ARTICLE pubs.acs.org/jchemeduc
Chemical Dosing and First-Order Kinetics Paul W. Hladky* Department of Chemistry, University of Wisconsin—Stevens Point, Stevens Point, Wisconsin 54481-3897, United States
bS Supporting Information ABSTRACT: College students encounter a variety of first-order phenomena in their mathematics and science courses. Introductory chemistry textbooks that discuss first-order processes, usually in conjunction with chemical kinetics or radioactive decay, stop at single, discrete dose events. Although single-dose situations are important, multiple-dose events, such as taking prescription and nonprescription medicines over a period of time, are familiar, practical applications of kinetics that can be presented to science and nonscience majors without increasing the mathematical demand. Continuous dosing events, such as intravenous drips, are readily accessible to students who have completed a semester or more of calculus. Examples of multiple-dose problems are presented that are appropriate for students taking introductory, general, and physical chemistry courses. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Analogies/Transfer, Applications of Chemistry, Drugs/Pharmaceuticals, Kinetics, Mathematics/Symbolic Mathematics, Rate Law
Students encounter first-order processes, also known as exponential growths and decays, in a variety of disciplines. Algebra and calculus courses draw examples from topics such as earnings on savings accounts, population growth, and radioactive decay.13 Biology and microbiology courses discuss population growth in more detail.4,5 Physics courses, especially calculus-based ones, discuss damped oscillations, charge on a capacitor in a direct-current RC circuit, and current in a directcurrent LR circuit.6 General chemistry texts devote at least one chapter to chemical kinetics and often include a chapter on nuclear processes.7 All of these topics, as they are typically presented, can be characterized as single-dose events; some initial or time-zero quantity, Ai or A0, of money or bacteria or substance or amplitude of displacement or electrical charge is initially present and students are taught how to calculate the quantity remaining at some later time. By the time the topic appears to be exhausted, students are expected to be able to calculate any one of the four items; initial quantity, final quantity, rate constant (or interest rate or doubling time or half-life), or elapsed time from the other three, based on the underlying assumption that only one dose occurred. An interesting practical application of first-order chemical kinetics that is important in the health sciences involves what can be called drug, or more generally, chemical dosing.8 Although the detailed study of drug dosing is a topic in the pharmaceutical sciences known as pharmacokinetics,8,9 chemical dosing has been experienced by everyone who has taken medicines or been exposed to chemical substances over a period of time. Tablets, injections, and inhalers deliver drugs in sudden discrete doses that can be repeated at regular or irregular intervals depending on Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
the nature of the substance and one’s medical needs. Administering drugs through an intravenous drip over an extended period of time is an example of continuous dosing. Ingesting substances and breathing air-born compounds, either intentionally or unintentionally, also constitute cases of chemical dosing and these events may entail continuous exposure for a period of time followed by no exposure or different levels of exposure during other time periods. Although the exposures are not first-order events, the mechanisms that clear substances out of an organism —metabolism, excretion, or both—can often be described using first-order kinetics.8,10 Although single-dose situations are the obvious starting point in introductory-level chemical kinetics, they are also the ending point. This is unfortunate because dosing problems can be kept simple enough for nonscience majors, they can be made challenging enough for chemistry majors at all levels, and they correspond to familiar and important real-world situations. The following sections cover the mathematical background and present examples of multiple-dose problems that are appropriate for the entire spectrum of college chemistry students. To keep the problems as simple as possible, the following assumptions are made: the relevant volume of the system (organism) is constant, all of the dosed substance is available and distributed throughout the system instantaneously, and the clearance pathways are not saturated. More realistic treatments of drug dosing810 and halflife values of many drugs911 are available.
Published: March 25, 2011 776
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’ DISCRETE DOSING First-order processes are well-known so only a brief summary of their equations will be given after which the modifications needed for continuous dosing will be presented. The instantaneous rate for a first-order process is given by dA ¼ (kA dt
ð1Þ
where A A(t) is some quantity (molar concentration at constant volume, mass of substance [see note 1], number of bacteria, money in an account, ...) at time t and k is a positive constant that corresponds to the process of interest. More specifically, k is related to the time that is required for the value of A to double in growth processes (þ sign) or halve in decay processes ( sign). Because chemical dosing problems involve first-order decays, the negative sign is appropriate, and the most general form for the integrated rate law is obtained by separating variables and integrating to obtain Af Aðtf Þ ¼ Aðti Þ exp½kðtf ti Þ Ai expðkΔtÞ ð2Þ Figure 1. (A) A substance (t1/2 = 8 h) is administered according to the dosing schedule in the table. Students are asked to calculate the mass of substance, Apost, that is present immediately after the hour 48 dose is given. See text for an explanation of the fonts. (B) Calculations with another dosing schedule. Note that this schedule leads to a similar longtime repeat pattern as the example in A. (C) Calculations with another dosing schedule. Note that the long-time repeat pattern in this case differs from those of the first two examples even though all three cases have the same average dose of 16 mg in an 8 h period. (D) Students calculate the mass of the dose that was given at hour 24 to account for the pre-dose mass of A at hour 48. This problem illustrates that one can go backward in time and that the mass of each dose can change.
which allows one to calculate the quantity at the final time, tf, from the quantity that was present at the initial time, ti. The rate constant, k, is related to the half-life of the process, t1/2, by t1=2 ¼ lnð2Þ=k which allows eq 2 to be rewritten as " # ðtf lnð2Þðtf ti Þ 1 Aðtf Þ ¼ Aðti Þ exp ¼ Aðti Þ t1=2 2
ð3Þ
ti Þ=t1=2
ð4Þ
The four examples that follow have been presented in lectures to introduce discrete-dosing problems to students enrolled in a one-semester chemistry course for nonscience majors, general chemistry, and physical chemistry. For this latter group, the presentation may be new material if dosing problems had not been covered in an earlier course or a review of material that was covered a couple of years earlier. Although the introduction is essentially the same for these vastly different student audiences, there are significant differences in the pacing of the presentation and homework problems that are assigned to each group. Each of the following four examples is presented in a table and has the same general form; the substance of interest has a known half-life, a dosing schedule (Adose = dose) is given, and a specific missing quantity is calculated. For convenience in this article, all of the information for a problem is contained in one table. The bold values define the problem (the rest of the table would initially be blank), the italicized values are calculated in the order indicated with the arrows, which leads to the solution indicated by the underlined italicized value in the double-line-bordered cell (Figure 1). Sketches of mass versus time, such as Figure 2, help students visualize the effects of dosing and decay. When regular dosing schedules, such as the ones in tables AC (Figure 1), are extended over a long enough period of time, the mass of substance eventually varies regularly between a limiting maximum, Amax, and a limiting minimum, Amin. Although the maximum and minimum levels of the substance can be estimated by carrying the calculation out for a long period of time, it is also possible to calculate the two limits just knowing
When the initial time is chosen to be zero and A(0) A0, the integrated rate law is written in the familiar forms " # t=t1=2 lnð2Þt 1 kt A AðtÞ ¼ A0 e ¼ A0 exp ¼ A0 t1=2 2 ð5Þ When the times, t, are limited to integer multiples, n, of the halflife, eq 5 reduces to n 1 ð6Þ An Aðnt1=2 Þ ¼ A0 2 This last equation, or its equivalent, can be found in chemistry textbooks that present radioactivity to nonscience majors even when chemical kinetics is not addressed.12 The preceding equations apply to single-dose events in which a quantity A(ti) is known at some time ti and nothing more is added to, or removed from, the system. Equations 2 and 46 can also be used to go backward in time (t < 0) in which case the assumption is that a single dose occurred in the past and its quantity is calculated from the currently known or measured quantity. Furthermore, in the case of solutions when A represents a molar concentration, the volume of the system is assumed to be constant and the mixing is assumed to be fast compared to the half-life. 777
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In the simplest case, v(t) = v0 is a constant for the entire time interval under consideration, which could describe a drug administered by an intravenous drip or an exposure to a substance that is present in the environment. A slight modification of this simplest case, which is easy to imagine in the case of drug dosing, includes a discrete dose (injection or other rapidly absorbed delivery method) that is also given at t = 0. Another modification is to have discrete doses occurring along with the continuous dosing. This scenario may not seem practical by itself but it bridges the gap between cases having a single rate of continuous dosing and realistic situations involving exposure at one constant rate during some time periods and at a second constant rate during other periods. For example, an intravenous drip is turned on and off for prescribed periods of time or the drip is changed. All of these situations start with eq 10, which is solved for the interval or intervals that are defined by discrete dosing events or periods of constant continuous dosing. The resulting integrated rate laws are derived below. More advanced techniques, such as Laplace transforms, are required to obtain integrated rate laws for situations in which v(t) varies with time; these kinds of problems are beyond the scope of this article. Before integrating eq 10, it is useful to obtain an expression for the long-time or steady-state limit of the quantity, A¥, that results from the continuous-dosing aspect of these problems. Setting the derivative in eq 10 to zero leads to
Figure 2. Graphical representation of the example in Figure 1A. The pre-dose and post-dose masses (dots) are connected with the doses (vertical lines) and decays (curved lines).
the half-life, time interval between doses, and dose mass, Adose. In general, there are two equations that connect Amax, Amin, and Adose to each other. In the three dosing schedules being considered AC in Figure 1), the maximum level occurs immediately after a dose so that Amax ¼ Amin þ Adose ð7Þ and the minimum level occurs just before the next dose so that eq 6 leads to Amin ¼ Amax =2 ð8aÞ for dosing schedules A and B, or Amin ¼ Amax =4
A¥ ¼
ð8bÞ
and
Amin ¼ Adose
ð9aÞ
Amin ¼ ð1=3ÞAdose
ð9bÞ
for dosing schedules A and B, and Amax ¼ ð4=3ÞAdose
and
for the third dosing schedule. In the case of medicines, two practical considerations are that the minimum mass is effective whereas the maximum mass does not cause undesirable side effects. All of the dosing schedules in the preceding examples were tied to half-life intervals. For courses aimed specifically at nonscience majors, all of the homework problems are also tied to half-life intervals and the Amin Amax problems are essentially the same as the examples shown above. For general chemistry and physical chemistry students, the dosing schedules for shorttime problems are not tied to half-life time intervals and regular dosing schedules for Amin Amax problems are more complex, that is, two different dosing levels that alternate or the dosing pattern “dosedoseno dose” repeats.
ð12Þ separating variables Z
AðtÞ
A0
dA lnð2Þ ¼ ðA A¥ Þ t1=2
Z
t
dt
ð13Þ
0
integrating and rearranging to the final form A AðtÞ
’ CONTINUOUS DOSING Physical chemistry students have the mathematical background to go beyond discrete dosing problems. Equation 1 can be modified to account for continuous dosing by adding a term, v(t), which is the rate at which the substance of interest, in quantity per unit time, is added to the system so that eq 1 becomes dA ¼ kA þ vðtÞ dt
ð11Þ
This result, which is independent of any discrete initial dose, reveals an advantage of administering a drug intravenously; the drug’s level can be set as high as needed and kept there as long as necessary. Because A is of primary interest, the constant rate of addition, v0, will be replaced with kA¥ in later equations. The integrated rate law for a discrete dose followed by constant continuous dosing is obtained by substituting eq 11 into eq 10, rewriting eq 10 dA v0 lnð2Þ ¼ k A ðA A¥ Þ ¼ kðA A¥ Þ ¼ dt t1=2 k
for dosing schedule C. Solving eq 7 and the appropriate form of eq 8 gives Amax ¼ 2Adose
v0 t1=2 v0 ¼ k lnð2Þ
"
# ( " #) lnð2Þt lnð2Þt ¼ A0 exp þ A¥ 1 exp t1=2 t1=2
ð14Þ
where A0 is the initial discrete dose, which may be zero, at time t = 0. The limits A = A0 and A¥ at t = 0 and ¥, respectively, are readily confirmed. Equation 13 can be adapted to cases in which two or more discrete doses occur in addition to a single constant rate of
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A0 = A(0) = 0, A1 = A(t1), A2 = A(t2), ..., Aj = A(tj), Ajþ1 = A(tjþ1), ... be the levels of the substance that are present immediately before the discrete doses occur at the times indicated. In these circumstances, eq 13 becomes Z Z AðtÞ dA lnð2Þ t ¼ dt ð15Þ t1=2 tj Aj þ Ad, j ðA A¥ Þ
continuous dosing. Suppose that the constant rate of dosing, characterized by v0 or A¥, starts at t = 0 and that discrete doses occur according to the schedule Ad,0 at t = 0, Ad,1 at t = t1, Ad,2 at t = t2, ..., Ad,j at t = tj, Ad,jþ1 at t = tjþ1, and so on. Furthermore, let
and eq 14 becomes A AðtÞ " ¼ ðAj þ Ad, j Þexp
# ( " #) lnð2Þðt tj Þ lnð2Þðt tj Þ þ A¥ 1 exp t1=2 t1=2
ð16Þ for each time interval tj e t e tjþ1. Figure 3 shows plots of mass versus time for four cases involving constant continuous dosing with and without discrete dosing. The long-time repeating pattern for case iv in Figure 3 is characterized by Amax and Amin values that can be calculated by two different but equivalent approaches. The first approach recognizes that eq 7 is valid and that eq 16 can be
Figure 3. Mass of substance versus time for a single continuous dosing rate, v0, to which discrete doses may have been added. Four cases are shown: (i) no discrete doses, (ii) an initial discrete dose of 4 mg, (iii) an initial discrete dose of 12 mg, and (iv) discrete doses of 4 mg at 16 h intervals. The solid dots indicate points that students can calculate so that they can sketch smooth curves and visualize their solutions to the problems.
Figure 4. (A) Mass of substance present during a two-part continuous dosing schedule. The schedule starts with a low rate for 16 h followed by a high rate for 8 h and then repeats for several days. The values of Amin and Amax are 4.4 and 7.2 mg, respectively. The solid dots indicate points that students can calculate so that they can sketch smooth curves and visualize their solutions to problems. (B) The same substance and dosing schedule as A except that a 12 mg discrete dose also occurs at t = 0 h. This schedule tends to the same long-time limits as A. (C) A substance with a shorter half-life following the same dosing schedule as B. The relatively short half-life, compared to the duration of the dosing intervals, means that the min and max levels, 3.2 and 9.8 mg, respectively, nearly reach the high and low limits during their respective intervals. (D) A substance with a longer half-life following the same dosing schedule as B and C. The relatively long half-life, compared to the duration of the dosing intervals, leads to min and max levels, 5.2 and 6.2 mg, respectively, that are closer to each other then they are to the high and low limits during their respective intervals. For very long half-lives, the min and max values both approach the time-weighted average of the high and low dosing levels; 5.7 mg in this case. 779
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and
" # ( " #) lnð2ÞΔt lnð2ÞΔt exp þ A¥ 1 exp t1=2 t1=2
Amax ¼ Amin
"
# ( " #) lnð2ÞΔthigh lnð2ÞΔthigh exp þ A¥, high 1 exp t1=2 t1=2
ð17Þ
ð24Þ
where Δt is the time interval between successive doses. Equations 7 and 17 can be solved for Amax and Amin. Alternatively, one can treat A¥ as a constant background level that can be ignored and focus on the effects of the discrete dosing (indicated by primes) by solving eq 8, in the form,
which can be solved to give the Amin and Amax values shown in Figure 4AD.
0
0
Amax ¼ Amin þ Adose and the expression 0
ð18Þ
(
0
Amin ¼ Amax
lnð2ÞΔt exp t1=2
) ð19Þ
for the “primed” quantities. The actual maximum and minimum levels are then given by 0
Amax ¼ Amax þ A¥
and
0
Amin ¼ Amin þ A¥
ð20Þ
Equation 15 can also be adapted to situations in which an initial discrete dose at t = 0 is followed by two continuous constant dosing intervals, a low rate and a high rate, that alternate in a regular way. For each dosing interval, eq 15 becomes Z AðtÞ Z dA lnð2Þ t ¼ dt ð21Þ ðA A¥, j Þ t1=2 tj Aj where Aj A(tj) is defined as before, remembering that the only discrete dose occurs at t = 0, and A¥,j is the long-time limit corresponding to the continuous dosing rate occurring during this time interval. Integrating and rearranging leads to "
#
A AðtÞ (
" #) lnð2Þðt tj Þ lnð2Þðt tj Þ ¼ Aj exp þ A¥, j 1 exp t1=2 t1=2
ð22Þ
’ CONCLUSION Discrete-dosing problems can be incorporated into introductory-level chemistry classes for both majors and nonmajors. A suitable introduction, together with examples of the types shown in tables AD (Figure 1) and the long-time Amin Amax values can be presented within a 50 min class period. For nonscience majors in a one-semester chemistry course, the homework problems are kept at the same level as the examples in tables AD (doses at half-life intervals). For students in a general chemistry sequence, the presentation is essentially the same but the homework problems, decouple the dosing schedules from the half-life intervals and problems involving repeating dosing patterns, are more complex. Physical chemistry students receive a short in-class review of discrete dosing followed by a presentation of the continuous dosing cases such as the Ad,0 = 0 or 12 mg curves shown in Figure 3. They are then assigned homework problems that are similar to the other cases in Figure 3. Dosing problems such as those shown in Figure 4 can be used to challenge the more mathematically minded physical chemistry students, particularly those minoring or double majoring in mathematics. Experience over the past several years, based on performance on exam questions, suggests that most students understand dosing problems. Although the relative ease of dosing problems is attractive, their frequent occurrence in common real-life situations and students’ ability to relate to these problems is the main motivation for adding them to courses as Supporting Information. Although the examples considered in this article are much simpler than the situations considered in serious pharmacokinetics modeling,13 these kinds of problems give students an introduction to understanding the fate of chemical substances to which they are exposed during their daily activities.
which is similar to eq 16. The four plots (Figure 4) illustrate the effects that the initial discrete dose and the half-life have on the level of substance that is present in the system over an extended period of time. The continuous part of the dosing schedules repeats every 24 h and the durations and rates have been chosen so that the low-dose rate lasts for 16 h and has A¥,low = 3.0 mg, whereas the high-dose rate lasts for 8 h and has A¥,high = 11.0 mg. The minimum and maximum levels that are shown in Figure 4AD can be calculated by recognizing that these limits occur at the ends of the low and high dose-rate intervals, respectively. Adapting eq 22 to these two limits yields
’ NOTE 1. An extensive quantity, mass or moles, of a substance rather than its molar concentration can be used in the first-order chemical kinetics problems presented in this article for the following reasons. First, consider eq 1 written in terms of molar concentration
Amin
dn ¼ (kn dt
" ¼ Amax exp
#
(
"
lnð2ÞΔtlow lnð2ÞΔtlow þ A¥, low 1 exp t1=2 t1=2
dðn=V Þ ¼ (kðn=V Þ dt
ðN1Þ
where n is the amount in moles and V is the volume in liters. Since the volume is constant, it can be canceled out and one obtains
#)
ðN2Þ
which is now in terms of moles. Multiplying through by the formula weight converts this expression to one involving mass. Second, consider the following example. Suppose one takes a substance and makes the following two solutions: (a) 1.0 mol in
ð23Þ 780
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1.0 L and (b) 1.0 mol in 2.0 L. Furthermore, suppose that the half-life is 1.0 min so that after 1 min the concentration of the substance is 0.50 and 0.25 M in solutions (a) and (b), respectively, which corresponds to 0.50 mol in both cases. Now suppose that one starts with 1.0 mol of substance in 1.0 L and adds an additional 1.0 L of solvent during the first minute of reaction. Even though the volume is changing, the solution will contain 0.50 mol of substance after 1.0 min just as it did for the two constant volume solutions (a) and (b). Note also that first-order processes such as growth of bacterial populations and radioactive decay depend on numbers of bacteria4,5 and nuclei,7 respectively, and not on their concentrations.
’ ASSOCIATED CONTENT
bS
Supporting Information A set of worked problems and a spreadsheet for dosing problems are available. This material is available via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ REFERENCES (1) Collins, W.; Cuevas, G.; Foster, A. G,; Gordon, B.; MooreHarris, B.; Rath, J.; Swart, D.; Winters, L. J. Algebra 2—Integration, Applications, Connections; Glencoe/McGraw Hill: Columbus, OH, 2001; Chapter 10. (2) Aufmann, R. N.; Nation, R. D. College Algebra and Trigonometry— A Graphing Approach; Houghton Mifflin: Boston, MA, 1995; Chapter 4. (3) Smith, R. T.; Minton, R. B. Calculus: Early Transcendental Function, 3rd ed.; McGraw-Hill: New York, 2007; Chapter 7. (4) Campbell, N. A. Biology, 4th ed.; Benjamin/Cummings: New York, 1996; Chapter 47. (5) Maier, R. M.; Pepper, I. L.; Gerba, C. P. Environmental Microbiology, 2nd ed.; Academic Press: Burlington, MA, 2009; Chapter 3. (6) Serway, R. A.; Jewett, J. W., Jr. Physics For Scientists and Engineers, 8th ed.; Brooks/Cole: Belmont, CA, 2010; Chapters 15, 28, 32. (7) McMurry, J; Fay, R. C. Chemistry, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 2001; Chapters 12, 22. (8) Yang, J.; Lewandrowski, K. Therapeutic Drug Monitoring. In Clinical Chemistry—Laboratory Management and Clinical Correlations; Lewandrowski, K., Ed.; Lippincott Williams & Wilkins: Philadelphia, PA, 2002; Chapter 51. (9) Cannon, J. G. Pharmacology for Chemists, 2nd ed.; Oxford University Press: New York, 2007; Chapter 2. (10) Welling, P. G. Pharmacokinetics—Processes, Mathematics, and Applications, 2nd ed.; American Chemical Society: Washington, DC, 1997; Chapter 13. (11) Pharmacist’s Drug Handbook; Springhouse Corporation and American Society of Health-System Pharmacists: Springhouse, PA, 2001. (12) Snyder, C. H. The Extraordinary Chemistry of Ordinary Things, 4th ed.; John Wiley and Sons: Hoboken, NJ , 2003; Chapter 5. (13) Bonate, P. L.; Pharmacokinetic-Pharmacodynamic Modeling and Simulation; Springer: New York, 2006; Chapter 1.
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