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Calculus Applicability to Science
Transcript of Calculus Applicability to Science
- Proposing this class
- Prerequisites: Calculus 1, General Chemistry, General Biology
- Hands on and newer technology, critical thinking - Understand applications of calculus in the science fields
- Learn how to convert models into differential equations
- Develop critical thinking skills and learn to use new technology
- Explore careers and career outlooks Goals and Objectives Example:
If a patient at time 0 has a concentration of 25 micrograms/mL of a drug in their blood. At time 24 hours the concentration is 5 micrograms/mL, what is the concentration of the drug between these times?
How to solve these kinds of problems:
To solve this, we have to follow first order kinetics (an order of a chemical rxn that depends on the concentration of only one reactant and is proportional to it) Pharmacokinetics - Conversion of rate equation for "x" into an equation of "x" versus time (integrated equation)
- Integration = "sums the information from small time intervals and gives a total result over a larger time period" ex: used for drug efficiency/dosage forms by taking the area of the plasma concentration vs. time graph Pharmacokinetics Pharmacokinetics - Study of kinetics of administered drugs and the rate of their absorption and elimination
- How is calculus involved?
- Differential Calculus is used to study the
rates of processes
- Calculus comes in when we look at
small time intervals Pharmacokinetics This equation describes the rate of elimination where k= rate constant, x= amt of drug remaining -Differentiation = "breaking a process down to look at the instantaneous process" Pharmacokinetics - Many pharmacy, veterinary, or medical students learn the mathematical concept of geometric series in calculus II.
- Geometric series = the ratio of each term to the one after it is a constant, r (in this case).
- Another definition is a half life when half of the drug concentration will have declined. Initial concentration is mesaured immediately after injection and r is the fraction of the drug eliminated per unit time. We will be looking at half-lives with geometric series. C1 represents the concentration after the first half-life. Derivation of Drug Elimination Formula - Amount of drug remaining after 2nd half-life - Amount of drug remaining after "n" half-lives - Multiply both sides by "r" - "r"= 1/2 and 2Cinir=Cini and adding Cini to right side - Substitute this equation into the equation above - By substituting in r=0.5 - "b" is the fraction of the drug eliminated per unit time. - Any positive base
can be written as a
power of "e". "r" is the fractional
rate of drug removal Drug Accumulation Formula - The formula for drug accumulation is also a derivation process starting off with a geometric series: As "n" approaches infinity, the drug level levels off and can't get past a certain concentration Accumulation and Elimination Standard curve for determining dosage intakes and dosage control. You need 5 half-lives to eliminate a particular drug Trapezoidal Rule - In toxicology (study of the adverse effects of chemicals), AUC (area under the curve) can be used as a measure of drug exposure
- In Pharmacokinetics, AUC can be used to determine other parameters such as bio availability or clearance Trapezoidal Rule - One way to calculate the area under the curve is through the trapezoidal rule (similar to Riemann sums) Euler's Method - Point slope method and can be used to integrate
- Initial amount or concentration of the drug is the point, and the differential equation is the slope <= Get slope from this equation
* Use this slope for a finite step size and use it as a starting point to calculate another step size Sample Problem - If we start with a dose of 100mg, k1 of .25/hour and a step size of .1 hour, the amount of drug remaining is 97.5mg - Both the trapezoidal rule and the Euler's method have error rates that put a limit to both methods, but it's an additional way to understand the concepts of drug absorption and elimination Careers Epidemiology = study of patterns, causes, and rates of diseases.
- Models can be converted to equations that can be used to solve important values such as equilibrium constants in the study of the rates of diseases.
Ex: Models were used to control the SARS epidemic (severe acute respiratory syndrome). Careers Ecology = study of relationships between organisms and their environment
- Modeling population dynamics in the Taihu lake ecosystem
- Predation of flatworms on mosquito larvae in rice fields was used as a biological control of disease transmission (connects ecology, medicine, and pharmacokinetics)
- Effects of lion predation in competition with wild dogs on cheetah population in the Serengeti. Biomedical Engineering Biomedical Engineering = Using engineering principles and applying it to biology and medicine
- Epitomizes the class that we are proposing
- Implantable cardiac pacemakers, defibrillators, joint replacement implants, biomedical imaging, tissue engineered skin for grafting, and novel drug
delivery (way to treat chronic diseases) Interview with Dr. Yukna - Computational Chemist by profession and a physics and chemistry professor at Maryville University
- Such a course would be interdisciplinary because it combines biology, chemistry, and math skills to understand a combination of concepts
- Would allow you to use technology while learning Interview with Dr. Yukna - To be able to study scientific problems correctly, you have you to understand the math behind it
- A course like this should be team taught by a math faculty with input from a science faculty member
- More hands on, independent, and provide work skills
- Beneficial for those students enrolled in the 3-2 year program for Biomedical engineering (with WashU) THE END Works Cited www.boomer.org/c/p3/c02/c0209.html
Rheinlander, Kim, and Dorothy Wallace. "Calculus, Biology and Medicine: A Case Study in Quantitative Literacy for Science Students." Numeracy 4.1 (2011): 3.