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Math and Medicine

Compute the mathematical factors utilized in concentrations/dosages of drugs.

Learning Objectives

After completing this section, you should be able to:

Compute the mathematical factors utilized in concentrations/dosages of drugs.
Describe the history of validating effectiveness of a new drug.
Describe how mathematical modeling is used to track the spread of a virus.

Concentrations and Dosages of Drugs

Consider any drug and the recommended dosage varies based on several factors such as age, weight, and degree of illness of a person. Hospitals and medical dispensaries do not stock every possible needed concentration of medicines. Drugs that are delivered in liquid form for intravenous (IV) methods in particular can be easily adjusted to meet the needs of a patient. Whether administering anesthesia prior to an operation or administering a vaccine, calculation of the concentration of a drug is needed to ensure the desired amount of medicine is delivered.

The formula to determine the volume needed of a drug in liquid form is a relatively simple formula. The volume needed is calculated based on the required dosage of the drug with respect to the concentration of the drug. For drugs in liquid form, the concentration is noted as the amount of the drug per the volume of the solution that the drug is suspended in which is commonly measured in g/mL or mg/mL.

Suppose a doctor writes a prescription for 6 mg of a drug, which a nurse calculates when retrieving the needed prescription from their secure pharmaceutical storage space. On the shelves, the drug is available in liquid form as 2 mg per mL. This means that 1 mg of the drug is found in 0.5 mL of the solution. Multiplying 6 mg by 0.5 mL yields 3 mL, which is the volume of the prescription per single dose.

A common calculation for the weight of a liquid drug is measured in grams of a drug per 100 mL of solution and is also called the percentage weight by volume measurement and labeled as % w/v or simply w/v.

Suppose you visit your doctor with symptoms of an upset stomach and unrelenting heartburn. One possible recourse is sodium bicarbonate, which aids in reducing stomach acid.

Calculating the Quantity in a Mixture

Try it.

How much sodium bicarbonate is there in a 250 mL solution of 1.58% w/v sodium bicarbonate?

Solution

$1.58\%\ \text{w}/\text{v}=1.58\ \text{g}$ sodium bicarbonate in 100 mL. If there is 250 mL of the solution, we have 2.5 times as much sodium bicarbonate as in 100 mL. Thus, we multiply 1.58 by 2.5 to yield 3.95 g sodium bicarbonate in 250 mL solution.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Validating Effectiveness of a New Vaccine

The process to develop a new vaccine and be able to offer it to the public typically takes 10 to 15 years. In the United States, the system typically involves both public and private participation in a process. During the 1900s, several vaccines were successfully developed, including the following: polio vaccine in the 1950s and chickenpox vaccine in the 1990s. Both of these vaccines took years to be developed, tested, and available to the public. Knowing the typical timeline for a vaccine to move from development to administration, it is not surprising that some people wondered how a vaccine for Covid-19 was released in less than a year’s time.

Lesser known is that research on coronavirus vaccines has been in process for approximately 10 years. Back in 2012, concern over the Middle Eastern respiratory syndrome (MERS) broke out and scientists from all over the world began working on researching coronaviruses and how to combat them. It was discovered that the foundation for the virus is a spike protein, which, when delivered as part of a vaccine, causes the human body to generate antibodies and is the platform for coronavirus vaccines.

When the Covid-19 pandemic broke out, Operation Warp Speed, fueled by the U.S. federal government and private sector, poured unprecedented human resources into applying the previous 10 years of research and development into targeting a specific vaccine for the Covid-19 strain.

Shibo Jiang

Dr. Shibo Jiang, MD, PhD, is co-director the Center for Vaccine Development at the Texas Children’s Hospital and head of a virology group at the New York Blood Center. Together with his colleagues, Jiang has been working on vaccines and treatments for a range of viruses and infections including influenzas, HIV, Sars, HPV and more recently Covid-19. His work has been recognized around the world and is marked with receiving grants amounting to over $20 million from U.S. sources as well as the same from foundations in China, producing patents in the United States and China for his antiviral products to combat world concerns.

Jiang has been a voice for caution in the search for a vaccine for Covid-19, emphasizing the need for caution to ensure safety in the development and deployment of a vaccine. His work and that of his colleagues for over 10 years on other coronaviruses paved the way for the vaccines that have been shared to combat the Covid-19 pandemic.

Mathematical Modeling to Track the Spread of a Vaccine

With a large number of people receiving a Covid-19 vaccine, the concern at this time is how to create an affordable vaccine to reach people all over the world. If a world solution is not found, those without access to a vaccine will serve as incubators to variants that might be resistant to the existing vaccines.

As we work to vaccinate the world, attention continues with tracking the spread of the Covid-19 and its multiple variants. Mathematical modeling is the process of creating a representation of the behavior of a system using mathematical language. Digital mathematical modeling plays a key role in analyzing the vast amounts of data reported from a variety of sources such as hospitals and apps on cell phones.

When attempting to represent an observed quantitative data set, mathematical models can aid in finding patterns and concentrations as well as aid in predicting growth or decline of the system. Mathematical models can also be useful to determine strengths and vulnerabilities of a system, which can be helpful in arresting the spread of a virus.

The chapter on Graph Theory explores one such method of mathematical modeling using paths and circuits. Cell phones have been helpful in tracking the spread of the Covid-19 virus using apps regulated by regional government public health authorities to collect data on the network of people exposed to an individual who tests positive for the Covid-19 virus.

Gladys West

Dr. Gladys West is a mathematician and hidden figure with a rich résumé of accomplishments spanning Air Force applications and work at NASA. Born in 1930, West rose and excelled both academically and in her professional life at a time when Black women were not embraced in STEM positions. One of her many accomplishments is the Global Positioning System (GPS) used on cell phones for driving directions.

West began work as a human computer, someone who computes mathematical computations by hand. Considering the time and complexity of some calculations, she became involved in programming computers to crunch computations. Eventually, West created a mathematical model of Earth with detail and precision that made GPS possible, which is utilized in an array of devices from satellites to cell phones. The next time you tag a photo or obtain driving directions, you are tapping into the mathematical modeling of Earth that West developed.

Consider the following graph ():

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

Compute volumes of prescription drugs in liquid and pill form.
Validate the effectiveness of a new drug.
Mathematical modeling can be used to describe and track the spread of a virus.

Practice (5)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

How much sodium bicarbonate is there in a 250 mL solution of 1.58% w/v sodium bicarbonate?

جواب کھوليں

$1.58\%\ \text{w}/\text{v}=1.58\ \text{g}$ sodium bicarbonate in 100 mL. If there is 250 mL of the solution, we have 2.5 times as much sodium bicarbonate as in 100 mL. Thus, we multiply 1.58 by 2.5 to yield 3.95 g sodium bicarbonate in 250 mL solution.
A doctor prescribes 25.5 mg of a drug to take orally per day and pills are available in 8.5 mg. How many pills will be needed each day?

جواب کھوليں

The prescription and the pills are in the same units which means no conversions are needed. We can divide the units of the drug prescribed by the units in each pill: $25.5/8.5=3$. So, 3 pills will be needed each day.
A patient is prescribed 2 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the patient weighs 45 kg, how many milligrams of the drug should be given per dose?

جواب کھوليں

Step 1: Calculate the total daily dose of the drug based on the patient’s weight (measured in kilograms):
$(2\ \text{mg}/\text{kg})(45\ \text{kg})=90\ \text{mg}$
Step 2: Divide the total daily dose by the number of doses per day:
$90\ \text{mg}/3=30\ \text{mg}$
The patient should receive 30 mg of the drug in each dose.
A patient is prescribed 2 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the drug is available in 20 mg/mL and the patient weighs 60 kg, how many milliliters of the drug should be given per dose?

جواب کھوليں

Step 1: Calculate the total daily dose of the drug (measured in milligrams) based on the patient’s weight (measured in kilograms):
$(2\ \text{mg}/\text{kg})(60\ \text{kg})=120\ \text{mg}$
Step 2: Calculate the volume in each dose:
$(120\ \text{mg daily total})/(3\ \text{doses a day})=40\ \text{mg per dose}$
Step 3: Calculate the volume based on the strength of the stock:
$\begin{array}{l}(\text{prescribed dose needed})/(\text{stock dose})=\text{volume} \\ (40\ \text{mg per dose})/(20\ \text{mg}/\text{mL})=2\ \text{mL}\end{array}$
The patient should receive 2 mL of the stock drug in each dose.
For the following exercises, use the sample contact tracing graph to identify paths ().
1. How many people have a path of length 2 from Jeffrey?
2. Find 2 paths between Kayla and Rohan.
3. Find the shortest path between Yara and Kalani. State the length and people in the path.
جواب کھوليں
1. 5 (Lura, Naomi, Kalani, Vega, Yara)
2. Answers will vary. Two possible answers are as follows:
  Kayla, Jeffrey, Rohan
  Kayla, Lura, Yara, Lev, Vega, Uma, Kalani, Rohan
3. Length is 4. People in path = Yara, Lev, Vega, Uma, Kalani

Symbols used here

factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\emptyset,\ |A|

empty set, cardinality
The set with no elements; the number of elements of A.

\forall,\ \exists

for all, there exists
Quantifiers: every x; at least one x.

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow

not, and, or, implies, iff
Logical connectives.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

O(n^2),\ \Theta,\ \Omega

big-O notation
Grows no faster than n² (up to a constant), for large n.

a \bmod n

remainder
What is left after dividing a by n.

How to: Math and Medicine

Compute the mathematical factors utilized in concentrations/dosages of drugs.
Describe the history of validating effectiveness of a new drug.
Describe how mathematical modeling is used to track the spread of a virus.
How many people have a path of length 2 from Jeffrey?
Find 2 paths between Kayla and Rohan.
Find the shortest path between Yara and Kalani. State the length and people in the path.
5 (Lura, Naomi, Kalani, Vega, Yara)
Answers will vary. Two possible answers are as follows:

Questions people ask

What makes mathematics "discrete"?

It deals with separate, countable objects — integers, graphs, statements — rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.

How does a proof by induction work?

Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.

آپ کے اپنے کو کوشش کریں

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

میں زیادہ Discrete Math & Logic

Truth tables Sums and induction Proof by induction Algorithms and growth of functions