Peak drug levels in linear pharmacokinetic systems—II. Conditions for a paradoxical injection rate effect with rectangular input

Bulletin of Mathematical Biology,

Printed in Great Britain.

0092-8240/81/060705-11$02.00]0

Vol.43, No. 6, pp. 705~715,1981.

Pergamon Press Ltd. © 1981 Society for Mathematical Biology.

PEAK D R U G LEVELS IN LINEAR P H A R M A C O K I N E T I C SYSTEMS--II. C O N D I T I O N S F O R A P A R A D O X I C A L INJECTION RATE EFFECT WITH R E C T A N G U L A R INPUTt$

• C. D. THRON Department of Pharmacology and Toxicology, Dartmouth Medical School, Hanover, NH 03755, U.S.A.

The preceding paper (Thorn, 1981) has shown that in a linear pharmacokinetic system with a multimodal impulse response the peak drug level may sometimes be smaller with slower rates of injection. This paper presents two theorems on this paradoxical i n j e c t i o n rate effect where the injection is a constant infusion of finite duration. The first theorem establishes a graphical method for determining whether a given impulse response will give a paradoxical injection rate effect; and the second establishes that the maximum paradoxical increase in peak drug level is by a factor of two. It is further shown that in order to approach this maximum paradoxical increase the impulse response must contain two isolated, sharp, narrow pulses of approximately equal area. Some examples of bimodal arterial dye-dilution curves from the literature are discussed as impulse responses; and there is also a discussion of the behavior of drug level maxima and minima at different injection rates.

1. Introduction. In the preceding paper (Thron, 1981) it was established that usually the peak drug levels resulting from an injection into a linear pharmacokinetic system are lower (or at least not higher) if the injection is made at a slower rate. On the other hand, it was shown that in a system with a multimodal impulse response the opposite, or paradoxical, behavior may sometimes occur, namely an increase in peak drug levels with decrease tThis work was supported in part by U.S. Public Health Service Research Grant GM 21269 from the National Institute of General Medical Sciences, and in part by Biomedical Research Support Grant $07 R R 05932 from the National Institutes of Health. ~:A portion of this work was presented at the Annual Meeting of the Society for Mathematical Biology at the Medical School of the University of Pennsylvania, Philadelphia, August 1976. 705

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C.D. THRON

in the rate of injection. This paper will discuss this paradoxical behavior in the special case where the input is a constant infusion of duration T and rate Q/T, and the total dose Q is fixed. As before, let ~ be the impulse response. Assuming O(t)=0 for t<0, the system response, by equation (1) of the preceding paper, is then t

Y( ) = 2 (

O(s)ds.

(1)

t-T

We shall assume that 0 is continuous and has a continuous derivative t)'; and therefore if y has a local maximum at tp, (dy/dt),, = (Q/T)[~(tv)- ~ ( t p - T)] = 0,

(2)

(d2y/dt 2)tp = (Q/T)[~' (tp) - ~0'(tp - T)] ~ 0.

(3)

and

We shall further assume that inequality holds in equation (3) (from which it follows that the output maximum at tv is not a plateau). 2. Conditions Where the Output Peak Increases with Time expansion of a Rectangular input. By time expansion of a rectangular input we mean slowing the infusion rate without changing the total dose Q, so that the duration Tis prolonged (cf. equations (2) and (3) of Thron, 1981). THEOREM 1. In a system obeying equation (1), a local output maximum at tp such that inequality holds in equation (3) will increase with time expansion (or decrease with time contraction) of a rectangular input of duration T if and only if tp

TO(tp) -

O (s)ds > 0.

(4)

tp - T

Proof According to equation (1), the local output maximum at tp is given by

Qff" Y(tP)-=T-

~k(s)ds.

(5)

Assuming inequality in equation (3), the implicit function theorem (Franklin, 1940, pp. 338-340) establishes that equation (2) defines tp

PEAK D R U G LEVELS IN LINEAR P H A R M A C O K I N E T I C SYSTEMS--II

707

implicitly as a differentiable function of T. We can therefore differentiate equation (5) with respect to T to obtain dy(tp)

aT

= -T-fQ ft, t)(s)dS + yLO(tp) Q[-[ dT--t)(tP-dtp T) d(tpT)1 aT ~p -

i

(6)

T

Since O ( t p ) = O ( t p - T ) by equation (2), we have

dy(tp) "~ ~ - - = ~-lQ-lt~_[-,.(.t. p. ) -

ft,

~k(s)ds ] .

(7)

tp - T

Setting this > 0 then gives equation (4). The inequality (4) has a straightforward geometric interpretation in terms of the impulse response, i.e. that the area under the impulse response on the interval ( t p - T , tp) must be less than the area of a rectangle on the same base, of height O(tp). Both top corners of this rectangle must lie on the curve ~k because tp(tp-T)=~p(tp) according to equation (2). In order for inequality (4) to hold, ~p must lie at least partly below the horizontal line joining 0(tp - T) and O(tp). lit follows that O is multimodal, as is already known from Theorem 1 of the preceding paper (Thron, 1981).] In addition, the assumption of inequality in equation (3) establishes that ~ lies partly above this line, because if ip'(tp--T)>0 then 0 rises above the horizontal line immediately at t p - T, while if ~'(tp- T)< 0 then, according to the inequality (3), ~p'(tp) must be negative, and ~ must therefore rise above the horizontal line as the argument decreases from tp. In terms of the inequality (3) we can distinguish three types of output peak, which are illustrated in Figure 1 for a bimodal ~: Type I occurs where both O'(tp-T) and ~b'(tp) are positive, Type II where ~'(tv) only is negative, and Type III where both are negative. The inequality (4) can then be seen as requiring that the horizontal line joining O ( t p - T ) and O(tp) cut off at least as much area above ~ as below it, i.e. that area A or A1 +A2 < area C in Figure 1. These considerations allow us to determine by inspection of the normal dye-dilution curve (Figure 2) that rectangular intravenous injections of substances confined to the plasma compartment will never produce an increase in peak arterial plasma level with decrease in the rate of injection. Maxima of Types I and II are impossible because a horizontal line through the second peak of the impulse response cuts off more area under the first peak than over the intervening valley. A Type III maximum is also impossible, because at all levels of the impulse response the second descent is less steep than the first descent, and therefore condition (3) cannot be satisfied. It is true, presumably, that the plasma level will eventually fall to

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C, D. THRON

I

TYPE I

F!

c

I

fp- T

fp

fp+T

TYPE rr

I

TYPE 1Tr

---Ctp-T

----C tp

tp-T

tp i

Figure 1. Bimodal impulse responses for which time expansion of a rectangular input of duration T will cause an increase in the peak output level. The effect will be seen if area A or A 1 +A2 < a r e a C. In addition, area F must exceed area D unless the paradoxical output peak is merely a local maximum. See text for further explanation.

z o n~ Fz t.LI u~ ZZ

O~ A

n~ Ill t-r~ I

0

,o

40 TIME,

SEe.

Figure 2. N o r m a l dye-dilution curve (Paton and Payne, 1968, p. 13). Coomassie Blue was injected into an antecubital vein at time 0. The subsequent time-course of arterial dye concentration was followed by ear oximeter. Since area C < a r e a A, output maxima of Types I and II (Figure 1) are impossible. M a x i m a of Type ]II are also impossible (see text).

PEAK D R U G LEVELS I N LINEAR P H A R M A C O K I N E T I C S Y S T E M S - - I I

709

zero, and therefore that it will at some time equal the minimum level below area C. As it approaches this point, the rate of decline will almost certainly be steeper than that near the minimum below C, since the latter approaches zero. However it is evident that any horizontalline joining the late final descent to the first descent will cut off more area" under the second peak than over the Valley at C, and hence inequality (4) will not hold. A survey of several abnormal dye-dilution curves (Marshall and Wood, 1966) showed that while the condition (4) is often met and produces a n . output maximum that rises with decreasing rate of injection, this often turns out to be only a local output maximum. For the global maximum, another condition follows directly from equation (1), namely that the interval T must not include a larger area anywhere else under ~b. For a Type I maximum (Figure 1), for example, this condition requires in particular that the area under ~k from tp to tp + T be less than the area under 0 from t p - T to tp, which in turn is less than the area of the T x 0(tp) rectangle. It follows that the second peak must have been reached before tp + T, and in fact by that time ~ must have dropped below O(tp) for long enough to compensate for the area cut off under the second peak by the horizontal line through O(tp), i.e. area F must exceed area D in Figure 1. More generally, if the T x O(tp) rectangle is slid back and forth along the horizontal axis its area must nowhere be exceeded by the area under the corresponding segment of ~. Numerical computations of equation (1) were carried out with each of two abnormal dye-dilution curves (Figure 3) as impulse response. The curves were from (A) a case of tetralogy of Fallot with severe pulmonary stenosis and (B) a case of atrial septal defect with pulmonary hypertension. The first curve (Figure 3A) gave rise to a local output maximum which increased about 35 ~ when the rectangular input was expanded from 12 to 25 sec; however the global output maximum always declined. The second curve (Figure 3B) gave a global output maximum which increased by about 0.5 ~ when the rectangular input was expanded from 19 to 21 sec, but otherwise decreased with time expansion of the input. In the first case (Figure 3A), the reason why the paradoxically rising output maximum is only a local maximum is clearly that the area under the crucial segment of the bimodal region of the dye-dilution curve is exceeded by the area under later unimodal segments of the same length. This would not be so likely to hold if the injected substance were not recirculated, and the arterial concentration Consequently fell rapidly to zero following the second peak. This situation could arise if the injected substance were rapidly metabolized or sequestered in the periphery. To explore the possible behavior of such substances, the dye-dilution curve of

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C, D. THRON

A z

o n,-

Cl

I I t

I

t t

I.fl-

"~. ~._

20

tO

~-=.

30

TIME,

_

,

40

50

40

50

SEC.

u.l<

_<

n" LLI I--rr

I1 II I] II

I ] f I

0

[ [ [ [

20 :30 TIME, SEC.

Figure 3, Two abnormal dye-dilution curves (Marshall and Wood, 1966, pp. 468, 432). (A) Tetralogy of Fallot with severe pulmonary stenosis. At time 0, indocyanine green was injected into the outflow tract of the right ventricle. The dashed curve shows a modification to simulate a drug which is eliminated in the periphery and is not recirculated. (B) Atrial septal defect with pulmonary hypertension. At time 0, the dye T-1824 was injected intravenously. Both curves give rise to maxima of Types I and II: Type I where area CI>AI+A2 and Type II where C1 + C 2 > A 1 +A3.

Figure 3A was arbitrarily modified to decline exponentially after the second peak. With this modification, the global maximum was found to increase by 6~o when a rectangular input was time-expanded from 17 to 20 sec.

3. Maximum Increase in the Park Output Level with Time expansion of a Rectangular Injection Function. THEOREM 2. In a system obeying equation (1), /f y~p and Yzp are the global output maxima for input durations T~ and T2, respectively, where T1 < T2, then

Ylp Proof

<

+

T2

(8)

Since T1 < T2, there is some intege r k > 0 such that

kT1 < r2 __<(k+l)T1.

(9)

PEAK DRUG

LEVELS IN LINEAR PHARMACOKINETIC

SYSTEMS--II

Let tp2 be any time when Y2 has its m a x i m u m value equation (1): Q

711

Y2p. Then

by

ftp2

y2/,=~22

O(s)ds.

(10)

tp2 -- T 2

In accordance with (9), the integration interval {tp2--T2, tp2] in equation (10) can be subdivided into k + l subintervals of length not exceeding T1. The hypothesis that all of the corresponding sub-integrals are less than 1) of the whole integral leads to the false conclusion that the sum of all the subintegrals is less than the whole integral. Therefore there is within the integration interval at least one subinterval [ t - T ~ , t] such that

1/(k+

1 k DU 1

f

,p2

O(s)ds<

O(s)ds,

tp 2 -- T 2

(11)

t-- T 1

where tp2- T2 d-T 1 ~t<=tp2. Combining equations (10) and (11), we obtain

yzp
~b(s)ds =

t- T 1

Tl(k+l)[~l ft T2

~k(s)ds1.

(12)

t- T 1

The quantity in brackets is the value of Yl at t, according to equation (1), and it cannot exceed the m a x i m u m value and therefore

Ylp;

y2p__
Tz/T1

Ylp.

(13)

Since k < by the first inequality in (9), (8) follows directly. It follows from (8) that the ratio of peak output levels is always less than two (since > 1), and that it can approach two only if T1 _ T2. In addition, there are sharp constraints on the form of 0, if is to approach two, namely that ~b contains two isolated, sharp, narrow pulses of approximately equal area. This can be shown as follows.

T2/T1

Y2p/Ylp

Yzp/Ylp

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C.D. THRON

Since Y2p> (Y2p/Ylp)Yl (tpz), we have -O(s)ds=> O(s)ds T2 o ~2 - r2 kYlp_] T1 ot,2 - Tt (14) a n d hence

ftp2-T1 I TlYlp~ftp2 O(s)ds> 1 O(s)ds. tp2 -- T2 T~p2p..J J tp2 -- T2 Similarly, from Y2p > (Y2p/Ylp)Yl(tp2-

;~f2 ,2 -r2 +rl

¢(s)ds>_ I 1

T2+ Zl ) w e

(15)

have

TlYtPlftp2 TzYzp A J tp2 - r2

¢(s)ds.

(16)-

If Y2p~2Ylp then by T h e o r e m 2 T2 ~ T~; it follows from equations (15) and (16) that approximately half the area under ¢ on the interval [tp2 - T2, tp2] is on a short interval [tp2-Tz, tpz-T~] at the left h a n d end, and approximately half is on a similar short interval [tp2- T2 + Tx, tp2] at the right hand end. Furthermore, since each of the two short intervals contains practically all the area that can be a c c o m m o d a t e d in the integral for y~p (i.e. about half the area for Y2p), it follows that ~ must essentially vanish outside the interval [tp2 - T2, tp2] for a distance of almost T1 (actually 2 T 1 - T 2 ) in both directions. In order for Y2p to approach 2yap, therefore, must contain two isolated, sharp, narrow pulses of approximately equal area. 4. Graphical Analysis of the Behavior of Output Maxima in Systems Obeying Equation (1). Several interesting features of the behavior of output maxima can be demonstrated by graphical analysis. We here describe three: the mutual extinction of an adjacent output m a x i m u m and minimum as T is changed, the appearance of an output plateau at a particular T, and the possible existence of a T interval where tp is not a single-valued function of T. In general, output maxima occur where the sign of dy/dt changes from positive to negative. Differentiating equation (1) [cf. equation (2)] shows that this is where the difference ~ ( t ) - ~ ( t - T ) changes from positive to negative, i.e. where ~ ( t ) = ~ ( t - T ) , assuming ~ is continuous. Therefore, if

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS~I

/ /

\

713

r'-, \

I:/I

/

', '

L

TIME Figure 4. Locating the extrema of the response to a rectangular input of duration T as the intersections of the impulse ~k (solid curve) and the displaced impulse responses ~br (dashed curve). The horizontal displacement of ~br from ~p is equal to the input duration T. The intersections at M 1 and M2 indicate output maxima at those times; the intersection at m indicates an output minimum. Within the small rectangle the two curves are exactly parallel (distance measured horizontally). The horizontal fines extending to the left of the small rectangle mark off areas under and over ~b as in Figure 1. See text for further explanation.

we define the function 0T by ~bT(t)=~b(t-T), then local output maxima occur where ~k intersects ~bT from above downwards, as t increases (Figure 4). The function fiT is just the function ~b displaced to the right a distance T; it is intuitively clear that as T is varied f i t is m o v e d back and forth horizontally, and its intersection points with ~b move in a continuous fashion. It is evident in Figure 4 that the adjacent intersections at m and M2 will move together as T is increased, merging momentarily into a point of tangency, and then ceasing to exist altogether when T is increased further. This means that in the o u t p u t the corresponding adjacent m a x i m u m and m i n i m u m come together and extinguish one another. Only local maxima, and not the global m a x i m u m , can be extinguished in this way, because a (local) m i n i m u m must always lie between two maxima, and necessarily will always be extinguished by the lesser of the two m a x i m a before it can extinguish the greater. (If the two m a x i m a are equal, then extinction of the intervening m i n i m u m will fuse the two maxima into one.) We turn next in Figure 4 to the portions of ~b and ~bT within the small rectangle. These segments are exactly parallel, in the sense that the horizontal distance between t h e m is constant. Therefore if T is increased just the right a m o u n t then these portions of the two curves will come to coincide over their entire length. This gives rise to a plateau m a x i m u m in

714

C.D. THRON

the output, because for this particular value of T the area under 0 from t - T to t does not change with t in this portion of the curve. Also, equality holds in equation (3). Obviously the slightest change in T will abolish the finite coincidence region of 0 and Or and the coincidence region will contract instantly to a point intersection at one end or the other, depending on whether T is increased or decreased. The output therefore has a point maximum which, as T is gradually increased, abruptly expands into a plateau, then immediately takes off again from the opposite end of the plateau; the time-coordinate tp of the output maximum therefore shows a jump discontinuity as a function of T. In Figure 4 horizontal lines have been extended to the left from the top and bottom ends of the region on 0 where Or may coincide. In accordance with Theorem 1 and Figure 1 the relative areas marked off under and over 0 by these horizontal lines determine whether the corresponding output maxima will rise or fall when T is increased. At the upper end of the coincidence region, the area under 0 and over the horizontal line (corresponding to area A in Figure 1, Type I) is less than the area over 0 and under the horizontal line (area C); but at the lower end this relationship is reversed. It follows that if T is such as to produce the coincidence region and the resulting output plateau then the output maximum increases if T is increased, but does not decrease if T is decreased. This illustrates why Theorem 1 cannot be freely generalized to cases where equality holds in equation (3). Finally, it is easily visualized that if 0 has a plateau maximum then Or and 0 will coincide over finite distances for a range of T values. Over this range of T, tp is not a single-valued function of T, and therefore differentiation to obtain equation (6) is invalid.

5. Discussion. It probably has not been widely recognized that in systems with multimodal impulse responses the peak output level may sometimes be paradoxically increased by slowing the rate of drug injection. Such paradoxical effects are perhaps not of very great importance most of the time, because multimodal impulse responses are not common, and when they do exist they often have a configuration which produces little or no paradoxical effect with the usual more or less rectangular injections. As with the normal dye-dilution curve (Figure 2), the fluctuations in most multimodal pharmacokinetic impulse responses are probably heavily damPed; this makes all three types of maximum in Figure 1 unlikely. Even the theoretical limit (Theorem 2) of a factor of two for the paradoxical increase with a rectangular input is not very large and impulse responses with two sharp, distinct peaks, necessary for reaching this limit, seem quite unlikely to be encountered in practice.

PEAK D R U G LEVELS IN LINEAR P H A R M A C O K I N E T I C SYSTEMS--II

715

If nonrectangular or discontinuous inputs are used, the Paracloxlcal effect may become more pronounced. Computations show that the normal dyedilution curve (Figure 2) can sometimes give rise to a small paradoxical effect if the input is a pair of pulses, or a pulse followed immediately by a rectangular wave. In other cases with non-rectangular inputs, the limitation of Theorem 2 on the output peak increase may be exceeded. For example, let 0 consist of three equally-spaced sharp pulses of equal amplitude, and let the input x be the same. The peak level of the resulting output will be where three output pulses superimpose, the first 0 pulse resulting from the last x pulse, the second ~ pulse from the second x pulse, and the last pulse from the first x pulse. A slight time contraction of x will displace all such superpositions and will reduce the peak output amplitude by approximately 2/3; re-expansion of the input will increase the peak output amplitude by a factor of three. The dye-dilution curves of Figures 2 and 3 probably represent noncompartmental pharmacokinetic systems (Zierler, 1963). The question whether multimodal impulse responses and paradoxical injection rate effects occur in compartmental systems will be the subject of the next paper (Thron, to be published). LITERATURE Franklin, P. 1940. A Treatise on Advanced Calculus. New York: John Wiley & Sons. Marshall, H. W. and E. H. Wood. 1966. "Diagnostic Applications of Indicator-Dilution Technics in Congenital and Acquired Heart Disease." In Intravaseular Catheterization, Ed. Zimmerman, H. A., 2nd ed., pp. 422-513. Springfield, Illinois: Charles C. Thomas. Paton, W. D. M. and J. P. Payne. 1968. Pharmacological Principles and Practice. Boston: Little, Brown. Thron, C. D. 1981. "Peak Drug Levels in Linear Pharmacokinetic Systems--I. Effect of Rate of Injection." Bull. Math. Biol. 43, 693-703. --. "Peak Drug Levels in Linear Pharmacokinetic Systems--III. Multicompartment Systems." Bull. Math. Biol. (to be published). Zierler, K. L. 1963. "Effect of Circulatory Beds on Tracer Experiments, or Noncompartmental Analysis." Ann. N.Y. Acad. Sci. 108, 106-116. RECEIVED 5-30-80 REVISED 2-3-81