In the Classroom
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Saving a Drug Poisoning Victim: A Kinetics Simulation Jodye Selco* Department of Chemistry, University of Redlands, Redlands, CA 92373-0999;
[email protected] Janet Beery Department of Mathematics, University of Redlands, Redlands, CA 92373-0999
In this project students are asked to pose as emergency room physicians who must save a child who has accidentally overdosed on the asthma medication theophylline (see structure). O H3C
O
N
N N
N
C H3
The progress of the drug through the child’s body can be modeled as a chemical kinetics problem involving first-order consecutive reactions. Unlike previous articles in this Journal that describe the kinetics of a drug in the body (1, 2), this project does not provide the students with the integrated form of the equation for the amount of drug in the bloodstream. Instead, students construct a system of linear first-order differential equations describing the medication’s absorption into and elimination from the child’s bloodstream using half-lives obtained from the Physician’s Desk Reference (PDR) (3). This approach forces students to pay careful attention to the construction of the mathematical model describing the kinetics and gives them practice with the useful tool of differential equations modeling. It also provides them with an example of how calculus, required for most chemistry degrees, really is used in chemistry. After using a computer to solve their differential equations numerically, students discover that the child will almost certainly die if they, as physicians, do not intervene. They then determine by how much they need to increase the drug’s elimination rate in order to save the child. This calculation dictates the appropriate medical action; students discover they need to use the more drastic treatment, extracorporeal filtering of the blood through charcoal, rather than simply administering oral doses of charcoal. We have found that this project appeals to a broad range of students in our courses, especially those who intend to enter the health professions. To complete the project, students need technology to implement Euler’s method or another numerical method for solving systems of differential equations (e.g., a spreadsheet program, computer algebra system, differential equations solver, or virtually any computer programming language including that available on a graphing calculator). The numerical results in this article were obtained using Excel. True Basic programs set up to analyze this model using Euler’s method are available in the supplemental material.W This assignment originally was developed as an interdisciplinary project in chemistry and mathematics and has been used in the kinetics portion of physical chemistry courses and in calculus courses. It also is appropriate for use in the kinetics portion of a biochemistry course or an honors general 578
chemistry course. Using a computer to model kinetics has been presented many times before in this Journal (2, 4–10); however, the medical setting is novel, as is the way in which the students are asked to model the situation. The situation presented here originally appeared in slightly different form in a physical chemistry textbook (11). Some of the questions for this project were inspired by the development of a susceptibleinfected-recovered (S-I-R) model for the spread of an epidemic, found in an innovative, applications- and computer-based calculus text (12). Setting the Scene You are a physician in a hospital emergency room. A child has just been brought to the emergency room by a frantic parent. The parent takes the asthma medication theophylline (see structure) in tablet form. Two hours before arriving at the hospital, the child ingested eleven 100-mg theophylline tablets. Like most oral drugs, theophylline is absorbed into the bloodstream at a rate proportional to the amount present in the gastrointestinal tract (stomach and intestines) and is eliminated from the bloodstream at a rate proportional to the amount present in the bloodstream. Your quick check of the Physician’s Desk Reference (PDR) (3) reveals that the brand of theophylline the child took has an absorption half-life of 5 hours and an elimination half-life of 6 hours. The PDR also warns that a blood-level concentration of 100 mg/L or more of the drug is seriously toxic and that a concentration of 200 mg/L or more is fatal.1 You estimate that the child has 2 L of blood. You also determine that, because of the 2-hour delay, the pills already have passed from the child’s stomach to his intestines, so that it is too late to eliminate the drug from the child’s body by inducing vomiting. Your task is to determine if the child is in danger, and, if so, to save his life.2 Building a Model The diagram below illustrates the progress of the drug through the child’s body. Gastrointestinal tract
k1
Bloodstream
k2
Eliminated drug
You are interested in the amount of theophylline in the child’s bloodstream over time. To determine this, you also need to determine the amount of theophylline still in the child’s gastrointestinal tract over time. (Actually, you are concerned about the concentration of theophylline in the child’s bloodstream. However, since you know only the amount and not the concentration of theophylline in the gastrointestinal tract, you decide to calculate the amount rather than the
Journal of Chemical Education • Vol. 79 No. 5 May 2002 • JChemEd.chem.wisc.edu
In the Classroom
concentration in the bloodstream.) You could also calculate the amount of theophylline eliminated from the bloodstream, but, since theophylline in this form is not dangerous, you decide not to keep track of the eliminated drug.
Part 1 First, predict the general shape of the graph of G(t), the amount of theophylline in the child’s gastrointestinal tract (in milligrams) after t hours, and of B(t), the amount of theophylline in the child’s bloodstream (in milligrams) after t hours. Using time t = 0 as the time at which the child ingested the theophylline, make separate rough sketches of the graphs of G(t) and B(t). On each graph, label the point at t = 0 h. (If t = 0 h is the time at which the child ingested the theophylline, what is the corresponding value for G ? for B?) Remembering that the half-life for absorption of theophylline from the gastrointestinal tract into the bloodstream is 5 h, label the points at t = 5 and t = 10 h on your graph of G(t). You need not label any other points on the graphs or mark any other values along their axes—yet. Part 2 Since you have more information about the rates of change of G and B than about G and B themselves, you decide to model the quantities G and B by writing equations for their rates of change (differential equations). The rates of change, or derivatives, of G and B with respect to time, t, are denoted dG/dt and dB/dt, respectively. Begin with the differential equation for G. Theophylline is absorbed into the bloodstream at a rate proportional to the amount present in the gastrointestinal tract. This means that theophylline is leaving the gastrointestinal tract at a rate proportional to the amount of the drug present there. Hence, taking k1 to be the rate constant, you have the differential equation dG/dt = ᎑k1G mg/h;
G(0) = 1100 mg
Use what you know about exponential decay, along with the fact that the absorption half-life of theophylline is 5 h, to write a formula for G(t), the amount of theophylline (in milligrams) in the gastrointestinal tract at time t. (That is, solve the differential equation for G(t), then solve for k1.) Record k1 to four places after the decimal point.3 You now should have both a formula for G(t) and a differential equation for G in which k1 has a numerical value.
Part 3 Now write a differential equation for B. Since theophylline is entering the bloodstream at one rate and leaving it at another, the differential equation for B is of the form dB/dt = absorption rate – elimination rate with units of milligrams per hour. Consider the first term, the absorption rate. Recall that theophylline is absorbed into the bloodstream at a rate proportional to the amount present in the gastrointestinal tract with an absorption half-life of 5 h. This should sound familiar; use your work from part 2 above to write an expression for the absorption rate. Now consider the second term, the elimination rate. Remember that theophylline is eliminated from the bloodstream at a rate proportional to the amount present in the bloodstream with a half-life of 6 h. In fact, if we ignore for
the moment that theophylline continues to enter the bloodstream from the gastrointestinal tract, we may assume that the amount of theophylline in the bloodstream is decaying exponentially. Use what you know about exponential decay to write an expression for the elimination rate. Determine the value of the rate constant, k2, and record it to four places after the decimal point. You now should have a differential equation for B involving the variables G and B and the rate constants k1 and k2. Using the Model Now that you have differential equations for G and B, you are ready to use them to determine if the child is in danger and, if so, how to treat him. Note that, unlike for the differential equation for G, there is not a simple closed form solution for the differential equation for B. That is, you may not be able to write an explicit formula for B(t) but instead may have to approximate values of B(t) using Euler’s stepahead method or another numerical method for solving differential equations. This means that you will start with the initial values for G and B and increment them using dG/dt and dB/dt. Your instructor will specify the degree of accuracy (number of significant figures) for your calculations.
Part 4 Determine the amount of theophylline in the child’s bloodstream at time t = 2 h, the time of his admission to the hospital. How does the value obtained depend upon the time interval used? How small must the time interval be so that the value of B(2 h) no longer changes (to the number of decimal places specified by your instructor?) Recalling that the child has 2 L of blood and that a blood-level concentration of 200 mg/L or more of the drug is almost always fatal, what amount of theophylline in his bloodstream, in milligrams, constitutes a lethal level for the child? Recalling that a blood-level concentration of 100 mg/L or more is seriously toxic, what amount constitutes a seriously toxic level for the child? What is his status at the time of his admission to the hospital? Does the amount of theophylline in his bloodstream pose any danger to him at this time? Part 5 Determine the amount of theophylline in the child’s bloodstream over several hours. Graph your results. Does the amount of theophylline in his bloodstream ever reach a lethal level for the child? If so, after how many hours? How many hours after the child’s hospital admission does this occur? Mark the lethal level and the time at which it occurs on your graph. After how many hours does the amount of theophylline in the child’s blood reach a seriously toxic level? How long before or after his hospital admission does this occur? Part 6 Determine the largest amount of theophylline the child ever has in his bloodstream, and the time at which this maximum level occurs. How much theophylline remains in his gastrointestinal tract at this time? Note that the largest value for B occurs when dB/dt = 0 or, equivalently, when the absorption rate is equal to the elimination rate. When you substitute your largest value for B and
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Part 7 Determine the maximum number of theophylline tablets the child could have taken without reaching the lethal bloodlevel concentration. Explain. How many could he have taken without reaching the seriously toxic blood concentration? Explain. Saving the Child Your results from parts 5 and 6 should have shown that the child who ingested the 11 theophylline tablets is in grave danger. What can be done? Fortunately, charcoal absorbs theophylline quickly, so it can be used to increase the rate at which theophylline is eliminated from the bloodstream (3, 13). For toxic levels of theophylline, the patient takes oral doses of charcoal, increasing the theophylline elimination rate to approximately twice the normal rate. For potentially fatal levels of theophylline, charcoal must be added to the bloodstream extracorporeally (outside the body) in order to remove the theophylline quickly enough. This procedure is risky, but may increase the theophylline elimination rate up to six times the normal rate, according to the Physician’s Desk Reference (3, 13).4
Part 8 In part 3, you expressed the rate of change of B as the difference between the absorption rate and elimination rate. Since it is too late to change the rate of absorption from the gastrointestinal tract, you must therapeutically change the elimination rate, which you have expressed as k2B mg/h. Find the smallest value for the constant k2 that ensures that the concentration of theophylline in the child’s bloodstream remains below the lethal level. For instance, would increasing k2 to 0.1200/h suffice? What about k2 = 0.1300/h? WARNING: Since you have an opportunity to increase the value of k2 only after the child has been admitted to the hospital, be sure to begin your calculations with larger values of k2 at the time of admission. To be safe, continue to increase k2 until you find the smallest value for k2 that causes the amount of theophylline in the child’s bloodstream to decrease immediately upon treatment. Assume that treatment is administered exactly at the time of hospital admission. Print or sketch a graph of the amount of theophylline in the child’s bloodstream from the time he ingests the pills to a few hours after his hospital admission and treatment. Your graph should show the effect of treatment on the child’s theophylline blood level. Note that you could have determined the smallest value for k2 that causes the amount of theophylline in the child’s bloodstream to decrease immediately upon treatment without any further computer work. Explain how. (HINT: See part 6, second paragraph.) Recall that you could have doubled the theophylline elimination rate by administering oral doses of charcoal. Would this treatment have been sufficient to cause an immediate decrease in the child’s theophylline blood-level? Explain. Remember that you may increase the elimination rate up to six times the normal rate by filtering the blood through charcoal extracorporeally. Do you need to increase the elimi580
1200
Theophylline / mg
your corresponding value for G into your equation for dB/dt do you get 0? Explain why you might not get 0.
1000
G (t )
800 600
lethal
400
seriously toxic
200
B (t )
0 0
4
8
12
16
Time / h Figure 1. Amounts of theophylline in the gastrointestinal tract and bloodstream as a function of time in the case of drug overdose without treatment.
nation rate this much in order to cause an immediate decrease in the child’s theophylline blood-level? Explain. Solution
Part 1 Although the graphs of G(t) and of B(t) should be sketched by hand on separate sets of axes, they should have essentially the shapes shown in Figure 1. The points (0, 1100), (5, 550), and (10, 275) should be labeled on the graph of G(t); the point (0, 0) should be marked on the graph of B(t). Students will have the opportunity to verify their predictions after they set up differential equations for G and for B. Part 2 Students should recognize the differential equation for exponential decay and write G(t) = 1100e᎑k1t mg. They then should use that G(5) = 550 mg to solve for k1, obtaining k 1 = ln 2 ≈ 0.1386/h 5h The formula for G(t) then is G(t) = 1100e᎑0.1386t mg. The initial value problem (differential equation, together with the initial value) for G(t) is dG/dt = ᎑ 0.1386G mg/h; G(0) = 1100 mg
Part 3 The absorption rate is given by the expression 0.1386G mg/h from part 2. To find an expression for the elimination rate, we assume that ᎑ dB/dt = k2B mg/h; B(6 h) = 1⁄2 B(0) mg and find that k 2 = ln 2 ≈ 0.1155/h 6
The elimination rate then is given by the expression 0.1155B mg/h, and the initial value problem for B(t) is dB/dt = (0.1386G – 0.1155B) mg/h; B(0) = 0 mg Students may conjecture a solution to this initial value problem of the form B(t) = ⫿1100e᎑0.1386t ⫾ B(0)e᎑0.1155t but this is not a correct solution. A correct solution, obtained
Journal of Chemical Education • Vol. 79 No. 5 May 2002 • JChemEd.chem.wisc.edu
In the Classroom
1
816.2
266.2
0.1
832.1
239.2
0.01
833.5
236.8
0.001
833.7
236.5
0.0001
833.7
236.5
1200
Theophylline / mg
Table 1. Values for G(2 h) and B(2 h) ∆ t/h G(2 h)/mg B(2 h)/mg
1000
G (t )
800 600
lethal
400
seriously toxic
200
B (t )
0 0
4
8
12
16
Time / h Figure 2. Amounts of theophylline in the gastrointestinal tract and bloodstream as a function of time in the case of drug overdose with treatment.
by matrix methods, is B(t) = ᎑6600e᎑0.1386t + 6600e᎑0.1155t Although this project could be modified to incorporate this analytic solution, we assume that students will solve the differential equation(s) numerically.
Part 4 Applying Euler’s method with a step size of ∆t = 0.0001 h to the initial value problem dG/dt = ᎑ 0.1386G mg/h; G(0) = 1100 mg dB/dt = (0.1386G – 0.1155B) mg/h; B(0) = 0 mg yields G(2 h) = 833.7 (800)5 mg and B(2 h) = 236.5 (200) mg. Computing G(2 h) using the formula for G(t) = 1100e᎑0.1386t mg yields G(2 h) = 833.7 (800) mg. One also could use the formula for G(t) to compute the values of G needed in the equation for dB/dt during the Euler’s method computations. Table 1 shows values for G (2 h) and B(2 h) obtained using Euler’s method with different time steps ∆t. Checking the accuracy of our model by comparing the obtained values of G and B with the values of the analytical solutions, G(t) = 1100e᎑0.1386t mg and B(t) = (᎑ 6600e᎑0.1386t + 6600e᎑0.1155t) mg, we see that if time steps of ∆t = 0.01 h are used, the values of G and B agree to 2.5 digits; if steps of ∆t = 0.001 h are used, the values agree to 4 digits; and if steps of ∆t = 0.0001 h are used, the values agree to 5 digits. Answers here are given to 4 significant figures so that faculty can help students diagnose problems with their models. Student answers should contain only a single digit because the information that they are given contains only one significant digit. Since the child has 2 L of blood, a lethal blood-level of theophylline for him would be 400 mg, well above his current 236.5 (200) mg blood-level. However, 200 mg would constitute a seriously toxic blood-level for him, and his current blood-level already has reached the seriously toxic range.
Part 5 By computing values of B from t = 0 to approximately t = 10 h, students can see that the amount of theophylline in the bloodstream increases and then decreases and that it does eventually exceed the lethal level (see Fig. 1). The amount of theophylline in the bloodstream just reaches B = 400.0 (400) mg after t = 4.866 (5) h (2.866 [3] h after the child’s admission to the hospital), and B = 200.0 (200) mg after t = 1.609 (2) h (at the time of the child’s hospital admission).
Part 6 A step size of ∆t = 0.0001 h yields G = 368.4 (400) mg and B = 442.1 (400) mg after t = 7.893 (8) h. Note that B = 400 mg is the lethal level for the child and that any value of B between 350 and 449.9, when expressed to one significant digit, is 400 mg; this might explain why physicians adopt a “wait and see” attitude when treating patients. Substituting G = 368.4 mg and B = 442.1 mg into the equation for dB/dt yields dB/dt = 0.0023 mg/h, indicating that our solution technique is approximate rather than exact and/or that round-off error has occurred. If a value of 400 mg is used for both G and B, dB/dt = 9. Students may use the formula for G(t) to obtain the value of G at the time at which the largest amount of theophylline in the child’s bloodstream occurs or to check the accuracy of their numerical approximations. Computing G(7.893 h) using the formula for G(t) yields G = 368.4 (400) mg. Part 7 If the child had taken only 9 tablets (G(0) = 900 mg), the largest amount of theophylline in his bloodstream would have been B = 361.7 (400) mg (∆t = 0.0001 h), which is below the lethal level according to the four-digit value yielded by the model. If he had taken only 4 tablets, his highest blood level of the drug would have been B = 160.8 (200) mg. Ten tablets and 5 tablets, respectively, would have resulted in maximum blood levels of just over 400 mg and 200 mg. When using one significant digit, it should be noted that 8 tablets yields a maximum of B = 300 mg, and 3 tablets yields a maximum of B = 100 mg. Part 8 Starting at the time of hospital admission (t = 2 h), students should increase k2. (They may need to use G(2) = 833.7 mg and B(2) = 236.5 mg from part 4.) Using k2 = 0.1442/h results in a maximum value for B of 399.9 mg, whereas using k2 = 0.1441/h results in a maximum value for B of 400.0 mg. Using k2 = 0.1908/h results in a maximum value of B = 349.9 (300) mg, whereas using k2 = 0.1907/h results in a maximum value of B = 350.0 (400) mg. Using k2 = 0.4886/h results in the maximum value for B of 236.5 mg, occurring at time t = 2.000 h, whereas using k2 = 0.4885/h results in the maximum value for B at t = 2.001 h. Or, students may notice that using k2 = 0.4833/h results in a maximum value for B of 236.5 mg, whereas using k 2 = 0.4832/h results in a maximum value for B of 236.6 mg.
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Note that B decreases when dB/dt < 0 or, equivalently, when the elimination rate is greater than the absorption rate. If we set dB/dt = k1G – k2B = 0 at time t = 2 h, we obtain 0.1386/h (833.7 mg) – k2(236.5 mg) = 0, so k2 = 0.4886/h. A graph showing the effect of treatment on the child’s theophylline blood-level is shown in Figure 2. Since 2 × 0.1155/h = 0.2310/h < 0.4833/h, administering oral doses of charcoal would not have been sufficient to cause the child’s theophylline blood-level to decrease immediately. Since 6 × 0.1155/h = 0.6930/h, we did not need to increase the elimination rate by six times the normal rate in order to cause B to decrease immediately. In fact, our value for k2 is approximately 4.2 × 0.1155/h. Note that the solutions presented above are correct when all four digits are assumed to be valid. However, this is not the case for this problem. Since half-lives for theophylline absorption and elimination are given to only one significant digit (owing to variations in human physiology), values computed from them are to only a single digit. This reflects the fact that each person responds differently to theophylline and other medications and treatments. As noted previously, these variations probably explain why physicians adopt a “wait and see” attitude when treating patients. Acknowledgments This project was originally developed as an Interdisciplinary Lively Application Project (ILAP) under the direction of NSF Division of Undergraduate Education grant DUE-9555414 for the Consortium for Mathematics and Its Applications (COMAP), Lexington, MA. For more information about ILAPs and COMAP, visit www.comap.com. We would like to thank the staff at Redlands Community Hospital and the Southern California Poison Center for medical information. We also thank the reviewers for their helpful suggestions. W
Supplemental Material
Basic programs set up to analyze this model using Euler’s method are available in this issue of JCE Online.
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Notes 1. These values are the concentrations at which 50% of the patients exhibit these symptoms. In the fatal case, the concentration of 200 mg/L, the lethal concentration for 50% of the population, is called the LC50 value. 2. In reality, a physician in this situation probably would contact the local poison center, which would provide information about which symptoms to watch for as well as the appropriate medical treatment. 3. Even though the half-lives are given to only one significant digit, more working digits are required to obtain reasonable values from the model. 4. Since the lethal concentration represents the response of 50% of the population and dialysis is both dangerous and expensive, physicians would resort to such drastic treatment only if the patient would not survive without it. 5. Here and in the rest of parts 4–8, the number in parentheses is the rounded value that is used.
Literature Cited 1. Calder, G. V. J. Chem. Educ. 1974, 51, 19–22. 2. Ball, David W. J. Chem. Educ. 1998, 75, 917–919, 1387. 3. Physician’s Desk Reference, 53rd ed.; Medical Economics Co.: Montvale, NJ, 1999. See, for instance, pp 875–878, 1457– 1471, 2563–2569, 2607–2609, and 3164–3170. 4. Andraos, John. J. Chem. Educ. 1999, 76, 1578. 5. Bozelli, J. W. J. Chem. Educ. 2000, 77, 165. 6. Bruist, M. W. J. Chem. Educ. 1998, 75, 372–375. 7. Goodman, Jonathan M. J. Chem. Educ. 1999, 76, 275. 8. Lo, Glenn V. J. Chem. Educ. 2000, 77, 532. 9. Pavlis, Robert R. J. Chem. Educ. 1997, 74, 1139. 10. Toby, Sidney; Toby, Frina S. J. Chem. Educ. 1999, 76, 1584. 11. Bromberg, J. Philip. Physical Chemistry, 2nd ed.; Allyn and Bacon: Boston, 1984; p 905. 12. Callahan, James; Hoffman, Kenneth; Cox, David A.; O’Shea, D.; Pollatsek, H.; Senechal, L. Calculus in Context; Freeman: New York, 1995: pp 1–69. 13. Physician’s Desk Reference Generics; 4th ed.; Medical Economics Co.: Montvale, NJ, 1998; see pp 2620–2633, especially pp 2626–2627.
Journal of Chemical Education • Vol. 79 No. 5 May 2002 • JChemEd.chem.wisc.edu
