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Contemporary Mathematics

13.3 Math and Medicine

Contemporary Mathematics13.3 Math and Medicine
Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
People are wearing face masks at an open market.
Figure 13.9 Shoppers wear masks during the Covid-19 pandemic. (credit: "True Covid Scene - Mask Buying" by Joey Zanotti/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

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

The pandemic that rocked the world starting in 2020 turned attention to finding a cure for the Covid-19 strain into a world race and dominated conversations from major news channels to households around the globe. News reports decreeing the number of new cases and deaths locally as well as around the world were part of the daily news for over a year and progress on vaccines soon followed. How was a vaccine able to be found so quickly? Is the vaccine safe? Is the vaccine effective? These and other questions have been raised through communities near and far and some remain debatable. However, we can educate ourselves on the foundations of these discussions and be more equipped to analyze new information related to these questions as it becomes available.

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.

FORMULA

Volume needed=(medicine dosage required)(weight of drug by volume)Volume needed=(medicine dosage required)(weight of drug by volume).

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.

Checkpoint

Note that the units for a desired dose of a drug and the units for a solution containing the drug or pill form of the drug must be the same. If they are not the same, the units must first be converted to be measured in the same units.

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.

Example 13.12

Calculating the Quantity in a Mixture

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

Your Turn 13.12

1.
How many milligrams of sodium chloride are there in 200 mL of a 0.9% w/v normal saline solution?

Example 13.13

Calculating the Quantity of Pills Needed

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?

Your Turn 13.13

1.
How many pills would be needed for a patient who has been prescribed 25.5 mg of a drug if each pill contains 4.25 mg.

Example 13.14

Calculating the Drug Dose in Milligrams, Based on Patient Weight

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?

Your Turn 13.14

1.
A patient is prescribed 1.4 mg/kg of a drug to be delivered intramuscularly, divided into 2 doses per day. If the patient weighs 60 kg, how many milliliters of the drug should be given per dose?

Checkpoint

Note that the units for a patient’s weight must be compatible with the units used in the medicine measurement. If they are not the same, the units must first be converted to be measured in the same units.

Example 13.15

Calculating the Drug Dose in Milliliters, Based on Patient Weight

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?

Your Turn 13.15

1.
A patient is prescribed 2.5 mg/kg of a drug to be delivered intramuscularly, divided into 2 doses per day. If the drug is available in 5 mg/mL and the patient weighs 52 kg, how many milligrams of the drug should be given per dose?

Who Knew?

Math Statistics from the CDC

The Centers for Disease Control and Prevention (CDC) states that about half the U.S. population in 2019 used at least one prescription drug each month, and about 25% of people used three or more prescription drugs in a month. The resulting overall collective impact of the pharmaceutical industry in the United States exceeded $1.3 trillion a year prior to the 2020 pandemic.

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.

People in Mathematics

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.

People in Mathematics

Gladys West

A portrait of Dr. Gladys West.
Figure 13.10 Gladys West (credit: "Dr. Gladys West Hall" by The US Air Force/Wikimedia Commons, Public Domain)

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 (Figure 13.11):

A graph represents contact tracing. The graph shows Alyssa at the center. Five lines from Alyssa lead to Suad, Sandra, Soren, Braeden, and Rocio. Two lines from Sandra lead to Nate and Munira. A line from Nate leads to Mikaela. A line from Braeden leads to Aarav. A line from Mikaela leads to Jose. A line from Jose leads to Lucia. A line from Rocio leads to Lucia.
Figure 13.11 Contact Tracing for Math 111 Section 1

At the center of the graph, we find Alyssa, whom we will consider positive for a virus. Utilizing the technology of phone apps voluntarily installed on each phone of the individuals in the graph, tracking of the spread of the virus among the 6 individuals that Alyssa had direct contact with can be implemented, namely Suad, Rocio, Braeden, Soren, and Sandra.

Let’s look at José’s exposure risk as it relates to Alyssa. There are multiple paths connecting José with Alyssa. One path includes the following individuals: José to Mikaela to Nate to Sandra to Alyssa. This path contains a length of 4 units, or people, in the contact tracing line. There are 2 more paths connecting José to Alyssa. A second path of the same length consists of José to Lucia to Rocio to Braeden to Alyssa. Path 3 is the shortest and consists of José to Lucia to Rocio to Alyssa. Tracking the spread of positive cases in the line between Alyssa and José aids in monitoring the spread of the infection.

Now consider the complexity of tracking a pandemic across the nation. Graphs such as the one above are not practical to be drawn on paper but can be managed by computer programs capable of computing large volumes of data. In fact, a computer-generated mathematical model of contact tracing would look more like a sphere with paths on the exterior as well as on the interior. Mathematical modeling of contact tracing is complex and feasible through the use of technology.

Example 13.16

Using Mathematical Modeling

For the following exercises, use the sample contact tracing graph to identify paths (Figure 13.12).

A graph represents contact tracing. The graph shows Jeffrey at the center. Five lines from Jeffrey lead to Leu, Aisha, Kayla, Nissa, and Rohan. Two lines from Kayla lead to Lura and Naomi. A line from Aisha leads to Lura. A line from Rohan leads to Kalani. A line from Lura leads to Yara. A line from Yara leads to Lev. A line from Lev leads to Vega. A line from Kalani leads to Uma. A line from Uma leads to Vega.
Figure 13.12 Contact Tracing for ECON 250 Section 1
  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.

Your Turn 13.16

For the following exercises, use Figure 13.12.
1.
List all the names of those who have a path length of three from Uma.
2.
What is the length of the shortest path from Naomi to Vega?

Check Your Understanding

8.
What two pieces of information are needed to calculate the volume of a prescription drug to be dispensed?
9.
Research on coronavirus vaccines began in 2020.
  1. True
  2. False
10.
What is mathematical modeling, and how is it used in the world of medicine with pandemics?

Section 13.3 Exercises

1 .
How many grams of sodium bicarbonate are contained in a 300 mL solution of 1.35% w/v sodium bicarbonate?
2 .
How many grams of sodium bicarbonate are contained in a 175 mL solution of 1.85% w/v sodium bicarbonate?
3 .
Using a saline solution that is 0.75% w/v, how many milligrams of sodium chloride are in 150 mL?
4 .
Using a saline solution that is 1.25% w/v, how many milligrams of sodium chloride are in 200 mL?
5 .
A prescription calls for a patient to receive 23 mg daily of a drug to be taken in pill form for 5 days. If the pills are available in 5.75 mg, how many pills will the patient need for the full prescription run?
6 .
A prescription calls for a patient to receive 21 mg daily of a drug to be taken in pill form daily. If the pills are available in 3.5 mg, how many pills will the patient need each day?
7 .
A patient is prescribed 4mg/kg of a drug to be delivered daily intramuscularly, divided into 2 doses. If the patient weighs 30 kg, how many milligrams of the drug would be needed for each dose?
8 .
A patient is prescribed 1.5 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the drug is available in 12.5 mg/mL and the patient weighs 50 kg, how many milliliters of the drug would be given per dose?
9 .
A patient is prescribed 0.5 mg/kg of a drug to be delivered intramuscularly, divided into 2 doses per day. If the drug is available in 2.5 mg/mL and the patient weighs 45 kg, how many milliliters of the drug would be given per dose?
10 .
A patient is prescribed 1.5 mg/kg of a drug to be delivered intramuscularly, divided into 3 doses per day. If the drug is available in 30 mg/mL and the patient weighs 54 kg, how many milliliters of the drug would be given per dose?
For the following exercises, use the mathematical modeling graph showing contact tracing for students in a particular class.
A graph represents contact tracing. The graph shows Justin at the center. Four lines from Justin lead to Nara, Luka, Kalina, and Tai. A line from Tai leads to Pasha. A line from Pasha leads to Javier. A line from Kalina leads to Javier. A line from Kalina leads to Luka. A line from Luka leads to Hani. A line from Hani leads to Nima. A line from Nima leads to Nara. A line from Nara leads to Aili. Two lines from Aili lead to Loise and Emmet. A line from Loise leads to Emmet.
Contract Tracing for Students in a Class
11 .
List the people who have a length of 2 from Justin.
12 .
Find 2 paths of with a length of 3 from Emmet.
13 .
Find the shortest path from Aili to Kalina.
14 .
Which people does the model show as directly in contact with Nara?
15 .
Find the shortest path from Tai to Hani.
16 .
Of the 12 people in the model, how many have a path of 2 or less from Justin?
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