Evolutionary Medicine

Content Objectives

This 2-3 week curriculum unit will cover the 7th-grade standards: 7.RP.A.2 Recognize and represent proportional relationships between quantities. 7.RP.A.2.a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.A.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships³.

The unit is organized into seven lessons concluding with a summative assessment that includes problems related to calculating medicine doses for adults and infants using ratios and proportions. It also includes real-world scenarios where students must apply their knowledge to solve dosage problems accurately. In the first part of the unit, students will learn key, content-specific vocabulary and build background knowledge about ratios and proportions. In the next part, they will begin to understand the concept of medicine dosing as an application of ratios and proportions and begin to understand how to calculate doses for adults and infants using ratios and proportions. Lastly, they will design a dosage chart for a specific medication, considering both adult and pediatric dosing.

For a review of the math concepts in the following sections, a textbook such as Illustrative Mathematics would provide a useful guide.

Ratio

A ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.

A ratio may be written as "a to b" or "a:b", or by giving just the value of their quotient a/b. A ratio may be written as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers⁴.

Notation and terminology

The ratio of numbers A and B can be expressed as:

• the ratio of A to B
• A:B
• A is to B as C is to D
• a fraction with A as numerator and B as denominator that represents the quotient (i.e., A divided by B, or A/B)

This can be expressed as a simple or a decimal fraction, or as a percentage, etc. When a ratio is written in the form A:B, the two-dot character is sometimes the colon punctuation mark. The numbers A and B are sometimes called terms of the ratio, with A being the antecedent and B being the consequent. A statement expressing the equality of two ratios A:B and C:D is called a proportion, written as A:B = C:D or A:B=C:D. This is often expressed as (A is to B) as (C is to D).

Ratios are sometimes used with three or more terms, e.g., the proportion for the edge lengths of a "two by four" that is ten inches long is therefore thickness : width : length = 2:4:10; a good concrete mix (in volume units) is sometimes quoted as cement : sand : gravel = 1:2:4. For a (rather dry) mixture of ⁴⁄₁ parts in volume of cement to water, it could be said that the ratio of cement to water is 4:1, that there are 4 times as much cement as water, or that there is a quarter (¹⁄₄) as much water as cement.

The meaning of such a proportion of ratios with more than two terms is that the ratio of any two terms on the left-hand side is equal to the ratio of the corresponding two terms on the right-hand side.

Number of terms and use of fractions

In general, a comparison of the quantities of a two-entity ratio can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is ²⁄₃ that of the second entity.

If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are ²⁄₃ as many oranges as apples, and ²⁄₅ of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in a ratio of 1:4, then one part of the concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is ¹⁄₄ the amount of water, while the amount of orange juice concentrate is ¹⁄₅ of the total liquid. In both ratios and fractions, it is important to be clear about what is being compared to what, and beginning learners often make mistakes for this reason.

Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities. A separate fraction can be used to compare the quantities of any two of the entities covered by the ratio: for example, from a ratio of 2:3:7, we can infer that the quantity of the second entity is ³⁄₇ that of the third entity.

Proportions and percentage ratios

If we multiply all quantities involved in a ratio by the same number, the ratio remains the same. For example, a ratio of 3:2 when multiplied by 4 is the same as 12:8.

If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, ²⁄₅ , of the whole is apples and ³⁄₅, of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion⁵.

Proportion

A proportion is a mathematical statement expressing the equality of two ratios. a:b=c:d where a and d are called extremes, and b and c are called means. Proportion can be written as a/b=c/d, where ratios are expressed as fractions. This type of proportion is known as geometrical proportion, not to be confused with arithmetical proportion and harmonic proportion⁶.

Fundamental rule of proportion. This rule is sometimes called Means-Extremes Property. If the ratios are expressed as fractions, then the same rule can be phrased in terms of the equality of "cross-products" and is called Cross-Products Property⁷.

• If , then ad = bc
• If , then
• If , then
• If , then
• If , then

Dose

A dose is a measured quantity of a medicine, which is delivered as a unit. The greater the quantity delivered, the larger the dose. Doses are most commonly measured for compounds in medicine. The term is usually applied to the quantity of a drug administered for therapeutic purposes but may be used to describe any case where a substance is introduced to the body⁸. Over-the-counter medicines such as Tylenol and Ibuprofen are examples of drugs that require a specific dose that differs for adults and children. 

Factors affecting dose

A 'dose' of any chemical or biological agent (active ingredient) has several factors which are critical to its effectiveness. The normalization of the adult dose according to age, body weight, or any other demographic covariate without prior evidence of how these factors contribute to differences in drug exposure may lead to poor and unsafe estimates of the pediatric dose⁹.

In contrast to the use of off-label prescription entrenched in clinical practice, the introduction of pediatric regulation by the European Union and the renewal of the Pediatric Rule by the  U.S. Food and Drug Administration on the requirements for pediatric labeling impose special attention on dose selection in pediatric clinical trials.

Dose scaling in pediatric trials remains an important issue, from both a clinical perspective and a drug development standpoint. Given that children may not be subject to dose-finding studies similar to those carried out in the adult population.

Probably, the most common method for dose adjustment in children in pediatric clinical practice is to normalize the adult dose by body weight (i.e., mg kg−1), assuming a linear relationship between weight and dose. This means that the dose doubles with a twofold increase in the weight of a child¹⁰.

Over-the-counter medications

In over-the-counter medicines, dosage is based on age. Typically, different doses are recommended for children 6 years and under, children aged 6 to 12 years, and persons 12 years and older, but guidance aside from these ranges is slim. This can lead to serial under or overdosing, as relatively smaller people (compared to average size) take more than they should, and larger people take less. Over-the-counter medications are typically accompanied by a set of instructions directing the patient to take a certain small dose, followed by another small dose if their symptoms don't subside. Under-dosing is a common problem in pharmacy, as predicting an average dose that is effective for all individuals is extremely challenging because body weight and size impact how the dose acts within the body.

Prescription drugs

Prescription drug dosage is based typically on body weight. Drugs come with a recommended dose in milligrams or micrograms per kilogram of body weight, and that is used in conjunction with the patient's body weight to determine a safe dosage. In single-dosage scenarios, the patient's body weight and the drug's recommended dose per kilogram are used to determine a safe one-time dose.¹ Medication under-dosing occurs commonly when physicians write prescriptions for a dosage that is correct for a certain time but fail to increase the dosage as the patient needs (i.e. weight-based dosing in children or increasing dosages of chemotherapy drugs if a patient's condition worsens).

History of dosing

Dose and time considerations in the development and use of a drug are important for assessing actions and side effects, and predictions of safety and toxicity. The importance of dose – the amount of anything – is apparent in everything biological, and in all other matters.

To understand drug actions, the locations and identities of specific targets need to be known. Questions of drug logistics that require consideration and answers include whether the target is reached, what amount is available and acting at the target over time, and how receptor-binding, metabolism, and excretion occur.

Developing a pharmaceutical product from the discovery phase to market introduction can take 15 years. For instance, starting from a screening of 10,000 discovery compounds, 250 compounds will be selected for further preclinical evaluation. Of these 250 compounds, only five will be considered drug candidates that will enter human clinical trials, of which one will finally be introduced into the market¹¹ (See Table 1 for examples)

Table 1- Example development of a pharmaceutical product from the discovery phase to market introduction.

Historically mouse studies are used to help determine how much and how often a drug should be taken.  Robert Gatenby, M.D., is a radiologist who directs the Cancer Biology and Evolution Program at the Moffitt Cancer Center in Tampa, Fla. Gatenby and colleagues published results in Science Translational Medicine showing that mouse models of aggressive human breast cancer live longer with fewer side effects if they get successively lower doses of chemotherapy instead of standard high dose treatment¹².

Randomized clinical trials with human volunteers are another way to test a new medicine. Randomized clinical trials deal with issues in a cumbersome and heavy, handed manner, by requiring many patients to balance the heterogeneous distribution of patients into the different groups. By observing known characteristics of patients, such as age and sex, and distributing them equally between groups, it is thought that unknown factors important in determining outcomes will also be distributed equally¹³.

Trade-off/Evolutionary Medicine 

It is estimated that more than 50 million animals are used in experiments each year in the United States.¹⁴  

Dogs have their hearts, lungs, or kidneys deliberately damaged or removed to study how experimental substances might affect human organ function. 

Monkeys are taken from their mothers as infants to study how extreme stress might affect human behavior.

Mice are force-fed daily doses of a chemical for two years to see if it might cause cancer in humans.

It is unclear whether these studies are ideally suited to usefully testing the equivalent biology in humans.

The animals most commonly used in experiments—are “purpose-bred” mice and rats (mice and rats bred specifically to be used in experiments). Chimpanzees have not been subjected to invasive experiments in the U.S. since 2015 when federal decisions were made to prevent their use. Many believe that animal experiments are time-consuming and expensive. Animal experiments do not accurately mimic how the human body and human diseases respond to drugs, chemicals, or treatments¹⁵.

In the United States and European Union countries, approximately 15 and 7 million laboratory rodents, respectively, are used annually for research and testing. Given the surprising and controversial nature of the data concerning mouse-to-human (in) compatibilities in tested inflammatory disease models, the findings of the PNAS paper, (The Proceedings of the National Academy of Sciences) were quickly publicized in the lay press. The initial account of the research in the New York Times entitled, “Mice Fall Short as Test Subjects for Some of Humans’ Deadly Ills”, led to a subsequent ripple effect in the form of several alarming follow-up editorials, posts, and/or blog. Their collective conclusion was clear and implied that decades of mouse-based research culminated in few scientific advances wasted precious research opportunities and were a poor use of taxpayers’ money¹⁶.

For decades, mice have been the species of choice in studying human diseases. As a result, years and billions of dollars have been wasted following false leads, they say.

The study’s findings do not mean that mice are useless models for all human diseases. But, the authors emphasized they do raise troubling questions about diseases like the ones in the study that involve the immune system, including cancer and heart disease¹⁷.