Growth Unit

UNIT 8—Growth and Decay

Teacher Materials


CLICK THE  SYMBOL OF EACH SECTION HEADER TO RETURN HERE.
TEKS Support
Teacher Notes
 Context Overview
 Mathematical Development
 Teaching Suggestions
 Preparation Reading—"dos... age (do¢ sij) n."
 Activity 1—First Steps
 Homework 1—We Need Input, Stephanie!
 Activity 2—Tell Me If It Hurts!
 Homework 2—Let’s Play Doctor
 Activity 3—The ‘K’ in ‘Decay’
 Homework 3—Grrrrrrr-Rate!
 Activity 4—You Grow Some, You Decay Some
 Homework 4—Stay Between the Lines!
 Activity 5—Close to a Final Form
 Assessment—Dosage Deliberations
 Handout 1—Prescription Model
 Transparency 1—Prescription Model
 Transparency 2—Prescription Model
Annotated Student Materials
 Preparation Reading—dos... age (do¢ sij) n.
 Activity 1—First Steps
 Homework 1—We Need Input, Stephanie!
 Activity 2—Tell Me If It Hurts!
 Homework 2—Let’s Play Doctor
 Activity 3—The ‘K’ in ‘Decay’
 Homework 3—Grrrrrrr-Rate!
 Activity 4—You Grow Some, You Decay Some
 Homework 4—Stay Between the Lines!
 Activity 5—Close to a Final Form
 Assessment—Dosage Deliberations
 Unit Summary
 Mathematical Summary
 Key Concepts
Solution to Short Modeling Practice
 Solution for Modeling Bacteria Growth
Solutions to Practice and Review Problems


TEKS Support

This unit contains activities that support the following knowledge and skills elements of the TEKS.

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The mathematical prerequisites for this unit are

The mathematical topics included or taught in this unit are

The equipment list for this unit is

Teacher Notes



"Simple Model"—Context Overview

The context of pharmacology and prescription medicine forms the basis for this unit. The modeling process is applied to identify and isolate various parameters that are relevant to understanding the context. These parameters include the minimum effective level, the maximum level of safe use, the rate of dissipation, the initial dosage, the response time and the deadline for the completion of the treatment. Students are asked to consider these parameters in various combinations, and led to increasingly complicated situations.




Mathematical Development

Linear and exponential functions are reviewed as the delivery of oral medication (pills) is an additive process, while the rate at which the body uses up the medication is a multiplicative process. The distinction between discrete and continuous functions is brought up by assumptions on how the pills dissolve. Sequence notation plays a big part in describing the understanding of how the two disjoint growth processes work in tandem. Working with exponents and roots are brought into consideration, and solving equations using logarithms is introduced. Graphs and tables form the basis for studying the changes in various growth processes over time.




Teaching Suggestions

Note: The development of the materials were made with the assumption that graphing calculators with "home-screen iteration" capability would be available to students for examining long-term projections. It would be preferable to show students how to build spreadsheets, and to explore a lot of those problems with spreadsheet models. As you go through these teacher notes, remember that as an option, if your class has access to computer work stations.




Preparation Reading—"dos... age (do¢ sij) n."

The context is one that is probably familiar to your students, but the chances are that they haven’t thought about where the determination of dosage and time interval come from. The entire first day’s class is devoted to moving the students from the background that they bring on prescription medication and dosage to becoming active modelers of the problem.

Given that goal for the first day, encourage students to participate with their own experiences. Continue to bring them back to the modeling process and the questions that must be addressed, but try to elicit information or ideas on the subject from them, rather than just tell them "This is how we’re going to do it!"




Activity 1—First Steps
   

This is a fairly structured group/class activity, which will guide the students into the first stage of the modeling process. For each question of Activity 1, students should be given the opportunity to read, think and discuss the question in their groups, and then draft a response. That provides a lot of potential input for a class discussion, which will provide students an opportunity to formulate their ideas and suggestions, and then allow them to share those in a forum environment. Finally, assist the class to resolve what the "best" answer would be, and what direction to take in modeling the problem.




Homework 1—We Need Input, Stephanie!
   

The purpose of this assignment is to provide additional clarification on some of the terms brought into the previous discussion, and to build a basis for understanding how the parameters relate to each other. Also, there is an opportunity for students to go find out more about the context area, and to network with professionals that might be involved in pharmacology or medical practice. Students that can bring in information and related facts should be given the opportunity to share what they’ve found ongoing throughout the unit.

Be sure to review the answers to Question 5 of Part 1 thoroughly. The drug amount affects the concentration levels, and the assumptions made in this problem establish a "unit conversion" that will be consistently applied throughout the model.

Some students will see that these dosage problems are proportional and simply multiply to obtain the answer. For example, 600 mg is 6 times the initial does of 100 mg. Therefore, the concentration is 6 times higher. Others will need to do the proportion.

Here is how the proportion will look for problem 5a):



Here is how the proportion will look for problem 5b):



Students may also ask why pills are less effective than shots. There are many answers to this question. The most important factor is that pills travel through the digestive system, most notably the stomach. Stomach acids, and the process of digestion limit the amount of active ingredient that makes it to the bloodstream. Therefore, the doses for pills are often higher than for shots to achieve the same concentraiton. This is also an opportunity to discuss how being a doctor is often a compromise between treating illness, and the comfort and convenience of patients.




Activity 2—Tell Me If It Hurts!
   

This activity focuses solely on the additive nature of the delivery of medicine by repetitive ingestion of pills. The modeling process begins by making unrealistic assumptions on the problem, simply so that a beginning reference can be established for understanding and describing the situation. In this context, we will assume (temporarily) that the chemicals don’t dissipate in the blood, and develop the mathematics that describes additive growth processes.

Various situations and strategies are considered, and students are encouraged to put themselves in the role of a doctor as they write out prescriptions for the various problem solutions. Remind students to associate given information in a particular problem with the problem parameters identified in the handout Prescription Parameters. Some of what the students will be doing in the unit is quite complex and can be confusing. As a strategy for you in helping students understand what they are doing or need to do, you can also draw the connection to those parameters. If students experience difficulty in answering parts of the assignment (or any assignments from the unit), remind them that the tools for exploration are tables and graphs, even if they are not specifically requested in the problems! Point out to students that the traditional names for arithmetic sequences are a0, a1, a2, a3, a4, a5, etc.

Note: 1 µg = 10–6g




Homework 2—Let’s Play Doctor
   

This set of problems still uses the assumption that the chemicals stay in the body indefinitely, and provides students more opportunity to solve problems in manipulating the various parameters. Question 2d) is a chance for your stronger students to apply the "Method of Generalization" and develop a solution in terms of the various parameters, rather than specific value solutions.




Activity 3—The ‘K’ in ‘Decay’
   

Activity 3 is where students focus their attention on the rate of dissipation as a parameter, and the mathematics behind exponential decay processes. Start with a discussion of the modeling process. The assumption that chemicals never dissipate was very unrealistic; students’ experiences will be to the contrary. Rather than add in an additional consideration, this activity tables the consideration of the dosage, and explores what happens to quantities over time once they are ingested. Make sure students understand the reasoning behind the decision that "Simpler is better", until you are able to put everything together.

There may be a need to go over how to convert from a percent of decline to a growth factor. In addition, recursive equations will be extensively used during the rest of the unit, so there should be an emphasis placed on students understanding what the subscripts do for the equation. Point out to students that the traditional names for geometric sequences are b0, b1, b2, b3, b4, b5, etc. A review of exponents, how they represent repeated multiplication operations and how to evaluate them, might be in order.

In debriefing Activity 3, be sure to go over Questions 5-8. Half-life is a useful way to describe exponential decay sequences, and all the various methods introduced for solving equations to find the half-life should be reviewed with the students.




Homework 3—Grrrrrrr-Rate!
   

Practice problems from Activity 3 form the majority of this set of questions. It can be an opportunity for you to find out if your students really understood the material brought out in the Activity. Question 6 asks students to think about how to mix additive growth and exponential decay processes together. In asking the question, it is setting the table for the understanding of how to describe those kind of patterns, and preparing students for the next Activity. Be sure to discuss the students’ findings with them on this.




Activity 4—You Grow Some, You Decay Some
   

In modeling the contextual problem, we’ve considered the effect caused by the oral prescription alone, and we’ve considered the effect caused by the body metabolizing the medication alone. In Activity 4, you now consider what happens when both are going on simultaneously. Be sure students understand that one action is being treated as a discrete process (assumption that the pill dissolves immediately causes "quantum" jumps in concentration), while the other is a continuous process (gradual dissipation of the medicine in the body). This shows up in the graphs as repeated spikes of increase, and classic exponential decay curves.

Mixed growth sequences form the basis for this activity. Students start off by building a sequence that balances the additive growth with the exponential decay. Then, a couple of examples in which the growth processes are not balanced are examined. Change of scale is introduced as a way of focusing on the top or bottom of the exponential curves over time, instead of having to deal with disjoint pieces. Finally, students arrive at the recursive expression defining mixed growth sequences; the logic that describes how to go from one time interval to the next is pretty nice, especially compared to the closed-form equations. Make sure your students understand the logic behind these expressions before moving forward. Also, make sure that the students understand the distinction between the pattern formed by the peak concentrations (pill first, then measure concentration) and the "bottoming-out" concentrations (measure concentration first).




Homework 4—Stay Between the Lines!
   

Students return to the model in this assignment as they juggle parameter values and try and solve problems based on the various conditions and constraints. Consideration for the boundary values of the partial sums is brought out in this set of problems, although not formalized. Again, remind students to refer to the Handout 1—Prescription Model. These problems are beginning to get conceptually more demanding, and there is a greater need for table building, home-screen iteration on the calculator, exploration using a spreadsheet template, and making graphs. Another tool that students can use is to sketch an x- and y- axes, and a rectangular "window" in the first quadrant, to represent the conditions for each of the problems. Encourage them to visualize how concentration would respond under each constraint condition.




Activity 5—Close to a Final Form
   

In this activity, a mathematical approach for finding the theoretical maximum value of a mixed-sequence growth is explored. The modeling approach is to try and establish a more general solution to the problem, than to explore each one as a "new" adventure. Point out to students that they are getting close to the crux of the problem, and that it is always more difficult sledding.

The idea is to analyze the computations for a mixed-sequence growth, in order to determine what’s going on. The first pill’s dosage goes through a decay process repeatedly with each time interval, and the second one as well, although it hasn’t been in the body as long. Each pill’s contribution to the total concentration is expressed using the algebra of exponents, since two time intervals would have a growth factor of k2 (if one time interval has a growth factor of k).

Depending on the students you have, you might want to discuss Questions 2, 3 and 4 in a "whole-class" forum, especially the parts requiring intensive algebra. Make sure the students see the intuitive development of the infinite series as a limit as the number of terms goes to infinity, and how that reduces the partial sums formula to the infinite series formula. Also, make sure they recognize that the infinite series formula has conditions on when it converges to that expression, namely that the growth factor must be less than one.




Assessment—Dosage Deliberations
   

This is the final stage of the work on developing the model. Students take the work done from Activity 5 in developing expressions for calculating the dosage and/or time interval from the theoretical maximum concentration allowed in the problem. They have to adjust their thinking to build in considerations for threshold and tolerance levels, and for response time. Strategies for generalizing the entire problem are hinted at, or at least improper strategies are examined and critiqued. Between the examples that students must work through, and the situations that students need to analyze, there should be enough foundation for students to arrive at a general process for determining dosage and time intervals.

Question 3, 4 and 9 should be discussed with the entire class. If students are able to formulate the process in English, help them to express it in mathematical notation. At any rate, after completion of this assignment, students should review the work done in the unit to try and summarize what was done. A writing assignment requesting that process work could be a nice way of finishing off the unit core material, prior to any assessment of the learning in the unit.




Handout 1—Prescription Model

 




Transparency 1—Prescription Model

 


Transparency 2—Prescription Model



Annotated Student Materials



Preparation Reading—dos... age (do¢ sij) n

Have you ever been given a prescription for some medication? It’s likely that the label on the bottle did not say, "Take two, and call me in the morning!" If it’s an oral medication (pills), probably the number of tablets in the bottle is somewhere on the label. There should also be some way of identifying what substance is in the container, and how much prescribed medicine is in each tablet. And most importantly, there should be instructions on how many pills to take each day, and how often. In other words, what is the dosage?

The American Heritage Dictionary of the American Language has the following definition for dosage: "1. The administration of a therapeutic agent in prescribed amounts. 2. The amount administered." Have you ever wondered how doctors come up with "the amount to be administered"? What kind of thinking and decision-making does it take to determine how to administer something in prescribed amounts? It may be as simple as looking up information in a table, or plugging some numbers into a formula that’s as old as medical knowledge. But it also is a rich problem context in which to explore and to practice mathematical modeling.

In this unit, you are going to place yourself in the role of a medical practitioner and practice the skills you’ve developed in learning to model. What is the question? What are some assumptions that need to be made? What are the variables? How should we begin this exploration? When you get through all that, you still have to figure out what kind of mathematics applies here. You may even need to learn more mathematics in order to solve your problem. That can be a lot to deal with for a single problem and may seem impossible, but there is one thing that is reassuring about the situation. Doctors solve this kind of problem on a daily basis, so it has to be possible. And their work can provide a kind of reality-check for our mathematical prediction, so we can test how effective our model is.

Take a few moments to think about the field of prescriptive medicine. How do they do that, anyway?




Activity 1—First Steps

 = 1, 10, 11

  1. Discuss the reading in your group; share experiences you may have had with the actual context. Then state a problem that would be "worthy" of modeling.


    2. Group:Answers will vary; refer to Teacher Notes for discussion ideas.

    Class: How can you determine the amount of medication to prescribe, and the time interval between dosages (or the number of dosages to give in a predetermined time period)?
  1. What are some of the factors to consider? What limitations must be considered?


    2. Group: Answers will vary; refer to Teacher Notes for discussion ideas.

    Class: Factors to consider include: the size of the patient, the kind of chemical being administered and its rate of absorption and dissipation in the body, whether an initial treatment is applied, whether the drug is administered orally or intravenously, whether the body manufactures the chemical.

    Limitations include: the amount of medicine needed to be effective, the amount that begins to cause harm to the patient, the time interval in which you must begin treatment, the time interval in which you must have completed treatment.
  1. In developing a model for this situation, what assumptions will be initially made?

    2. Group: Answers will vary; refer to Teacher Notes for discussion ideas.

    Class: The critical factor to monitor is the concentration level. Assume that the body weight and blood volume are constant, so that the amount of medicine present directly affects the concentration level. There’s no prior amount of the medicine present before administering the first dosage, and the body doesn’t manufacture the chemical naturally.
  1. What are the variables for the model?


    2. Group: Answers will vary; refer to Teacher Notes for discussion ideas.

    Class: Input (independent) variables will be dosage and time between doses, and the output (dependent) variable will be concentration.
  1. What might be a good "next step" to take in developing the model?


    2. Group: Answers will vary; refer to Teacher Notes for discussion ideas.

    Class: Consider a simpler situation in which only one of the parameters is considered at a time; i.e., where the chemicals don’t dissipate, where there isn’t an initial treatment, or where there is only an initial treatment.



Homework 1—We Need Input, Stephanie!
   

Part 1: Understanding the Problem

In the discussion that took place in Activity 1, input/output variables and parameters for the problem were introduced. For the following questions, Handout 1, Prescription Model, will be useful.

    1. What happens if the concentration level C(t) doesn’t get as high as Cmin?


      2. The amount of the drug won’t be great enough to do any good, and certainly won’t treat the condition.
    2. What happens if the concentration level C(t) gets higher than Cmax?


      3. There could be side effects (i.e., hives), overdose symptoms (i.e., falling asleep), and possibly a new condition brought on from toxic levels of the chemical (i.e., kidney failure).
    3. What does it mean for the response time to be fairly small, like say a couple of hours?

      4. It means that you’d better see the doctor really quickly, or there may be complications. For example, if you have pneumonia, antibiotics can treat the situation fairly easily. But if you go past the response time, you might get fluid in the lungs, which would be very serious.
    4. What does it mean when the deadline for treatment is fairly big, like say a month?


      5. It might mean that there’s no rush to treat the condition, as long as it’s taken care of during that time, but it may mean that the treatment needs to be stretched out over a long time period. What is definite is that the treatment needs to be finished by that moment in time.
    5. What would happen if Cmin were bigger than Cmax?


      6. The condition can’t be treated; or if it has to be treated, there will be some complications, since the smallest amount required to take care of the problem will be more than what the body can handle safely.
  1. Use the axes below to record your answers to parts a) and b). Introduce a scale and use numbers if necessary to draw the lines in the correct places, but indicate what numbers you are using for each of the values.

    1. Sketch in two vertical lines to represent the response time to and the treatment deadline time tf.
    2. Sketch in two horizontal lines to represent the threshold level Cmin and the treatment deadline time Cmax.
  1. If you were using the graphing calculator, what feature would represent the box that you just drew in the previous problem? What would to, tf, Cmin and Cmax represent?


    2. Basically, that’s establishing a WINDOW for viewing the important part of the problem, with xmin = to, xmax = tf, ymin = Cmin and ymax = Cmax. See the shaded part of the graph.
  1. There is a relationship between the response time, the treatment deadline time, the number of doses and the time between doses. For now, we won’t worry about the output concentration, so the actual amount in each pill is not important. Work out a couple of examples first:


    1. to = 6 hrs, tf = 36 hrs, and D = 12 pills. Find Dt.


      2. Note: Review the use and meaning of "D."

      Total treatment time = 36 – 6 = 30 hours. Break that time up into 12 intervals, and you get 2.5 hours in between pills.
    2. to = 2 hrs, tf = 36 hrs, and Dt = 2 hrs. Find D.


      3. 36 – 2 = 34 hours 1/2 2 hour intervals = 17 pills.

    3. Write a general relationship between to, tf, D and Dt.


      4. There will be various ways to express this, but they should all be equivalent to:
    4. What assumptions did you make in answering parts a) and b)? (Those become assumptions on using the relationship you just derived!)


      5. We assumed that the first pill would be given at the beginning of the treatment time interval (at to), and that the last pill to be given would "lose its effect) at the treatment deadline time. In other words, a pill would not be given at the deadline time, since its effects would be after the deadline.

  2. When considering prescriptions, doctors have to be concerned with two factors:

    First, it is not the amount of the dose that treats the problem, but the concentration of the substance that matters. The concentration of the drug is equal to the amount of the substance divided by the blood volume of the patient or. This concentration is usually measured in micrograms over milliliters (µg/ml). Blood volume is strongly related to weight, so most dosages increase with the weight of the patient. The doctor may also consider other factors like gender, metabolism, and general health when they prescribe a drug.

    Second, a part of any drug is used up in dissolving pills, digestion, and other factors. Therefore pills and injections require different dosages to reach the same concentrations in the blood stream. Generally injections are more effective than pills, and require smaller doses. Pills, however, are more convenient for patients to take on a regular basis.

    1. Let's assume that a typical adult male with a blood volume of 2,000 ml is given an injection of 100 mg. After losses getting to the blood stream, this causes a change in concentration of 50 µg/ml. What would the change be for a 600 mg shot?

      (600 mg)/C = (100 mg)/(50 µg/ml)

      C = 300 µg/ml

    2. Let's assume that the same adult male is given a 100 mg pill. After losses getting to the bloodstream, this causes a change in concentration of 10 µg/ml. What would the change in concentration be for a 250 mg pill?

      (250 mg)/C = (100 mg)/(10 µg/ml)

      C = 25 µg/ml

      Note: In developing this model, we will continually work with this patient and drug combination. Therefore, the following assumptions are needed:

      • The drug distributes itself evenly throughout the body.

      • A 100-mg shot will change the concentration of drug by 50 µg/ml.

      • A 100-mg pill will change the concentration of drug by 10 µg/ml.

      You may want to investigate how answers might change if you varied the type of drug administered (which will change the difference in effectiveness between the shot and the pill, the weight of the patient (which will change the blood volume), or other factors you might identify).

      Note:

      1 millogram =

      1 mg = (1/1,000,000)g = 10–6 g

      1 milligram = 1 mg = (1/1,000)g = 10–3 g

      1000 mg = 1 mg



Part 2: Gaining Familiarity With the Problem

It would be nice to know if our parameters, assumptions and variables form a "total package", so we need to check in with some relative experts on the subject. There are several sources of information.

Bring in whatever you can find to share with the class. Happy hunting!




Activity 2—Tell Me If It Hurts!
   

 = 7, 12, 13, 16

In the discussion on modeling the problem of prescription dosage that just took place, a lot of factors were brought in to consideration. A lot of times, it can be overwhelming to consider all of them at the same time, so we’re going to consider a simpler situation. For the duration of this activity, we’ll assume that the chemical never breaks down in the body, but simply accumulates over time as more of it is put into the body.

Remember the assumptions for our patient:

  1. Consider a situation in which a patient has a tolerance level of 100mg for a certain chemical, but a threshold level of 40mg is necessary to treat a condition. As the doctor, you decide to give the patient a single shot, and not bother with a prescription.


    1. How much of the chemical should be in the shot?

      2. Answers will vary, but should range from 80 to 200 mg. Probably the midrange value of 140 mg would be the safest answer.

      A/(70 µg/ml) = 100 mg/(50 µg/ml)

      A = 140 mg
    2. List the various parameter values that relate to this problem and their values. We’ll call this a "control panel" for the problem.

      3. D = 0; Cmin = 40; Cmax = 100; k = 0;

      Co = whatever the answer to part a) was = 70 µg/ml.

    3. Sketch a graph of the situation in the space provided above. Indicate where all the important information is located.

    4. Why is the input variable D not considered? Why is the input variable Dt not considered? Why are the parameters to and tf not considered?

      5. D isn’t considered because we’re assuming that the treatment will consist of a single shot, in this case containing 70 mg. Dt, to and tf are not important since we’ve assumed the chemical doesn’t dissipate over time, and the shot is being given at time t = 0.

    5. Write an equation describing this situation. What would the general equation for these assumptions look like?

      6. In this particular case, C(t) = 70.

      In general, C(t) = n, where n is some number in the interval [Cmin, Cmax].
  1. Let’s try a different approach with that same problem now. Suppose the treatment favors a gradual increase of the concentration level over time, until it just reaches the tolerance level at the treatment deadline time of 48 hours. Furthermore, being anxious to begin treating the symptoms of the condition, you decide on a single shot to bring the chemical concentration up to the threshold level, and then oral medication applied at regular time intervals to finish the job.

    1. How much medicine should be in the shot? How much more medicine needs to be provided from the pills?

      2. 80 mg in the shot; 600 mg from the pills
    2. If the patient were to take a pill every hour for the entire 48 hours of treatment, what would have to be the dosage (amount of each pill)? How did you determine that answer?

      3. Dpill = 12.5 mg. Divide the amount the patient still needs by the number of hours.
    3. It’s more convenient to have the patient take pills twice a day, or four times a day, than to be woken up in the middle of the night every hour, taking a pill. Have the patient take a pill every 6 hours. How many pills, and at what dosage, will the pharmacist provide? What’s the prescription going to read?

      4. Quantity: 8 pills. Dosage: 75 mg. Prescription: Take 1 pill every 6 hours.
    4. Make a table of values for the total amount of medication at 6-hour intervals. Then sketch the graph of concentration of medicine vs. time.


      5. Time
      (hr)

      Concentration
      (µg/ml)

      0

      40

      6

      47.5

      12

      55

      18

      62.5

      24

      70

      30

      77.5

      36

      85

      42

      92.5

      48

      100

    5. In describing this model, what is the closed-form equation for C in terms of t? For C in terms of n (the number of pills taken)?

      6. Closed form: C(t) = 1.25t + 40, where t is measured in hours and C in mg/ml.

      C(n) = 7.5n + 40
    6. Which equation makes more sense to use in this situation? Explain.

      7. It depends on the assumptions. The first one implies that the medication is being released at the same rate during each moment of the 48 hour period. The second one says that the concentration "jumps" every time we take a pill, which isn’t real, but we can "assume" that the pill dissolves instantly.

      The concentration was affected by the number of pills taken, and formed a pattern that was created by repeating an addition operation. That kind of pattern is called an arith-metic sequence; describing such a sequence involves associating the values with its place in the list of numbers.
    7. Fill in the table with the appropriate values. Notice the names used to describe each of the numbers in the sequence.


      8. Term Name

      c0

      c1

      c2

      c3

      c4

      c5

      c6

      c7

      c8

      No. of Pills Taken

      0

      1

      2

      3

      4

      5

      6

      7

      8

      Concentration

      40

      47.5

      55

      62.5

      70

      77.5

      85

      92.5

      100
    1. Now, describe the sequence in recursive form—using one statement to describe what the original value is, and the other statement to describe how to get any value from the previous term of the sequence.

      2. c0 = 40; cn = c n–1 + 7.5 (0 £ n £ 8)

    2. How could you have calculated the value of the 8th term?

      3. Start with 40 and add 7.5 to it a total of 8 times, or

      40 + (7.5)(8).

  1. There may not be a need for immediate treatment, or building up concentration over time with pills may be preferred to giving a shot to the patient. Let’s look at an example of that kind of administration strategy. Suppose that the threshold level of 180 mg must be reached at the response time, which is 10 hours, and that you want to reach the tolerance level of 450 mg by the treatment deadline, which is 36 hours.

    1. How much medicine needs to be provided per hour in order to reach the threshold level on time? Break into 2 time intervals:

      2. 0®10 to reach threshold

      10®36 to reach maximum

      1800 mg / 10 hr = 180 mg/hr
    2. Once you achieve that concentration, how much more medicine needs to be given per hour in order to reach the tolerance level on time?

      3. Amount = 4500 – 1800 = 2700 mg

      Rate = 2700 µg/26 hr = 10.38 mg/hr

    3. How are you going to have to address the fact that the two previous answers aren’t the same?

      4. You need to use two different prescriptions
    4. Let’s say you see the patient at 9:00 a.m., and you decide that taking a pill every two hours is desirable until you reach the threshold level, and after that you want a pill every 6 hours. Write out the prescription.

      5. Pink pills: quantity 5, dosage 360 mg, take one every 2 hours until gone, then switch over to the purple pills

      Purple pills: quantity 4, dosage 675 mg, take one every 6 hours until gone
    5. Explain how you arrived at the prescription you gave.

      6. Because the time interval for the first stage was determined to be 2 hours, I multiplied the hourly amount by two to get the dosage. The quantity was calculated by taking the number of total hours and dividing it by the time interval chosen. For the second stage, the time remaining (26 hrs) didn’t divide by the time interval (6 hours) evenly. The number of complete 6-hr time intervals became the quantity, and the amount that was left to be delivered was divided by the quantity to determine the dosage.
    1. Fill in the table, and then make a graph of C(t) vs. t.


      2. Term

      Time (hrs)

      Concentration
      (mg/ml)

      c0

      0

      0

      c1

      2

      36

      c2

      4

      72

      c3

      6

      108

      c4

      8

      144

      c5

      10

      180

      c6

      16

      247.5

      c7

      22

      315

      c8

      28

      382.5

      c9

      34

      450

    2. Write a recursive equation for how the pills increase the concentration for each part of the prescription strategy of this problem.

      3. c0 = 0, cn = cn–1 + 36 (0 £ n £ 5) and c5 = 180, cn = cn–1 + 67.5 (5 £ n £ 9)

    3. What assumptions have been made about the way in which the pills dissolve?

      4. The pills are completely dissolved by the end of the time period, and we either don’t know or don’t need to be concerned by what’s going on during each time period.
    4. Write closed form equations that describe how concentration changes over time for each part of the prescription strategy of this problem.

      5. C(t) = 18t + 0 (0 £ t £ 10) and C(t) = 11.25(t – 10) + 180 (10 £ t £ 34)

    5. What assumptions have we now made about the way in which the pills dissolve?

      6. The pills are "time-delayed" and dissolve a little bit at a time, and at a constant rate.
    6. Explain how the numbers that make up the closed-form equations are determined by the problem conditions.

      7. In the first equation, the domain is determined by to, the y-intercept by C0, and the slope by Cmin. In the second equation, the domain is determined by the expression tf – to, the y-intercept by C0, and the slope by both tf – to and Cmax – Cmin.



Homework 2—Let’s Play Doctor
   

You wish to compare the strategy of giving a shot to raise the concentration up to the threshold level immediately with one in which you use oral medication to bring the concentration up to the threshold level by the response time. Let’s use a threshold level of 200mg/ml and a tolerance level of 500 mg/ml. The response time will be 12 hrs and the treatment deadline will be 48 hrs.

  1. Consider the strategy of giving an initial shot first. Calculate the amount needed for the shot, and write a prescription for the oral medication.

    The amount needed by the shot is 400 mg, since C0 = Cmin = 200 µg/ml. The additional amount to be delivered by pills is 3000 mg. This is to be done over a 48-hr period, but at 6-hr intervals. Therefore, the prescription will read: Quantity: 8. Dosage: 375 mg. Take 1 every 6 hours.
  2. Now, consider the strategy of treating the condition only with pills. Write the prescription for the oral medication.

    3. The threshold level of 200mg/ml should be reached within 12 hours, in 6-hr doses, so the dosage for the pills has to be 1000 mg each. At that rate, the tolerance level will be reached after 5 pills. So, the prescription will read:

    Quantity: 5. Dosage: 1000 mg. Take 1 every 6 hours.
  3. Let’s change the time between doses to 4 hours. How do the prescriptions change for each of the strategies?

    4. Strategy 1 (with initial shot): Amount of shot: 400 mg. Prescription: Quantity: 12. Dosage: 250 mg. Take 1 every 4 hours.

    Strategy 2 (no initial shot): Prescription: Quantity: 12. Dosage: 416.7 mg. Take 1 every 4 hours.
  4. Now, let’s put everything together. Describe the prescriptions for each of the two strategies in the general case, where the time between doses is Dt, the amount of dosage D, the number of pills n, the amount of the shot C0, the threshold level Cmin, the tolerance level Cmax, the response time to and the deadline time tf.

    5. Strategy 1 (with initial shot): Amount of shot C0 = Cmin

    Quantity: n = integer value of (tf / Dt)

    Dosage: D = (Cmax – Cmin) / n

    Take 1 every Dt hours

    Strategy 2 (with no shot): Initial amount of medicine C0 = 0

    If Cmin / to £ Cmax / tf, then:

    Quantity: n = integer value of (tf / Dt )

    Dosage: D = Cmax / n

    Take 1 every Dt hours

    If Cmin / to > Cmax / tf, then:

    Quantity: n = integer value of ( (Cmax / Cmin) * (to / Dt ) )

    Dosage: D = Cmax / n

    Take 1 every Dt hours




Activity 3—The ‘K’ in ‘Decay’
   

 = 2, 4, 6

Our model of how medicine is introduced into the bloodstream was based on the assumption that the medicine stayed in the body forever. Now we want to focus on how the body uses up, or metabolizes, medicine. We will make two assumptions at this time. The first assumption is that no additional medicine is taken. This assumption will be changed in a later activity.

The second assumption is that a constant fraction, or constant percentage of medicine is metabolized per unit time. This fractional rate of change, which we will designate as r, is constant. The fraction of medicine remaining after a unit time interval is also constant and is designated by K. The two constants are related as:

K = (1 – r)

  1. Let’s start with a situation in which there is 800 mg/ml of a certain medicine present at first, and it decays at a rate of 10% per hour.


    1. What would the concentration be after 1 hour? Explain how you got that answer.

      2. r = 0.10
      K = (1 – r) = 0.90
      C1 hr = 800 µg/ml (0.90) = 720 µg/ml
    2. What would the concentration be after 2 hours? Explain how you got that answer.

      3. At the start of the second hour we have 720 µg/ml. We will lose another 10% the second hour. We still have r = 0.10 and K = 0.90.

      C2 hr = 720 µg/ml (0.90) = 648 µg/ml
    3. Fill in the table with concentrations for the first 10 hours, then make a graph of C(t) vs. t.


      4. Term

      Time
      (hr)

      Concentration
      (mg/ml)

      c0

      0

      800

      c1

      1

      720

      c2

      2

      648

      c3

      3

      583.2

      c4

      4

      524.88

      c5

      5

      472.39

      c6

      6

      425.15

      c7

      7

      382.64

      c8

      8

      344.37

      c9

      9

      309.94

      c10

      10

      278.94

    1. Write a recursive expression for this sequence that describes how it decays each hour.

      2. C0 = 800; Ct = Ct–1 – (0.10) C t–1 or Ct = C t–1 * 0.90
    2. Fill in the table with the values that describe how the chemical decays over 2-hour time periods, and write a recursive expression for this sequence.


      3. No. of 2-hr Time Intervals (n)

      0

      1

      2

      3

      4

      5

      Concentration (C)

      800

      648

      524.88

      425.15

      344.37

      278.94

      C0 = 800; Cn = Cn–1 – (0.19)Cn–1 or Cn = Cn–1 * 0.81
  1. Fill in the table with the values that describe how the chemical decays over 5-hour time periods, and write a recursive expression for this sequence.


    2. No. of 2-Hr. Time Intervals (n)

    0

    1

    2

    Concentration (C)

    800

    472.39

    278.94

    C0 = 800; Cn = Cn–1 – (0.40951)Cn–1 or Cn = Cn–1 * 0.59049
    1. Explain how you determined the numbers describing the last two decay processes.

      2. I took repeated subtractions of 10%, and then determining the percent of the original amount, or I just multiplied 0.90 by itself as many times as the number of hours in the time interval, or refer to table in 1c.

      K5 hr = b5/b0 = 472.39/800

      K5 hr = 0.59

    2. Here’s a closed-form equation for the concentration drop-off on an hourly basis:

      3. C(t) = 800*(0.90)t.

      The order of operations is to raise (0.90) to the power specified by the value of the variable t, and then multiply that answer by 800. Verify that the answer you wrote as the value for the concentration at the end of the 8th hour is correct, by using t = 8 in the closed-form equation.

      C(8) = 800 * (0.90)8 = 800(0.90)(0.90)(0.90)(0.90)(0.90)(0.90)(0.90)(0.90) = 344.37. (It’s the same!)
    3. Write closed-form equations for the other two patterns, which keep track of the concentration in 2-hour and 5-hour intervals.

      4. C(t) = 800*(0.81)t and C(t) = 800*(0.59049)t.
  1. We’re going to go back to the original problem and change the rate of dissipation to see how that affects the concentration level over time. This time, the 800 mg/ml of medicine that is in the body decays at only 3% per hour.


    1. What’s the growth factor, the number by which you multiply the previous hour amount to get the next hour amount?

      2. 0.97
    2. Fill in the table and sketch the graph of concentration C(t) vs. time t.


      3. Time
      (hr)

      Concentration
      (µg/ml)

      0

      800

      1

      776

      2

      752.72

      3

      730.14

      4

      708.23

      5

      686.99

      6

      666.38

      7

      646.39

      8

      627

      9

      608.18

      10

      589.94

    3. Describe this decay pattern in both recursive-form and closed-form expressions.

      4. Recursive: C0 = 800; Cn = Cn–1 * 0.97

      Closed Form: C(t) = 800 (0.97)t
  1. Now, repeat the same calculations as in the previous problem, only allow the medicine to decay at 25% per hour.


    1. What is the growth factor in this case?

      2. 0.75
    2. Fill in the table, and sketch the graph of concentration vs. time.


      3. Time
      (hr)

      Concentration
      (µg/ml)

      0

      800

      1

      600

      2

      450

      3

      337.5

      4

      253.13

      5

      189.84

      6

      142.38

      7

      106.79

      8

      80.09

      9

      60.07

      10

      45.05

    3. What are the recursive and closed-form equations for this decay process?

      4. Recursive: C0 = 800; Cn = Cn–1 * 0.75

      Closed Form: C(t) = 800 (0.75)t
  1. Describe what the various values of the growth factor did for the table and graph of the three situations that were examined.

    2. In each case, the table values for concentration got smaller over time; the graph was a curve that started at 800, dropped toward the x-axis and curved in the direction of the x-axis. As k, the decay rate, got larger, the more quickly the graph curved towards the horizontal axis; the table values got smaller faster as well.
  1. Sometimes, the rate of dissipation (or any other decay process that behaves like this) is described in terms of its half-life, the time it takes for half of the amount to decay. Refer back to the work on the medicine that decays at the rate of 10% per hour.


    1. How many hours it will take for half of the original concentration to be used up? Explain how you got that answer.

      2. Between 6 and 7 hours. Look in the table for how long it takes the 800 mg/ml to decay down to 400 mg/ml.

    2. If the rate of dissipation is larger than 10%, will the half-life go up or down?

      3. It will go down, since the concentrations will get smaller faster.
  1. Let’s examine one of the ways in which you could determine the half-life for a substance, if you knew the rate at which it was being consumed. Let’s assume that the 800 mg/ml of medicine under study is used up in the body at the rate of 15% per hour.


    1. You could get the half-life (at least a "ball-park" figure) by extending the table until you have half the amount that you started with. Complete the table for this decay sequence, and determine the half-life:


      2. Time (t)

      0

      1

      2

      3

      4

      5

      6

      7

      Concentration (C)

      800

      680

      578

      491.3

      417.6

      355

      301.7

      256.5

      The half life would be somewhere between 4 and 5 hours, since that’s the interval in which the substance decreases to only 400 mg/ml.

    2. You could even narrow down the answer, if you figured out a growth factor over a different interval of time. Earlier, we calculated growth factors over two or five hours by (growth factor)2 and (growth factor)5. If we broke down each hour into tenths, what would the growth factor be for each of those intervals?

      3. It’s a solution to the equation x10 = 0.85 (the growth factor in this case).

      So, the new growth factor would be: » 0.9839
    3. Now, fill in the table for this same situation between t = 4 and t = 5 hours, using the growth factor you just calculated. What is your new estimate for the half-life?


      4. Time (t)

      4.0

      4.1

      4.2

      4.3

      4.4

      4.5

      4.6

      4.7

      4.8

      4.9

      5.0

      Concentration (C)

      417.6

      410.9

      404.3

      397.8

      391.3

      385

      378.8

      372.7

      366.7

      360.8

      355

      The half-life for this medicine will be between 4.2 and 4.3 hours.
  1. Another way to determine the half-life would be to use the calculator’s ability to graph functions. Enter the closed-form equation describing this decay sequence as your first function:

    2. y1 = 800(0.85)x,

    and enter the condition for determining when half the substance has decayed as your second function:

    y2 = 400.



    1. What are the WINDOW settings?

      2. Xmin = 0, Xmax = 6, Xscl = 1, Ymin = 0, Ymax = 1000, Yscl = 50
    2. Now, GRAPH and TRACE to the intersection point. What’s the half-life?

      3. Somewhere around 4.28 hours.
    3. Now, use the INTERSECT feature from the CALC menu. Select the first curve and press <ENTER>; select the second function and press <enter>; then move near the intersection point and press <ENTER> a third time. What’s the half-life?

      4. Somewhere around 4.265 hours.



Homework 3—Grrrrrrr-Rate!
   
  1. Let’s look at a table of values that describe a medication that starts out with a concentration level of 600 mg/ml and is used up at a rate of 8.00% per hour.


    2. Term

    Time
    (t) (hr)

    Concentration
    (C) (µg/ml)

    c0

    0

    600

    c1

    1

    552

    c2

    2

    508

    c3

    3

    467

    c4

    4

    430

    c5

    5

    396


    1. What does the ratio c1 / c0 equal?

      2. c1 / c0 = K = 0.92
    2. What does the ratio c3 / c2 equal?

      3. c3 / c2 = K = 0.92
    3. What does the ratio c4 / c3 equal?

      4. c4 / c3 = K = 0.92
    4. What do your answers for parts a)–c) have to do with the original problem?

      5. Reducing the concentration by 8% per hour means that r = –0.08 and K = 0.92. Calculating the ratios of successive terms verifies that the same number was used each time in the multiplying.
    5. What does the ratio c5 / c0 equal? What does that value have to do with the original problem?

      6. 0.6591. Successive multiplying by 0.92 for 5 hours is the same as multiplying by (0.92)5, which equals 0.6591.
    6. Why do you think the rate of dissipation is given the variable ‘k’?

      7. The concentration from one period to the next is found by multiplying the previous concentration by K.
  1. A certain medication decays in the body at the rate of 20.0% per hour. Suppose we start with an initial concentration of 360 mg/ml.


    1. Fill in the table of concentrations for the first 10 hours, then sketch the graph.


      2. Time
      (t) (hr)

      Concentration
      (C) (µg/ml)

      0

      360

      1

      288

      2

      230.4

      3

      184.3

      4

      147.5

      5

      118

      6

      94.4

      7

      75.5

      8

      60.4

      9

      48.3

      10

      38.7

    2. What is the growth factor for this decay sequence?

      3. 0.80
    3. Write the recursive and closed-form equations for this decay sequence.

      4. Recursive: C0 = 360 mg/ml. Ct = Ct–1 * 0.80

      Closed Form: C(t) = 360 (0.80)t
    4. Predict the concentration after 15 hours.

      5. 360 (0.80)15 » 12.67 mg/ml

  1. A certain type of medicine has a half-life of 9 hours. What is the rate of dissipation? (What percent of the amount decays in each hour?)

    2. If you assume that there is 100 µg/ml initially, the equation to solve becomes 50 = 100(x)9.
    50/100 = x9
    = x
    x = 0.926, which corresponds to a drop of around 7.4%.

  1. If a shot with 1200 mg of medicine is administered, and the kind of medicine used decays at a rate of 8.00% per hour, how many hours will it take for the concentration level to drop down to 200 mg/ml?

    2. Build a table, and find that it reaches 200 mg/ml sometime during the fourteenth hour.

    Graph y1 = 600(0.92)x and y2 = 200, and trace to the intersection point, and find that it takes around 13.2 hours to reach 200 mg/ml.

    Time
    (hr)(t)

    Concentration
    (C) (µg/ml)

    0

    600

    1

    552

    2

    508

    3

    467

    4

    430

    5

    395

    6

    364

    7

    335

    8

    308

    9

    283

    10

    261

    11

    240

    12

    221

    13

    203

    14

    187


  1. Find the rate of dissipation for a certain type of medicine if 640 mg are administered at the start, and 100 µg/ml are still remaining after 5 hours.

    2. The equation describing the problem is: 100 = 320 (k)5

    Divide by 320: 0.3125 = (k)5

    Take a 5th root: = k

    k » 0.792.
  1. A patient is given a prescription that reads as follows: Quantity 8, Dosage 500 mg, take 1 pill every 6 hours. The medicine is metabolized in the body at the rate of 3% per hour. Assume that the medicine doesn’t begin to decay until after the last pill is dissolved. Make a table and a graph of this growth and decay sequence.


    2. Time
    (t) (hr)

    Concentration
    (C) (µg/ml)

    0

    0

    6

    50

    12

    100

    18

    150

    24

    200

    30

    250

    36

    300

    42

    350

    48

    400

    54

    333.19

    60

    277.54

    66

    231.18

    72

    192.57

    78

    160.40

    84

    133.61

    90

    111.29




Activity 4—You Grow Some, You Decay Some

 = 3, 8, 10

Our first step at understanding the model that describes concentration over a period of time assumed that there was no rate of dissipation. This allowed us to "concentrate" on the effect of continually adding medicine, and allowed us to describe it mathematically. The second step was to focus our attention on the rate of decay, but it was also assumed that there was no medicine being added. We learned to describe how the body consumes an initial shot of medicine, but still leaves us the task of putting it together.

  1. Let’s begin by working with a situation in which a patient receives a shot of medicine in the amount of 1000 mg. The medicine decays at the rate of 5% per hour, but the patient is given a prescription for 1324.5-mg pills to be taken every 6 hours.


    1. Fill in the table with the amount present each hour. Then sketch the graph.


      2. Time
      (t) (hr)

      Concentration Loss From Decay
      (C) (µg/ml)

      Concentration Gain From Dose
      (µg/ml)

      Concentration Level
      (C) (µg/ml)

      0

      ---

      ---

      500

      1

      25

       

      475

      2

      23.75

       

      451.25

      3

      22.56

       

      428.69

      4

      21.43

       

      407.26

      5

      20.36

       

      386.90

      6

      19.35

      132.45

      500

      7

      25

       

      475

      8

      23.75

       

      451.25

      9

      22.56

       

      428.69

      10

      21.43

       

      407.26

      11

      20.36

       

      386.90

      12

      19.35

      132.45

      500

    2. There are three patterns in the table (or graph). Explain what is going on in each of them.

      3. First, there is the decay of the chemical over a 6-hour period, in which 5% is removed each hour. Second, there is the increase from taking the pill every 6 hours. Finally, there is the long-term pattern that will be a concentration level of 500 µg/ml every 6 hours until the patient runs out of pills.
    3. How could you determine that adding 1324.5 mg would just cancel out the effects of the dissipation of the chemical?

      4. Calculate 500*(0.95)6, which gives 367.55 µg/ml. That’s the concentration that’s left after six hours. Then subtract that from the original amount of 500 µg, to figure out how much to add (in order to restore it to the original value).
  1. The pattern discovered in the previous problem can be called an "equilibrium" state, since it maintains a balance between the dissipation of the chemical in the body and the increases caused by taking pills. Use the same situation as before, except change the prescription to take the pills every 4 hours instead.


    1. Fill in the table with the amount present each hour. Then, sketch the graph.


      2. Time
      (t) (hr)

      Concentration Loss From Decay
      (C) (µg/ml)

      Concentration Gain From Dose
      (µg/ml)

      Concentration Level
      (C) (µg/ml)

      0

      ---

      ---

      500

      1

      25

       

      475

      2

      23.75

       

      451.25

      3

      22.56

       

      428.69

      4

      21.43

      132.45

      539.71

      5

      26.99

       

      512.72

      6

      25.64

       

      487.08

      7

      24.35

       

      462.73

      8

      23.14

      132.45

      572.04

      9

      28.60

       

      543.44

      10

      27.17

       

      516.27

      11

      25.81

       

      490.46

      12

      24.52

      132.45

      598.39

       

       

       

       

    2. Does the result you found in part a) agree with your "intuition" about what should happen if you shorten Dt?

      3. Yes, because the amount contained in the pill just balanced out the effects of chemical dissipation over a 6-hour period. Not as much would dissipate in a 4-hour period, so you’re slowly going to increase the concentration over time.
    3. What would the dosage have to be, in order to achieve that "equilibrium" pattern again?

      4. 500 – 500(0.95)4 = 92.75 µg/ml, so the dosage would be 927.5 mg.
  1. Look over the work done in Question 2 again. Refer back to the table to answer the following questions:


    1. How much did the concentration go up during the first four-hour interval (from t = 0 to t = 4)? During the second four-hour interval? During the third four-hour interval?

      2. 39.71 µg/ml; 32.33 µg/ml; 26.35 µg/ml.

    2. If the pills continue to be taken at 4-hour intervals, does it seem likely that the concentration will continue to go up? Explain.

      3. It will continue to increase, but the amount of increase will get smaller over time. Eventually, it will reach a "top" value or upper limit on the sequence.
    3. Do you think the dissipation rate affects whether or not the concentration will continue to go up? Explain.

      4. If the dissipation rate is a fairly large percentage, so that the amount being added from taking pills is smaller than the amount being used up by the body, then the sequence won’t go up at all. And if the dissipation rate is really small, then the growth will seem like our original additive model.
  1. Work with the same situation as before, except change the prescription to take the pills every 12 hours instead.


    1. Fill in the table with the amount present every 2 hours. Then, sketch the graph.


      2. Time
      (t) (hr)

      Concentration Loss From Decay
      (C) (µg/ml)

      Concentration Gain From Dose
      (µg/ml)

      Concentration Level
      (C) (µg/ml)

      0

      ---

      ---

      500

      2

      48.75

       

      451.25

      4

      44.0

       

      407.25

      6

      39.71

       

      367.55

      8

      35.84

       

      331.71

      10

      32.34

       

      299.37

      12

      29.19

      132.45

      402.63

      14

      39.26

       

      363.37

      16

      35.43

       

      327.94

      18

      31.97

       

      295.97

      20

      28.86

       

      267.11

      22

      26.04

       

      241.07

      24

      23.50

      132.45

      350.02

      26

      34.13

       

      315.89

      28

      30.80

       

      285.09

      30

      27.79

       

      257.30

      32

      25.09

       

      232.21

      34

      22.64