Calculus in Medicine
Pharmacokinetics is the study of how drugs (or any other substances that can be consumed) are processed within the body.
Pharmacokinetics can be broken down into five general steps in which a drug takes its course:
- Liberation – the drug is released from its pharmaceutical formulation
- For example, this would be the point in which the outer core of a pain-relieving gel capsule (ie., Advil) disintegrates to release the medicinal components inside.
- Absorption – the drug enters the body through blood circulation
- This would occur when the medicinal components of the gel enter the bloodstream.
- Distribution – the drug is dispersed through the body
- During this point, the pain-relieving gel provides pain relief as it is being spread through the blood stream by the body.
- Metabolism – the drug is processed and broken down by the body
- During this point, the pain relieving effects begin to wear off slightly.
- Excretion– the drug leaves the body
- The drug no longer provides pain relief and is excreted by the body.
In order for doctors to prescribe the correct dosage of a drug and provide a regimen for treatment (ie., “take 2 capsules twice a day”), the drug’s concentration over time must be tracked. This prevents under and over-dosing.
The way that a drug’s concentration over time is calculated is using calculus! In fact, a drugs course over time can be calculated using a differential equation.
In applications of differential equations, the functions represent physical quantities, and the derivatives, as we know, represent the rates of change of these qualities. Therefore, a differential equation describes the relationship between these physical quantities and their rates of change.
In order to create a differential equation that describes the physical quantities of a drug in relation to the rate at which they change, let’s consider the following variables:
$d$ – drug dosage in milligrams
$c$ – the concentration of the drug at any time t
$t$ – time in hours since consumption of drug
In addition to the above variables which seem relatively obvious, the equation also requires the use of :
$k_a$– the absorption constant of a drug
$k_e$ – the elimination constant of drug
$v$ – volume of drug in body
$b$ – bioavailability*
*bioavailability is the amount of the drug which has already been absorbed divided by the total amount of drug available
Calculating the rate of absorption:
In order to calculate the rate of absorption, the equation must include the drug dosage, the absorption constant, and the drug’s bioavailability.
Absorption = $(k_a)(d)(b) \times e^{-at}$
Calculating the rate of elimination:
In order to calculate the rate of elimination, the equation must include the elimination constant, the volume of drug distributed throughout the body, and the concentration of the drug that is left.
Elimination = $(k_e)(c)(v)$
Formulating a differential equation:
In order to model our concentration over time, the elimination must be subtracted from the absorption in order to calculate the amount of drug that is left in the body at various time points.
$\dfrac{dc}{dt}$ = $\dfrac{k_a}{k_a-k_e}$ $[(k_a)(d)(b) \times e^{-at}$ $- (k_e)(c)(v)]$
Drug Concentration vs. Time
The graph above represents a solution to the differential equation.
When actual numbers corresponding to a drug are inserted into the above equation, the graph will tend to generally form as a curve that has a steep positive slope during absorption, levels off once peak drug concentration is reached, and a has a negative slope during elimination.
Consider the following variables
$\dfrac{da_g}{dt}$– the amount of drug being absorbed over time
$\dfrac{de_g}{dt}$ — the amount of drug being eliminated over time
Absorption phase:
In the absorption phase, the drug is being absorbed faster than it is being eliminated, causing the drug concentration to increase.
$\dfrac{da_g}{dt}$ > $\dfrac{de_g}{dt}$
Elimination phase:
The drug is no longer being absorbed and the rate of elimination exceeds the rate of absorption.
$\dfrac{da_g}{dt}$ < $\dfrac{de_g}{dt}$
Medical professionals need calculus!
Without drug specialists in the pharmaceutical industry testing drug concentrations over time and modeling them using calculus, we would not have labels on medication that provide instructions for dosage use.
Although the equation for each drug looks unique depending on its properties and the patient’s anatomy, calculus is necessary for medical professionals to have the ability to map the relationship between drug concentration in the body over time.
Sources:
http://pharmacy.unc.edu/files/2015/06/PK-Book-2014.pdf (pg. 59-62)
https://math.stackexchange.com/pharmacokinetics-differential-equations
https://www.boomer.org/c/p1/Ch08/Ch0802.html
https://www.ausmed.com/cpd/articles/pharmacokinetics-and-pharmacodynamics
This is an excellent article and analysis of how calculus impacts the medical industry. We take for granted the work that goes into supplying citizens with medicine and we fail to recognize the importance of directions such as, how many pills you can take. My naive self has always thought it was doctors themselves that claimed how many pills one can take and yet, in reality, I am very wrong. I hope our society can encourage individuals to continue to enter into a drug specialist profession because without their research and application of calculus to their work we would not be able to utilize medicine properly.
I agree! Although I had some previous sense of the calculations involved in pharmacokinetics, I, too, had no idea that it was this calculus-based. We all benefit from the calculations involved in tracking drug concentrations and your last comment about encouraging people to explore such professions is particularly significant as not many are aware of such fields. Thanks for reading!
Nice post, Nejla!
I discovered—this morning—that there is actually a section in the textbook that talks about drug concentration! (Section 4.8.)
Drug concentration can be modelled using a so-called surge function, which is of the form $f(t) = ate^{-bt}$ (where $a$ and $b$ are positive constants). See here for an example of a graph of a surge function.
The surge function model is a lot simpler (so probably less accurate) than what you talk about in your post, but you can kind of see how to translate between them: if $f(t) = ate^{-bt}$ then the product rule gives
$$f'(t) = (a)(e^{-bt}) + (at)(-be^{-bt}) = a e^{-bt} – abte^{-bt}$$
Writing $c=f(t)$, this becomes
$$\dfrac{dc}{dt} = ae^{-bt} – bc$$
which looks remarkably like the differential equation you discuss in your post (with different constants, of course).
The equation from the textbook matches the one in my post! I think they condensed some of the variables due to it being in a calculus textbook and not meant for a medicine/pharmacy audience and also because its a surge function model, as you mentioned. Regardless it’s nice to know the equation is explored by many fields.
This was a super interesting application of calculus that I’d never really thought about before! It’s amazing how math factors into our everyday lives without us even realizing it. One thing that I was thinking about while reading was how much individual factors affect the accuracy of the equation? For example differences in metabolism, age, weight, gender, etc. I could be totally wrong and these things have no impact, but it seems like an interesting thing to look at. Specifically, I was thinking about the different doses of medication suggested for adults vs. children for most over-the-counter medications. Is it that the children absorb the drug differently or that they simply cannot handle the same concentration as adults? Either way, great post!
Thats a very good (and valid) point actually. Things like weight and age both influence the rate of metabolism dependent on the individual. This is actually reflected in recommended doses on the back of medication. Directions on the back of medication will often recommend to use the smallest effective dose. To continue my original Advil example, the recommended dose is 1 tablet every 4-6 hours and if pain does not respond to 1 tablet, 2 tablets may be used. The 4-6 time range and 1-2 tablet range reflect the metabolic differences (and pain tolerance) in different people. But in order to obtain this general range, the different drug concentrations had to be originally mapped by the drug company and modeled using calculus.
Thanks for the article! Where does the (ksuba/(ksuba-ksube)) come from in the final expression for dc/dt? If dc/dt measures the net rate of change, why isn’t it just da/dt – de/dt?