weight and rate of change of height. For instance, the derivative of Ehrenberg’s model for the average girl reveals that as the weight increases, the rate of change in height decreases.
This equation was actually applied in a case study of children at a public school in New Delhi. It was found that the equation was a good fit for the school children and corroborated earlier studies about the influence of nutrition in children’s growth. The simplicity and practical usefulness of calculus is revealed here once again.
A similar pattern can be followed to find the relationship between skull size and height and determine which one grows faster by using the principles of derivation.
Growth rate
Medical teams at laboratories often deal with a bacterial colony to investigate and test the characteristics of particular bacteria. Differential calculus comes in handy when tracking the change in the population of a bacterial culture in reaction to manipulated variables such as toxin, nutrients, or temperature.
For instance, scientists can project the rate at which the population is changing at a certain time after a nutrient is administered. They can also know if the bacterial culture declines or grows at a specific time, and when the largest population will form by using optimization. Because calculations enable them to make predictions and compare and contrast the effects of different variables, scientists rely on calculus to facilitate their work.
Similar applications include modeling the growth rate of blood cells.
Calculus joke:

A definite integral walks into a bar and orders five pints of Guinness. The bartender pours them, and the definite integral finishes them one after the other. “Can I have five more?” he asks.
The bartender says, “Don’t you think you’ve had enough?”
“Don’t worry about me,” says the integral. “I know my limits.”
Sources:

Mehrara, Esmaeil. Quantitative Analysis of Tumor Growth and Response to Therapy. Göteborg: Department of Radiation Physics, U of Gothenburg, 2010. Web. 28 May 2014.
Vandiver, R. "Biological Applications to Incorporate into Calculus." Biological Applications to Incorporate into Calculus. Bryn Mawr College, n.d. Web. 28 May 2014.
Mahaffy, Joseph M. Calculus for the Life Sciences II - Lecture Notes - Linear Differential Equations. N.p.: n.p., 2012. PDF.
Mahaffy, Joseph M. Calculus for the Life Sciences II - Lecture Notes - The Derivative of e^x and ln(x). N.p.: n.p., 2012. PDF.
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