The Time Evolution of Drugs in the Body G. V. Calder lowo State University Ames, IOWO 50010
An application of the principles of chemical kinetics
The time evolution of drugs in the human body is an annlication of the nrincinles of chemical kinetics that is -~== ~-~ especially appropriate in physical chemistry courses designed for students of the life and medical sciences; however, this application is not commonly encountered in physical chemistry texts and is not familiar to the typical instructor of physical chemistry. The purpose of this article is to present a few basic kinetic concepts as they apply to the absorption and utilization of drugs by the hody. The absorption and elimination of ethanol is chosen to illustrate these principles because of its widespread usage and because an understanding of its action might mntrihUte to a decrease in its misusage. Drug absorption commonly proceeds either by zeroorder kinetics or by first-order kinetics. Zero-order absorption occurs when the drug is administered by a continuous, constant intravenous infusion or when the drug is ingested orally in the form of a "time release" capsule. First-order absorption occurs when the drug is administered by a simple intramuscular injection or by oral ingestion of a single tablet or capsule.' The first-order kinetics result because the complex distribution process is often governed by a slow transport of the drug across some hiological memhrane. In simple cases this transport is governed by Fick's Law which states that the rate of diffusion of a compound across a concentration gradient is proportional to the concentration difference
d(B)/dt
-
k,(A) - kdB)
d(C)ldt
-
(B) (Adk,/(k8 k,Nexn(-k,t) ... . . . . - exd-k.t)) .. - ., (3) ..
kLB)
(C) ( A d 1 + (k, - kJYk* exp(-k,t) - k, exp(-kd))) (4)
~
The utilization (or elimination) of a drug, in actual practice, often follows a complex mechanism; however, in a large number of cases the overall rate of utilization or elimination can he realistically described by a first-order or zero-order rate law. First-order kinetics occur when the rate of utilization is governed by memhrane transport, while zero-order kinetics occur when the pathway for utilization becomes saturated. Saturation takes place, for example, when the regeneration of the enzyme responsihle for metabolizing the drug is the rate determining step in the utilization process. This is the case in the elimination of ethanol by the body.'
-
-
-
In these equations (A) represents the concentration of the drug in the gastrointestinal tract, or in the site of injection; (B) represents the concentration of the drug in the blood; and (C) represents either the amount of the drug eliminated by various metabolic functions, e.g., urine, sweat, breath, etc., or the amount of the drug utilized by various active sites in the hody. Either or both interpretations of (C) may be relevant for a particular drug. A typical plot of (A), (B), and (C) shown in Figure 1, illustrates several aspects of drug kinetics. First, it should be noted that the ordinate scales for (A), (B), and (C) are not generally the same since body fluids dilute the drug in various sites of the body. Secondly, only one pathway of elimination has been assumed. Other pathways for elimination, e.g., pa :ng though the intestine without being adsorbed, may :.reduce parallel mechanistic steps not included in the mechanism presented above. These complications do not affect the conclusions materially, however. For most drugs there is a minimum concentration that must be present in the hlood stream or body tissue in order for the drug to be therapeutically effective. This minimum effective drug concentration (EDC) is characteristic of the drug, and hence requires a certain input dosage to he attained. In terms of Figure 1 this means that if (Ao) is not sufficiently large, (B) can never achieve an effective concentration. Also note that there is an induction period between the time of administration and the attainment of the minimum effective dose. The only adjustable parameter for shortening the induction period is the increase of the initial dosape (Ao). This fact is reflected in the instructions commonly found on certain prescriptions: "Take two tablets a t meals and a t bedtime for the first two days, then one tablet a t meals and at bedtime." It is evident from Figure 1 that once the effective
First-Order Absorption-First-Order Utilization If a single dose of a drug is ingested orally or injected intramuscularly, the concentration in the blood is governed by the well known equations for consecutive firstorder reactions k
A 1 ,B
d(A)/dt
-
B -k,(A)
first-order
AC
first-order or (A) = (Ad exp(-k,t)
(2)
1 Goldstein, A., Amnow, L., end Kalman, S.M., "Principles of Drug Action: The Basis of Pharmacology," Harper and Raw, New York, 1968.
, m e I*///
Figure
1. A
plot of eqns.
(2-4)
assuming k , = 0.1
hr-' and kr = 0.05
hr-'.
Volume 51. Number
1. January 1974
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19
dosage concentration has been achieved, i t can be maintained hy administering smaller doses. This "plateau" effect can be treated quantitatively using certain reasonable simplifying approximations. If the drug is absorbed rapidly and continuously so that the elimination of the drug obeys a simple exponential law
then at a time t' just prior to the administration of a second dose of the same size as the first, the concentration of drug in the blood is (B), = iB),.,
exp(-kd')
(6)
Shortly after the administration of the second dose, the concentration of the drug in the hlood is
that is, simply the sum of the new dose and the amount remaining from the first dose. After a second time interval of equal length has elapsed, the hlood level of the drug is (B), (B).., exp(-k,.,t') (B)... exd-k,.,2t3) (8). Similarly, just after the third dose, the drug level is
"
Letting exp (-k,,,t') n"'dose
+
= X, it is easy to see that after the
which is simply the geometrical progression whose sum is given by
SinceX < 1, in the limit that n B
=B
-X
-
= B
-e
x - k t
(12)
Alternatively, the number of doses, spaced a t equal time intervals t', required to attain a given fraction of the plateau concentration is
Once the first-order rate constant for elimination and the effective dosage concentration have been determined clinically, eqn. (14) is of obvious practical value in establishing prescription standards. The appropriate rate constants k , , and k,,, are ohtained in much the same way that rate constants for any chemical reaction are determined. A measured amount of the drug is administered orally or injected intramuscularly and the concentration of the drug in the hlood is monitored by appropriate analytical methods. From these data the order of the absorption process and k , , are determined in the usual way. The elimination order and rate constant are often determined by monitoring the hlood concentration after direct intravenous injection. This avoids complications due to slow absorption times from the gastrointestinal tract ioto the hlood. 2Harger, R. N., "Toxicology. Mechanisms and Analytical Methods," (Editors: Stewart, C. P., and Stolman, A,), Academic Press, New York, 1961, Vol. 11, Ch. 4.
/ Journal 01 Chemical Education
Utilization
The Time Evolution of Ethanol The principles introduced in the previous section are illustrated very effectively by the time evolution of the drug ethanol. This example is especially pertinent because of the ubiquitous use of ethanol in our society. Indeed, it is probably the most widely consumed drug in the history of man. The absorption of ethanol through the stomach and intestinal wall is a rapid first-order p r o c e ~ s .The ~ kinetic order is consistent with a diffusion controlled process governed by Fick's Law, eqn. (I),in which CA,the concentration in the gastrointestinal tract, is much larger than CB, the concentration of the hlood. As will be seen shortly, the rate of absorption of ethanol into the hlood stream is considerably faster than its elimination. Thus a short time after ingestion, the ethanol is essentially quantitatively absorbed ioto the hlood. Ethanol is eliminated from the body by oxidation to acetaldehyde and ultimately to acetic acid in a series of enzyme controlled reactions that take place primarily in the liver. The rate of oxidation of the ethanol is governed by the rate at which the active form of the enzyme is regenerated after the oxidation step. Since this latter process does not depend on the amount of ethanol present, but rather on other independent biochemical functions, the rate of elimination of ethanol is independent of its concentration, i.e., the elimination is zero-order.',z The over-all mechanism for the metabolism of ethanol is then G
*'"- B
B
A
and m
where (B), is the ultimate steady state or plateau concentration that will he attained by administration of the drug over a long period of time. The fraction of that ultimate concentration obtained after the nth dose is
20
First-Order Absorption-Zero-Order
d(G)/dt d(B)ldt d(E)ldt
first.order
E zero.order
-
-
-k,.(G) kAG) - k.., k..,
Integration of the expression for the concentration in the blood (B), assuming (B) = 0 at t = 0, yields
The condition of maximum hlood concentration implies that d(B)/dt = 0 which can he expressed in alternative forms
The general character of the time dependence of the blood alcohol concentration after a single rapid ingestion with a linear deis a? exponential increase for t < t,, crease fort > t,.,. The determination of precise values of the rate constants applicable to all individuals is not possible because the rate of elimination of ethanol can vary by as much as 25% in different individuals, and the rate of absorption depends on nonreproducihle factors such as the quantity of food in the stomach. Nonetheless, several facts about the physiology of ethanol permit reasonable, typical, estimates for these constants. First, the rate of absorption of ethanol from the gastrointestinal tract is so fast that within 30-60 minutes after ingestion absorption is virtually complete. Thus the half life for the first-order absorption must be much less than 1 hr. For computational simplicity a value of k , , = 10 hr-1 was assumed. This corresponds to a tliz = 0.069
lu.While this value may be in error by even 100% the kinetics are not sensitive to the value of kt, but are governed by the fact that the rate of absorption is much faster than the rate of elimination. Second, the rate of elimination of ethanol, which is controlled by an enzymatic oxidation, is a con~tant,'.~,and although the rate can vary by as much as *30% from one individual to another, it is always much slower than the rate of absorption. A typical value of the rate of metaholism is 10 ml ethanol/hr. This rate of metaholism can he converted to more convenient units using the following considerations. The transport of ethanol from the blood to various aqueous body fluids is among the most rapid of all chemicals except water, hence the concentration of ethanol in various hody fluids is essentially uniform.3 Since there are typically 40 1 of aqueous fluid in an adult, a rate of metabolism of 10 ml/hr corresponds to a rate constant k,,, = 0.192 g ethanol/l of body fluid hr. The validity of various blood, urine, and breath tests for intoxication rests on the fact that all of the aqueous fluids of the body serve essentially like a sponge for ethanol. Using these constants it is possible to plot the time evolution of ethanol in the blood. For convenient reference certain conversions used in the subsequent discussion have been tabulated in the table. The curves in Figure 2, which assume essentially instantaneous ingestion of the ethanol, illustrate a number of points regarding the kinetics of drug Ltion mentioned in the previous section. First, for ethanol there is a very short induction period before minimum effective concentration is achieved. In this example, an effective concentration of 1 g/l has been selected since i t corresponds to the legal definition of intoxication, and as such is a useful reference point where Go is the initial dose of ethanol expressed in g/l hody fluid. The induction time for legal intoxication is always less than about 25 min. It is also evident from Figure Conversion Factors and Values of Constants Used in the Analysis 01 the Time Evolution of Ethanol in the Body
2 that the time at which the maximum blood concentration is attained is not a sensitive function of the dosage t.., k,.-I In (G,k,./k..,) (19) and furthermore, the slow rate of elimination dictates that the maximum hlwd concentration is essentially equal to the amount of ethanol ingested. There are two characteristic times associated with the elimination pmcess-the sobriety time, t,,~, and the elimination time, tel. The sobriety time is that time required for the alcohol level to decrease below 1 g/l after an initial dose in excess of that amount is ingested.
-
t,& 2 (Go - l)k..,
tratlon of
Ethanol (g)
1
20
Equivalent volume of 3 0 p m f (4Wh ulv ethanol1 liquor ml oz
40
0.6 1.0
60
1.5
190
6.4
80
2.0 2.5
254 317
3.6 10.7
1W
k,. = lo b-1.;..k
bodyfluid =401.
63
m
2.1 4.3
-
C
vneonaciausnesa
(20)
-
k.3-
G
= 0.192 g othsnol/l body fluid hr = 10 ml ethanollhr; volume of
- 4.3)
t,, Cok,.,-' OZ X 1.22 (a) This time is the residence time of the ethanol in the body and roughly characterizes the recuperation period or hangover time. Approximately 5 hr are required to clear the ethanol from a minimally intoxicated individual. While the above concentrations illustrate the chemical kinetics of ethanol metabolism, they are unrealistic from a social viewpoint in that the dosage is ingested essentially instantaneously. The chemical kinetics of a cocktail party are not usually represented by that mechanism except for the most compulsive drinkers. "Experimentally," it has been observed that the consumption of alcohol by a given individual at a cocktail party, occurs roughly a t a constant rate, after which consumption ceases. So a more appropriate mechanism is
Qualitative p h y r i d a p i d ~ d pycholopicai rospons mnscdeuphoria mild lo medium disturbam inmafo.cmrdina&m. legally infozicotad. overt lack of cmrdination and d u n i n g o f r p e h amnesia indueedd?epor
1.22(0Z
where G o is expressed in g/l and for comparison OZ is expressed in ounces of 80 proof liquor consumed. Thus, for every ounce of 80 proof liquor ingested in excess of the 4.3 oz, about 1 hr and 15 min is required to become legally sober once more. Exoressed in another wav. . . once the effective concentration ( 1 g I) has been achieved only a very small additional dosaae is necessary to maintain the b l ( ~ ) d concentration at the same level-s~ecifically, 10 ml ethanol/hr = 0.85 oz of 80 proof liquor/hr. This is only 20% of the initial dosage! The time required to completely eliminate the ethanol is given by
C0"ce"~ ethanol in body fluid
-
B
G variable zero-order
A *,
B rapidfirst-order
E slow zero-order
where, as before, ( G ) is the concentration of alcohol ingested, and ( B ) is the hlwd alcohol concentration. Solving the appropriate differential equations for this mechanism one obtains for ( B ) B =k - k t kIk,,h - e x - k t f o r t < t...
-
(22)
Because the rate of transport of ethanol across the gastrointestinal membrane is usually rapid compared with the dose rate during the ingestion period, this expression for ( B ) can he approximated accurately by B
-k
k
t fort
< t...
(21)
Fort > t,,,, that is, when ingestion is terminated (B)
-
-
( B ) m a , k0",t
This behavior is illustrated in Figure 3 for two different ingestion rates, 1 g/l hr and 1.5 g/l hr for a period of 1 hr and is compared with the corresponding blood alcohol #>me, " X I
Figure 2. The time evolution of ethanol in the body: assuming instantaneous ingestion.
Harger, R. N., Hulpieu, H. R., and Lamb, E. B., J. Bid. Chem., 120,689(1937). Volume 51, Number 1, January 1974
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21
Figure 3. The time evolution of ethanol in the body assuming a constant rate 01 ingestion tor 1 hr and subsequent termination.
concentration curves for the instantaneous ingestion. Several interesting conclusions can be drawn. The maximum blood concentration of ethanol is not significantly decreased by the prolonged ingestion period unless it is of the order of tSob.As a result, the compulsive
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Journalof Chemical Education
drinker and the more patient drinker who "nurses" his alcohol for one hour both will become equally intoxicated, provided both ingest the same amount of alcohol and insofar as the psychological response of the two individuals is the same. On the other hand the length of time that the blood alcohol concentration remains above 1 g/l is signifcantly reduced by prolonged ingestion. This shortening of the period of intoxication occurs in the early stages however and does not result in a reduction oftsob which is essentially governed by the rate of metabolism which is invariant. In short, the patient drinker will become equally intoxicated more slowly but will require the same amount of time to regain sobriety as the compulsive drinker. Summary
The tune evolution of drugs is a timely and interesting application of the principles of chemical kinetics that is particularly suitable for students in the biological and life sciences. In particular the time evolution of ethanol is an especially enlightening example in view of its widespread use in our society.
