Peak drug levels in linear pharmacokinetic systems—III. Multimodal impulse responses in multicompartment systems

Bulletin of Mathematical Biology Vol. 44, No. 5, pp. 609~35. 1982. Printed in Great Britain.

0092-8240/82/050609-27503.00]0 Pergamon Press Ltd. (~) 1982 Society for Mathematical Biology

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--Ill. MULTIMODAL IMPULSE RESPONSES IN MULTICOMPARTMENT SYSTEMSt • C. D. THRON Department of Pharmacology and Toxicology, Dartmouth Medical School, Hanover, NH 03755, U.S.A.

This paper deals with the unimodality of the 'impulse response' in compartmental systems, where the 'impulse response' in any given compartment is the time course of the amount of diffusing substance in that compartment after an initial instantaneous injection of the substance into that or some other compartment. It is shown that in certain compartmental structures, with injection in certain compartments, the impulse response is always unimodal or monotonic in all compartments, regardless of the numerical values of the various transfer rate coefficients. Structures with this property are here named 'UM structures', and they include the familiar mammillary and catenary structures. In this paper, the set of all UM structures is described. Structures which are not UM (NUM structures) are identified by showing that, by removal of certain connections and compartments according to certain rules, they can be reduced to small structures which can be shown to be NUM by numerical computation. Computations on two systems with bimodal impulse responses show that with constant infusions of a fixed amount of substance the peak level may increase paradoxically with decrease in the infusion rate over a certain range. This effect is extremely small, however.

In preceding papers (Thron, 1981a, b) it was established that 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, provided the impulse responses (the time courses of the drug levels after an initial instantaneous injection) are unimodal. On the other hand, examples were provided to demonstrate that if the impulse response is not unimodal, the peak drug level obtained may sometimes be at least slightly greater with a slower (time-expanded) input. All the real examples given of multimodal impulse responses were dye-dilution curves, whose shapes arise from the finite time-delay in transporting blood from one point to another in the circulation. Apart from the initial distribution phase, however, the pharmacokinetics of most drugs seem to fit multicompartment models; and since ideal multicompartment systems have no time-delays, the question arises whether they can have multimodal impulse responses like the dye-dilution curves and, if so, whether tSupported 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 S07 RR 05392 from the National Institutes of Health. 609

610

c.D. THRON

these can give rise to p e a k drug levels which increase with timeexpansion of the input, particularly a rectangular input (i.e. a constant infusion of finite duration).

1. Multicompartment Systems With Bimodal Impulse Responses.

A

multicompartment system with a bimodal impulse r e s p o n s e can easily be constructed as follows. Set up a c o m p a r t m e n t with rapid elimination, so that direct injection into that c o m p a r t m e n t p r o d u c e s a rapidly-declining drug level. Call this compartment No. 4 (cf. Figure 1, inset). A slowly rising and falling drug level can then be p r o d u c e d in c o m p a r t m e n t No. 4 as the q u a s i - s t e a d y state resulting from an infusion at a slowly rising and falling rate. As a source of such an infusion, set up a train of two c o m p a r t m e n t s (Nos 2 and 3) such that an impulse injection into No. 2 is released slowly into No. 3, and is then slowly lost from No. 3, so that there is a slowly rising and falling level in No. 3. F e e d the outflow from No. 3 to the original compartment (No. 4), and the level in No. 4 must then follow this slow rise and fall. N o w add a fourth c o m p a r t m e n t (No. 1) which receives an impulse injection and rapidly distributes it partly to No. 4 and partly to No. 2. The effect is a p p r o x i m a t e l y equivalent to impulses p r e s e n t e d simultaneously at N o s 2 and 4; and the resulting drug level in No. 4 is therefore approximately the sum of the two impulse responses: a brief peak from injection directly into No. 4, and a slowly rising and falling peak from injection into No. 2. The bimodal impulse r e s p o n s e of this system is illustrated in Figure 1.

c30cnuJ .05 63 '¢ .04 -~ .~ .03

Zl.-

~ ~ _02

~

.01

Injection

-

o

Observation

Q: TIME UNITS Figure I. Bimodal impulse response of a 4-compartment system with two paths from the point of injection to the point of observation. The compartmental model is shown in the inset. Parameter values for the curve shown were a =0.865, b =0.112 reciprocal time units. This curve and those in Figure 2 were computed and plotted on the Dartmouth Time-Sharing System.

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--Ill

611

The model of Figure 1 has two paths from the site of injection to the point of observation; but such multiple paths are not always n e c e s s a r y for a bimodal impulse response. An example of a 'tree' s y s t e m t with bimodal impulse response was found by considering the convolution of a bi-exponential function on itself. Suppose this function is initially positive and declines rapidly to a v e r y low level, but t h e r e a f t e r declines v e r y slowly. The early part of this function is close to a simple exponential, and therefore it gives rise to a unimodal self-convolution. The later part of the function, h o w e v e r , is close to a constant function, which would have a monotonically increasing self-convolution with no upper bound. This suggests that the self-convolution of our function will show a s e c o n d a r y rise to a second maximum. C o m p u t a t i o n shows this to be the case; and it follows that a system with a bimodal impulse response can be constructed by connecting two identical t w o - c o m p a r t m e n t systems by a o n e - w a y path, as illustrated in Figure 2.

.12[

o

~~ o ,.3

~-

II~.z

Injection

~ o~

Observation '

'1

'

2' TIME

UNITS

Figure 2. B i m o d a l impulse r e s p o n s e s of a 4 - c o m p a r t m e n t tree s y s t e m . T h e c o m p a r t m e n t a l m o d e l is s h o w n in the inset. P a r a m e t e r s for c u r v e A w e r e a = d = 3.1198, b = e = 1.3636, c = [ = 1 r e c i p r o c a l time units_ W i t h impulse r e s p o n s e A, the o u t p u t p e a k rose v e r y slightly (0.027%) o n t i m e - e x p a n s i o n of a r e c t a n g u l a r i n p u t f r o m 2.80 to 2.99 time units. P a r a m e t e r s for c u r v e B were a = d = 24.57, b = e = 3.027, c - - - f = 3.0027 r e c i p r o c a l time units. C u r v e B s h o w s a m o r e m a r k e d b i m o d a l i t y , b u t the a r e a u n d e r a n y s e g m e n t of t h e b i m o d a l r e g i o n of t h e c u r v e satisfying the criteria of F i g u r e 1 of a p r e v i o u s p a p e r ( T h r o n , 1981b) is always e x c e e d e d b y the area u n d e r some later u n i m o d a l s e g m e n t of the same length. Conse,quently, the only m a x i m a w h i c h i n c r e a s e d with t i m e - e x p a n s i o n of the i n p u t w e r e local m a x i m a , a n d the global m a x i m u m always d e c r e a s e s .

t A 'tree' system is a connected compartmental system without multiple paths and without 'cycles'. A 'cycle' is a set of 3 or more compartments connected in a ring, so that material can pass through all of them in succession and then directly back to the first.

612

C.D. THRON

For each of these multicompartment systems (Figures 1 and 2) a set of parameters could be found such that a rectangular input would produce an output whose peak level increased with time-expansion of the input over some limited range. The effect was extremely small at best, however. Attempts were made to find parameter values that maximized this effect, i.e. maximized the total relative increase in the output peak over the whole range of input durations where the output peak increased with time-expansion of the input. For the model of Figure 1, use was made of a computer program based on the flexible simplex method (slightly modified after Himmelblau, 1970), but for the model of Figure 2, only a trial-and-error estimate was obtained. For the model of Figure 1, the largest increase in the output peak was about 1%, which occurred as the input was time-expanded from 9.5 to 13.5 time units' duration. In the model of Figure 2, the largest increase was about 0.027%, over input durations from 2.80 to 2.99 units.

2. A Class of Compartmental Structures in which the Impulse Response is Always Unimodal or Monotonic. Two well-known compartmental structures are the mammillary and catenary structures (Sheppard and Householder, 1951). The first consists of several peripheral compartments, all connected to one central compartment and not to each other; the second is a chain of compartments, each connected only to its immediate predecessor and successor in the chain. Here we shall consider a class of structures, which we shall call CM structures, formed by adding mammillary peripheral compartments to the last compartment of a catenary structure (Figure 3). It is taken as part of this definition that injections are made only into the first compartment. This class includes pure mammillary and catenary structures as special cases.

Figure 3. The general CM structure.

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--III

613

The rate equation for an n - c o m p a r t m e n t system, in matrix form (Hearon, 1963; Thron, 1972), is d y / d y = - Ky,

(1)

where y is the c o l u m n vector of drug quantities in the n c o m p a r t m e n t s , and for a CM system,

al -

-bl

cj

0

0

a2

-

-- c2

b2

a~

• "

• "

K=

0

a=_2

-

b.,-2

0

0

0

a.,_~

-- bm-I

0

0

o o

• -

-c,~-2

• -

0

• "

0

0

-- C=+l

am+l

0

0

• "

0

0

-- Cm+ 2

0

a~+2

0

•.

0

0

--C~

0

0

-

c.,-a

am

-

b=+l

-

b=+2

-- b.

(2) The explanation of the elements of this matrix is as follows. L e t kij be the rate c o n s t a n t for transfer f r o m the j-th to the i-th c o m p a r t m e n t ; let koi be the rate c o n s t a n t for loss f r o m the i-th c o m p a r t m e n t to the outside world; and let m be the index n u m b e r of the central c o m p a r t m e n t of the m a m m i l l a r y part of the system. The entries in equation (2) are then as follows. F o r i < m : bi=k~.i+l, c i = k ~ + L , and a~=ko~+b~ ~+c~ (except al = kol + cl). For i > m" b~ = k,~i, c~ = k~,,, and a~ = ko~ + b~. Finally, a,~ = kom + b,,_l +

E

c~. If m = n or n - 1 we have a pure c a t e n a r y structure;

i=m+l

and if m = 1 or 2 we have a pure m a m m i l l a r y structure with injection into the central c o m p a r t m e n t or a peripheral c o m p a r t m e n t , respectively. T h r o u g h o u t this paper we shall a s s u m e c~# 0, since there is no need to c o n c e r n ourselves with c o m p a r t m e n t s which c a n n o t be reached f r o m the point of injection.

1. The impulse response is unimodal or monotonic in every compartment o f a C M system. In particular, the impulse response is everywhere unimodal or monotonic in a pure mammillary system with injection into any one compartment, or in a pure catenary system with injection into the first compartment.

THEOREM

614

C.D. THRON

Proof. We first p r o v e the theorem for the case w h e r e the system is strongly-connectedt ( b i # 0, c i # 0), and then r e m o v e this restriction. For a strongly-connected C M system, K is s y m m e t r i z a b l e (Hearon, 1963, pp. 52-53) and hence is similar to a diagonal matrix of its characteristic roots (Bellman, 1970, pp. 50-54). The solution of equation (1) can therefore be written in the form yi = ~

r,j e ~",

(3)

i=1

where the a~ are the distinct characteristic roots of K, which we take in the order AI > - " > a.,, n ' -< n, and R = {r~j} is an n × n' matrix w h o s e columns are characteristic v e c t o r s of K for the c o r r e s p o n d i n g aj (Coddington and Levinson, 1955, pp. 75-78). The number of maxima in the impulse r e s p o n s e y~ can be determined from the number of sign changes in the s e q u e n c e of coefficients r,, r~2. . . . , ri., b y the following lemmas. LEMMA 1.

The n u m b e r of zeros o f E rj e-*J'(a~ > . , • > A, >- O) does not j=l

exceed the n u m b e r o f sign changes in the sequence rj . . . . , r,. Proof.$

Let r1--1

f(t) =

j=l

ri e-**'/e-*"' = ~, r, e ¢*'-*")' + r,.

(4)

j=l

The zeros of f are precisely those of Z rj e -*j'. L e t Z ( f ) be the n u m b e r j=l

of zeros of f. The mean value t h e o r e m (Franklin, 1940, p. 113) implies that between any two zeros of f lies a zero of f ' ( = d f / d t ) ; so Z(f') ~ Z(I)-

1.

(5)

We can make a stronger s t a t e m e n t if f, and r,-i have the same sign. Since A1 > . . . > A,, the series in equation (4) is d o m i n a t e d at large t by the term r,-i e-~"-~-~")'; and it follows that at large t the sign of f' is opposite to that of r._~. On the other hand, since lira f ( t ) = r,, if t~ is the largest zero of f then f'(tz) must have the same sign as r,. T h e r e f o r e , if r, and r,-i are of like sign, then f' must undergo a sign change b e t w e e n tz tA strongly-connected system is one in which it is possible for the diffusing substance to pass from any compartment directly or indirectly to any other compartments. ¢The proof of Lemma 1 was kindly supplied by C. P. Thron.

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--Ill

(where it has the same sign as r,) and t ~ that of r,_~). It follows, then, that

Z(f') >- Z ( f )

615

(where its sign is opposite to

(r.r._, > 0).

(6)

Let S(f) be the n u m b e r of sign changes in the s e q u e n c e r ~ , . . . , rn, and let S(f') be the n u m b e r in r ~ , . . . , r,_~. If !", and r,_t are of like sign then S(f') = S(f); otherwise S(f') = S ( f ) - 1. In either case, b y equations (5) or (6), Z ( f ) > S(f) implies Z(f') > S(f'). Therefore, the existence of a sum of the form E ri e-XJ'(,t~ > ... > ,~, -> 0) with n terms and more zeros than i=1

sign changes in the s e q u e n c e of coefficients implies the existence of a sum of the same form (namely f') with n - t terms which also has more zeros than sign changes. Since no such sum exists for n = 1, this establishes the lemma. LEMMA 2. F o r any c o m p a r t m e n t a l system obeying equation (3), if p~ is the m i n i m u m p a t h lengtht f r o m the initially-injected c o m p a r t m e n t to c o m p a r t m e n t i, and if s~ is the number o f sign changes in the sequence ril,. • .,r~,,, then the number o f extrema in the impulse response Yi does not exceed sl - pi + 1. Proof. W e consider the higher derivatives of the y~, using the notation y(h) for dhy/dt h. If yi has k extrema, then yt~) has at least k zeros, and must have k + 1 e x t r e m a if yl~(0)= 0, since y~l) vanishes at infinity. Similarly, if y~h-1) has k + h - 1 extrema, then y~h) has at least k + h extrema, provided y~h)(o)= 0. H e a r o n and L o n d o n (1972, L e m m a s 4-6) have s h o w n that ylh)(0) = 0 for h = 1 , . . . , P i - 1, w h e r e p~ is the minimum path length from the initially-loaded c o m p a r t m e n t . W e have then in general that y~pi-~) has k + p~ - 1 extrema. It follows that y~P'>has k + p, 1 zeros. N o w , b y equation (3), the coefficient s e q u e n c e in y~P') is r , ( •t~) ~'. . . . , ri,,(-A,,) p', which has the same n u m b e r si of sign changes as r~ . . . . , r~n,. B y L e m m a 1, then, k + pi - 1 -< s, and L e m m a 2 follows directly. The n u m b e r of sign changes in the s e q u e n c e r , , . . , r,-,, in equation (3) is established b y the following lemma. LEMMA 3. In a strongly-connected C M system obeying equations (1), (2) and (3), the coefficient sequence f o r the i-th c o m p a r t m e n t r , , . . . , r~, has at m o s t pi sign changes, where p~ is the m i n i m u m p a t h length f r o m the initially-injected c o m p a r t m e n t to c o m p a r t m e n t i. Proof. Since the columns of R are characteristic vectors of K for the tThe minimum path length from compartment a to compartment b is the minimum number of intercompartmental transfers which must take place in getting from a to b.

616

C . D . THRON

corresponding roots As, we have K R = RA, w h e r e A = diag {Aj}. F r o m this (i.e. from the first m - 1 rows) we get (m > 1)

r2s = ( a , - blAs)r~ s

(ai-I

-

Aj)ri-l'J

riJ =

-

ci-2ri-2'J

(2 < i --< m),

(7)

(8)

bi-i

and from the last n - m rows, if As# a~, r~s =

cirmj

a~ - Aj

( m < i <-- n ) .

(9)

From equations (7) and (8) it follows that rq = r u ~ i _ l ( A S )

(2 -< i -< m),

(lo)

where ~bi ~(Ai) is a polynomial in Aj of degree i - 1 . The polynomial $~_~(A) has at most i - 1 roots; hence it changes sign at most i - 1 times as A decreases continuously, and therefore the sequence ~b~ I(A0, . . . , ~bi_~(A,,), w h e r e /~1 > " ' " > An', has at m o s t i - 1 sign changes. Hearon (1979) has s h o w n that the r~s are positive, and t h e r e f o r e it follows from equation (10) that the s e q u e n c e ril tin, has at most i - 1 ( = p~) sign changes (1 - i < m). From equations (9) and (10) we have . . . . .

r~j = c'rtJ~bm-~(Ai)

a~ - Aj

( m < i <-- n ) ,

(11)

assuming h i # ai. The function ~b,, l ( A ) / ( a i - h ) has at m o s t m - 1 real roots and one pole, and evidently changes sign at these points only; hence it has at most m sign changes. F r o m equation (11), therefore, the sequence r , , . . . , ri,, has at m o s t m ( = pi) sign changes (m < i -< n). If there is some Ak = a~, then equations (9) and ( l l ) do not hold for / = k. Since the As are distinct, As# a~ for j # k; hence equations (9) and ( l l ) hold for j # k. T h e r e f o r e , the n u m b e r of sign changes in the s e q u e n c e (~rn l ( A l ) , - . -, S m - m ( A k - l ) , ( ~ m - l ( A k + l ) . . . . , ~b,,_m(A,,) does not e x c e e d m - 1. The corresponding s e q u e n c e r , , . . . , ri.k-l, r~.k+~, . . . , ri,, has the same sign sequence, except for the effect of the d e n o m i n a t o r in equation (11), which changes sign b e t w e e n ri.k-m and rl,k+t. The terms r~,k-i and ri, k+l t h e r e f o r e differ in sign iff there is no sign change b e t w e e n t[)m_l(Ak_l)and ~bm_l(Ak+l).

P E A K D R U G L E V E L S IN L I N E A R P H A R M A C O K I N E T I C S Y S T E M S - - I I I

617

Therefore, either the s e q u e n c e ril, • •., ri,k-1, ri,k+l, tin' has a sign change b e t w e e n ri, k-~ and ri,k+, and has at m o s t m sign changes in all, or it has no sign change b e t w e e n r~.k ~ and ri,k+~, and has at most m - 2 sign changes in all. W h a t e v e r the sign of r~k, its insertion b e t w e e n r~,k ~ and r~,k+~ introduces no additional sign change in the first case, and at most 2 additional sign changes in the s e c o n d case. In both cases, therefore, the s e q u e n c e r ~ , . . . , ri,, has at most m ( = Pi) sign changes. This c o m p l e t e s the p r o o f of L e m m a 3. It follows directly from L e m m a s 2 and 3 that the number of e x t r e m a in Yl does not e x c e e d 1. This p r o v e s the t h e o r e m for the case w h e r e the system is strongly-connected. W e n o w extend the t h e o r e m to s y s t e m s which are not stronglyconnected. W e define a ' U M matrix' for a given initial state y(0) as a matrix K in equation (1) such that y is unimodal or monotonic. •

•

-,

LEMMA 4. The set o f U M m a t r i c e s f o r a given y(0) is closed, i.e., f o r a given y(0), if a s e q u e n c e o f U M m a t r i c e s H~, H 2 , . . . converges to K (in the sense that the elements o f Hi converge to the c o r r e s p o n d i n g elements o f K) then K is UM. Proof. If d z / d t = - H z and equation (1) holds, then d(y - z) _ dt

K(y - z) + (H - K)z.

(12)

W e here use the s y m b o l '*' to denote c o n v o l u t i o n with the integration carried out from 0 to t : f * g = f d f ( t - s ) g ( s ) d s . W e have from equation (12) y- z = e K'*(H-K)z,

(13)

assuming z ( 0 ) = y(0). Since the functions on the right are b o u n d e d , z ( t ) a p p r o a c h e s y ( t ) at any finite timer as H a p p r o a c h e s K. It is clearly impossible for a multimodal y to be the limit of a s e q u e n c e of nonmultimodal z's (cf. G n e d e n k o and K o l m o g o r o v , 1968, pp. 160-161, T h e o r e m 4), and the lemma follows. To c o m p l e t e the p r o o f of T h e o r e m 1, note that the matrix K for a system which is not s t r o n g l y - c o n n e c t e d is a p p r o a c h e d b y a s e q u e n c e of matrices H for s t r o n g l y - c o n n e c t e d s y s t e m s c o n s t r u c t e d b y inserting arbitrarily small rate coefficients in appropriate off-diagonal positions of K and adding them to the appropriate diagonal terms. L e m m a 4 then establishes the t h e o r e m for s y s t e m s not strongly-connected. t T h i s conclusion does not extend to infinite time because the convergence of y - z to zero as H ~ K m a y not be uniform on 0 -< t -< ~. In general, if K is singular and H is not, then the y s y s t e m is at least partially closed, while the z s y s t e m is completely open. T h e n z(~z)= 0, but y(ao) will be nonzero if the closed part of the s y s t e m is reachable from the site of injection.

618

C.D. THRON

3. Some Alternative and Partial Proofs of Theorem 1. W e summarize these methods briefly, in case they should be useful in related problems. Another p r o o f of L e m m a 3 is along the following lines. L a p l a c e transformation of equation (1) (cf. Schoenfeld, 1963) gives L(y) = (K + Is)-~y(0),

(14)

where I is the identity matrix. The inverse of K + Is can be e x p r e s s e d in terms of its determinant A(s) and c o f a c t o r s A~j(s); and if all y~(0)= 0 except yl(0)= 1, then

a , , ( s ) _ c ( u , + s ) . . . (uk + s) L ( y ~ ) - A(s) (,~ + s ) . . . (h. + s) '

(15)

where if i -< m then c = ( - c O . . . ( - Ci--1), k = n - i, and the /z~ are the characteristic roots of the trailing k × k principal minor, while if i > m then c = ( - c l ) . . . ( - c m - 0 ( - c~), k = n - m - 1, and the p.~ are the a~ for > m, with a~ omitted. E x p a n d i n g b y partial fractions in the usual w a y (Churchill, 1972, pp. 70 ft.), and assuming all the hj are distinct, we get

L(y,) = ~

rq

i=] hi + s'

(16)

where

c ( g ~ - x j ) . . . (~k - hi) r,j = (X, - h i ) . . . (hi_, - hj)(,~j+, - ,~j)... (h. - hi)"

(17)

It is easily verified that the /.L, are always a subset of the characteristic roots of the ( n - 1)× ( n - 1) principal matrix obtained b y deleting the min{i, m}-th row and column of K, and in view of the symmetrizability of K (Hearon, 1963, pp. 52-53) and the root separation p r o p e r t y (Browne, 1930; Bellman, 1970, pp. 117-118), it follows that at least one A~ lies b e t w e e n every two p.~, i.e. there is at most one /zv b e t w e e n any Aj and Ai+~. Consider now the change in the n u m b e r of negative factors in the numerator and denominator of equation (17) as we go from rij to ri.j+~. W e have: (a) rijri,j+~ < 0 iff there is no p., b e t w e e n Aj and hj+~; and (b) the number of times this occurs in the s e q u e n c e A~. . . . , A, is m i n { i - 1 , m} (= p,). For a pure catenary system, the unimodality p r o p e r t y has b e e n given by Sheppard (1971, pp. 500--501),~- w h o used a stochastic model and an elegant proof based on the variation-diminishing p r o p e r t y of totally t I am indebted to one of the reviewers for drawing my attention to this reference.

P E A K DRUG LEVELS IN L I N E A R P H A R M A C O K I N E T I C SYSTEMS--III

619

positive kernels (Karlin, 1964, T h e o r e m s 2.6, 4.3(i); 1968, p. 233, T h e o r e m 3.1). This line of p r o o f apparently will not do for any but pure catenary systems, since the total positivity of the transition density matrix implies that K is a Jacobi matrix (Karlin, 1968, p. 115, T h e o r e m 3.4).

4. Other Compartmental Structures.

B y a 'structure' we mean a designation of which the intercompartmental and exit rate constants can be nonzero, and also which c o m p a r t m e n t receives the injection.t W e shall say a c o m p a r t m e n t a l structure is U M if the impulse r e s p o n s e in e v e r y c o m p a r t m e n t is always unimodal or monotonic, w h a t e v e r values are assigned to the rate constants. C o n v e r s e l y , we say a structure is N U M if there exists a s y s t e m of that structure in which the impulse r e s p o n s e in some c o m p a r t m e n t is not unimodal or monotonic. W e begin b y establishing two principles for establishing the N U M p r o p e r t y b y reducing complicated structures to simpler ones. Intuition suggests that a s y s t e m with a v e r y slow t r a n s f e r b e t w e e n two compartments b e h a v e s v e r y m u c h like one with no transfer at all b e t w e e n 'those c o m p a r t m e n t s , at least for some time. Intuition also suggests that a c o m p a r t m e n t which v e r y rapidly transfers its contents to another comp a r t m e n t b e h a v e s almost like no c o m p a r t m e n t at all, in that it immediately passes on all that it receives. These notions are formalized in the following two principles. Principle 1. (Deletion of connections.) If a c o m p a r t m e n t a l structure $1 is reducible to a structure $2 b y setting some of the rate constants equal to zero, and if $2 is N U M , then S 1 is N U M . Principle 2. (Removal or fusion of compartments.) If in a structure Sj there is a c o m p a r t m e n t a which transfers material only to one other c o m p a r t m e n t b and not to the outside world, and if $1 is reducible to a structure $2 b y removing a and r o u t i n g to b all transfers originally going to a (and the injection, if originally made in a, is n o w made in b), and if $2 is N U M , then $1 is N U M . Proof. Principle 1 follows directly from the negative c o n v e r s e of L e m m a 4, since the matrix K for any system of structure $2 is app r o a c h e d b y a s e q u e n c e of matrices Hi for s y s t e m s of structure S~. For Principle 2, if $2 o b e y s equation (1) then S~ o b e y s dz m

dt

- K z + u,

(18)

w h e r e z is the column v e c t o r of quantities in the c o m p a r t m e n t s other ~We might call this an 'oriented structure' because we specify which compartment receives the injection, but we will use 'structure' for brevity.

620

C.D. THRON

than c o m p a r t m e n t a, and u is a c o l u m n v e c t o r w h o s e t e r m s are all zero except

Ub = -- ~ [kojzj] + kb~Za i

(19)

= ~ [koj(zj*kb,, e -kb°' - zj)] + kbazo(O) e -k~°'. J

F r o m equations (18), (19) and (1) w e get y - z = e - K t [ y ( 0 ) - z(0)] - e K ' * u.

(20)

The last term in e q u a t i o n (20) is a c o l u m n v e c t o r v w h o s e e l e m e n t s v~ are the terms 4% in the b - t h c o l u m n of e -K', e a c h c o n v o l u t e d with the right-hand side of e q u a t i o n (19): v, = ~ {ko,[(4J~b *zi)*kbo e

--kbat

- (¢,,b*zj)]} + zo(O)4%*kb~ e -k~°'.

(21)

J

(The ~0ib are in f a c t the i m p u l s e r e s p o n s e s in S 2 w h e n the i n j e c t i o n is made into c o m p a r t m e n t b.) F o r a n y t w o f u n c t i o n s f and g with derivatives f' and g', integration by parts shows that f*g' = f ( O ) g ( t ) - f ( t ) g ( O ) + f ' * g . A p p l y i n g this e q u a t i o n to the c o n v o l u t i o n s of kbo e -k~°t = g' in e q u a t i o n (21) w e get

v~ = - Y. [k,,j($~b*zj)'*e -kb°']

• It r * ~ - zo(O)[q,~b(O) e -kb°'- 4~b + ~,~b ~

kbat]

J.

(22)

If the injection is given into the s a m e c o m p a r t m e n t in S 1 and 82, then y(0) = z(0), za(0) = 0, and w e h a v e f r o m e q u a t i o n s (20) and (22) lira kba--~

( y i - z~) =

lira ( -

v~) = O.

(23)

kba--*~

If the injection is given into c o m p a r t m e n t b in $2 and into a in Sl, then zo(0) = yb(0), so za(O)$ib = yb(0)q% = Yi in e q u a t i o n (22). F u r t h e r m o r e ,

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--Ill

621

z(0) = 0 in equation (20); so from equations (20) and (22) we have y~ - z, = ~ , [kai(tl%*z~)'*e -k~°'] + y,(O)e -k~°' + y'i*e -k~°t,

(24)

J

and lim (y, - zi) = 0

(t > 0)

kba--+~

(25) = y,(O)

(t = 0).

In determining w h e t h e r a structure is N U M , we need only consider times at and near the e x t r e m a of the impulse response. T h e r e f o r e this p r o o f that y - z ~ 0 for t > 0 establishes Principle 2. The effect of reduction b y Principle 2 is to 'fuse' c o m p a r t m e n t a into c o m p a r t m e n t b. Evidently, any c o m p a r t m e n t can be made suitable for r e m o v a l or 'fusion' under Principle 2 b y first deleting all but one efflux 'according to Principle 1. H o w e v e r , this need to delete effiuxes is not a trivial restriction, and certain reductions are not permissible under Principles 1 and 2 (cf. Figure 4). To p r o v e that a given structure $1 is N U M , it suffices to show that it is reducible b y Principles 1 and 2 to an N U M nucleus $2. T w o such nuclei have already b e e n identified (Figures 1 and 2), and Figure 5 presents nine

5.

~

rr. ~

l

\\

rrr~

P~

Figure 4. Illustration of the limitation of 'fusion' of compartments by Principle 2. Structure I can be reduced to II or V by Principle I(P1), and thence to III or VI respectively by Principle 2 (P2). Structure IV is formed by fusing compartment 1 directly to compartment 3 without first deleting the connection from 1 to 2, but this is not a valid reduction under Principles 1 and 2.

622

C.D. THRON

10.5

(4)

~ 50

(5.)~

,1 ,0

,o

10

~

I0 [ ]

2 ~, 0.5

JILl[

5~

io~

Figure 5. NUM compartmental nuclei_ Responses for the parameter values shown were computed by numerical integration by a 4th-order Runge-Kutta procedure. Each compartment which has a non-unimodal, non-monotonic impulse response contains a sketch of its response. The time scale on these sketches runs from 0 to 5. The vertical scale in Type 4 runs from 0.04 to 0.06. The other vertical scales all start at 0, the upper ends being as follows: Type 1, 1.0; Types 2, 3, 5, 6, 0.06; Type 7, 0.0001; Type 8, 0.02; Type 9, 0.01.

N U M nuclei which are sufficient for testing all c o m p a r t m e n t a l systems, as will presently be shown. The T y p e 2 nucleus is like that of Figure 1, and the Type 3 nucleus is like that of Figure 2, e x c e p t that it is closed. Note that five of the nuclei (Types 2, 4, 7, 8 and 9) have straightforward bimodal responses, three (Types 3, 5 and 6) are closed structures with one maximum and one minimum, and one (Type 1) is a closed structure with only one e x t r e m u m , a minimum, in the impulse response. It is to be emphasized that the rate constants in these nuclei are not permitted to take on zero values, c o n t r a r y to what we allow with structures generally. For example, a non-zero exit rate constant in the T y p e 2 nucleus is essential for the N U M property, and if a larger structure reduces by Principles 1 and 2 to a nucleus like T y p e 2 but with a zero exit rate constant, then the larger structure is not t h e r e b y p r o v e d N U M . We now survey c o m p a r t m e n t a l structures for the N U M property, beginning with the class of, s t r o n g l y - c o n n e c t e d tree structures. We identify as c o m p a r t m e n t rn the central c o m p a r t m e n t of the m a m m i l l a r y part of a CM structure. In a pure c a t e n a r y structure, m m a y be taken as either

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--III

623

the last or the next-to-last c o m p a r t m e n t . Given any CM structure, the addition of a n e w c o m p a r t m e n t reversibly c o n n e c t e d to c o m p a r t m e n t m p r o d u c e s another C M structure; b u t addition of a new, reversiblyc o n n e c t e d c o m p a r t m e n t a n y w h e r e else p r o d u c e s - a structure reducible (by Principles 1 and 2) to a T y p e 3 N U M nucleus (Figure 5). Since all structures with three or f e w e r c o m p a r t m e n t s are C M structures, and since all strongly-connected tree structures can be c o n s t r u c t e d b y successive additions of r e v e r s i b l y - c o n n e c t e d c o m p a r t m e n t s , it follows that CM structures are the only U M strongly-connected tree structures. Structures with cycles can all be r e d u c e d to a T y p e 1 N U M nucleus b y Principles 1 and 2; t h e r e f o r e t h e y are all N U M . It follows that C M structures are the only s t r o n g l y - c o n n e c t e d U M structures. Turning n o w to structures that are not strongly-connected, we first show that if the only input of a c o m p a r t m e n t j is a unidirectional flux from a c o m p a r t m e n t i with a unimodal or m o n o t o n i c impulse r e s p o n s e , then j also has a unimodal or m o n o t o n i c impulse response. Differentiation of the rate equation gives /

¢

¢

y'; = ki, y , - ~. k.jyj,

(261

v

w h e r e the primed s u m m a t i o n sign means s u m m a t i o n with omission of u = j. Since yj is initially zero, its first e x t r e m u m must be a maximum. F r o m equation (26), with y~ = 0, this must o c c u r w h e r e Y'i < 0. Since Yi is unimodal (or monotonic), Y'i does not later b e c o m e positive; so all e x t r e m a of yi must be maxima, and there can be at most one. N o w , given a U M structure with one or more ettluxes to the outside world, one can f e e d (irreversibly) any of those effluxes into one or more 'tails', straight or branched, in which each c o m p a r t m e n t receives material only from its immediate p r e d e c e s s o r and gives it only to its immediate successor(s) or the outside world. B e c a u s e of the irreversibility of the connections, the 'tails' will not affect the b e h a v i o r of the rest of the structure. It follows, therefore, from the preceding paragraph that the structure with 'tails' added is still U M . With general structures, we determine the behavior of the w h o l e structure b y determining that of certain substructures, which we shall call 'primary c o m p o n e n t s ' and which we define in the following way. First, organize the structure as a set of strong c o m p o n e n t s t c o n n e c t e d b y o n e - w a y transfers (Thron, 1980) and consider the digraphS: of strong tA strong component is a maximal set of strongly-connected compartments, i_e. a set not contained within a larger strongly-connected set (Harary et al., 1965, p. 54). z~The digraph (directed graph) of strong components is the set of all these components and the unidirectional transfers between them.

624

C.D. THRON

components. Identify the 'terminal' strong c o m p o n e n t s as those which have no path to any other strong c o m p o n e n t . To e a c h terminal strong component there corresponds a 'primary c o m p o n e n t ' which comprises all the strong components on all the paths from the origin (site of injection) to that terminal strong component. The behavior of a primary c o m p o n e n t is clearly unaffected by the rest of the structure, b e c a u s e it receives no transfers from the rest of the structure. It is evident that a structure is UM if and only if all its primary c o m p o n e n t s are UM. Take now a U M tree structure and consider the digraph of strong components. Starting at the origin, a path can go through any n u m b e r of one-compartment strong c o m p o n e n t s and any n u m b e r of pure c a t e n a r y strong components, provided the continuation of the path f r o m each of the latter is from its last c o m p a r t m e n t . H o w e v e r , once we r e a c h a catenary strong c o m p o n e n t with a continuation of the path f r o m other than the last c o m p a r t m e n t , or o n c e we r e a c h a strong c o m p o n e n t that is not a pure catenary structure (i.e. it is a CM structure with m a m m i l l a r y compartments), the r e m a i n d e r of the path must consist entirely of irreversibly-connected single c o m p a r t m e n t s , because otherwise the structure would be reducible to a T y p e 3 N U M nucleus. It follows that a tree structure is U M if and only if each of its p r i m a r y c o m p o n e n t s is a CM structure with or without a 'tail'. As a whole, of course, a U M tree structure m a y contain more than one strongly-connected structure with mammillary c o m p a r t m e n t s ; but each such strongly-connected C M structure must lie in a different primary component. Finally, we consider non-cyclic structures other than trees, n a m e l y structures for which the digraph of strong c o m p o n e n t s m a y have multiple paths from the origin to the terminal strong c o m p o n e n t s . A reachable closed c o m p a r t m e n t (i.e. a reachable c o m p a r t m e n t with no exit, either to the outside world or to another c o m p a r t m e n t ) necessarily has a monotonically increasing impulse response. Since there are no transfers from this c o m p a r t m e n t to other c o m p a r t m e n t s , the behavior of the rest of the structure is unaffected by removing this c o m p a r t m e n t and letting its inputs go to the outside world. It follows that if a structure has multiple paths w h i c h . c o n v e r g e precisely at a closed c o m p a r t m e n t b, then the structure is UM, provided that the structure p r o d u c e d by removing compartment b is UM. We turn then to multiple paths converging at an open c o m p a r t m e n t b. We have four cases to consider. Case 1. A path exists f r o m b to a c o m p a r t m e n t which precedes b on a path from the origin but is not an immediate predecessor. In this case a cycle exists, and the structure is reducible to a T y p e 1 N U M nucleus.

PEAK DRUG LEVELS IN LINEARPHARMACOKINETICSYSTEMS--III

625

Case 2. A path exists from b to an immediate p r e d e c e s s o r compartment j. In this case, if the recurrent path has more than one step, then it forms a cycle. If j is the c o m m o n origin of two or more distinct paths to b, then a cycle is again f o r m e d b y even a one-step recurrent path (since at least one of the distinct paths f r o m j to b m u s t have length of at least 2). If j is not the c o m m o n origin of multiple paths, it must lie on one of two or more distinct paths f r o m a more distant p r e d e c e s s o r a to b. The structure is then reducible to a T y p e 6 N U M nucleus if either (a) the path on which j lies has three or more steps from a to b, or (b) that path and an alternate path from a to b b o t h have two or more steps. This leaves for a U M structure only the possibility that b is c o n n e c t e d to an immediate p r e d e c e s s o r lying in a two-step path from a to b, with a one-step alternate path (Figure 6A). Case 3. A recurrent path exists from b to itself, a path which does not pass through any p r e d e c e s s o r of b. If this path has three or more steps, then it forms a cycle; otherwise the structure r e d u c e s to a T y p e 5 N U M nucleus. Case 4. C o m p a r t m e n t b transfers material to the outside world or to a c o m p a r t m e n t f r o m w h i c h there is no path b a c k to b. In this case the structure is reducible to a T y p e 2 N U M nucleus if any of the multiple distinct paths to b has three or more steps. This leaves for U M structures the possibility of multiple paths of one or two steps (Figure 6B, C). At this point we have not yet proved that the structures in Figure 6 are in fact UM. N e v e r t h e l e s s , the preceding discussion shows that if there are a n y U M structures with multiple paths to an open c o m p a r t m e n t , then in such structures each pair of multiple paths must form one of the three structures in Figure 6. E v e r y such U M structure can therefore be c o n s t r u c t e d by adding c o m p a r t m e n t s and connections to one of the three s t r u c t u r e s in Figure 6. Since we need only consider the p r i m a r y corn-

A.

B.

J

"x

I --... I

j

I I

C.

Figure 6. The three basic two-path UM structures. In each case, two paths run from compartment a to compartment b.

626

C . D . THRON

ponents, we proceed to find all the primary components which can be so constructed and which are not provably NUM, i.e. which are not reducible to one of the N U M nuclei in Figure 5. Consider the digraph of strong components of any UM primary component. Proceeding out from the origin, there is a first branch point where multiple paths arise. Call the first compartment where multiple paths arise compartment a. Continuing outward, all multiple paths eventually converge. Call the last compartment where multiple paths converge compartment b. There must be two distinct paths from a to b, because otherwise the structure would be reducible to a Type 9 N U M nucleus. We can therefore take as a starting point for a reconstruction of the UM primary component compartments a and b and the compartments making up any two distinct paths from a to b. These compartments must form one of the structures in Figure 6. Compartment a can be preceded only by a pure catenary structure, because if there were any preceding mammillary compartments, then the structure would be reducible to a Type 8 N U M nucleus. Likewise, compartment b cannot be succeeded by anything but a 'tail', or the structure would reduce to a Type 5 N U M nucleus. We turn, therefore, to the reconstruction of the ' a - b section' between compartments a and b. All compartments in this section have paths coming from compartments a and paths going to compartment b. Therefore, in our reconstruction the compartments can be added as chains of irreversiblyconnected compartments (reversibility can be introduced in later steps of construction), running from one to another (or the same) compartment in the already-completed part of the a - b section. In fact, we need not add any chains of more than one compartment, for the following reasons. First, a chain which starts and ends at the same compartment cannot have more than one other compartment, or it will form a cycle. Second, our method of construction will assure that every compartment is open and reachable from the origin, so the addition of a new bridging chain forms multiple paths to an open compartment, and if the added chain has more than one compartment, then the structure will be reducible to a Type 2 NUM nucleus. The construction can therefore proceed by adding, one at a time, (a) a transfer from one existing compartment to another, (b) a new compartment reversibly connected to an existing compartment, or (c) a new compartment forming a bridge between two existing compartments, receiving from one and giving to the other. We temporarily defer adding exits to the outside world. In order to enumerate all the structures that can be produced by this method of construction, we begin by cataloguing some structures which

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETICSYSTEMS--Ill

627

are 'unchangeable' (i.e. the addition of any n e w transfer or c o m p a r t m e n t , according to the a b o v e rules, p r o d u c e s a structure reducible to an N U M nucleus), and also s o m e classes of structures which are 'invariant' (i.e. e v e r y e l e m e n t a r y c o n s t r u c t i o n step p r o d u c e s either another structure of the same class or an N U M structure). Referring to Figure 7, we shall temporarily use the letters P, Q, R, S, T and T' to d e n o t e just the structure from c o m p a r t m e n t a to c o m p a r t m e n t b, inclusive, with all exits s u p p r e s s e d e x c e p t that from c o m p a r t m e n t b. Then a s y s t e m a t i c exploration of the possible construction steps s h o w s that P, Q and R are unchangeable and S is invariant. F u r t h e r m o r e , a structure of the general class T (except structure C, Figure 6) can be changed only to one of class T or T', and a structure of class T' ( e x c e p L s t r u c t u r e B) can be changed only to one of class T' or (if there is but one bridging C M structure) S. Turning to the starting structures, we find that structure A is unchangeable. Structure B is changed in one step to a structure of class S or T', or to a structure B' which is like A e x c e p t that there is an exit from c o m p a r t m e n t b. These are all the possibilities for construction from B. Like A, B' is unchangeable. Structure C is changed in at m o s t two steps to structure P, Q, R, T or T', and these are all the possible constructions from C. As potential U M structures we have then structures A, B, B', C, P, Q, R, S, T, T' and s o m e structures derived from C b y one step. Since structures A, B, B', C and the structures derived from C are contained within structures P, Q, R, S, T or T', it will be sufficient to p r o v e the U M p r o p e r t y for structures P, Q, R, S, T and T'. W e generalize these b y

Figure 7. The set of all multiple-path UM primary components (excepting CM structures and structures with 'tails' or multiple paths to a closed compartment). Each circle marked CM represents the first compartment of a CM substructure.

628

C.D. THRON

adding exits and catenary p r e d e c e s s o r s , and arrive at the structures in Figure 7. These and the C M structure (Figure 3) form the whole set of potentially U M primary c o m p o n e n t s , e x c e p t for those with 'tails' or multiple paths to a closed c o m p a r t m e n t . We must prove that the structures in Figure 7 are in fact U M , since so far we have only shown that they are not reducible to one of the N U M nuclei in Figure 5. In structure P, the c o m p a r t m e n t s up through the (b - 1)-th form a C M structure; hence we have 0nly to consider c o m p a r t m e n t b. A s s u m e that the compartments up through the ( b - 1)-th are strongly-connected, and that k0b does not equal any of the characteristic roots of the stronglyconnected part of the system. Then the system o b e y s (3). Since there are at most b terms in the s e q u e n c e rb~,..., rbb, there are at most b - 1 sign changes. The minimum path from the origin to c o m p a r t m e n t b has length b - 2; hence Yb has at m o s t 2 extrema, b y L e m m a 2. Since yb(0) = yb(o~) = 0 and Yo -> 0, there cannot be an e v e n n u m b e r of extrema, and therefore there is but one. The a s s u m p t i o n of strong c o n n e c t i v i t y and the restriction on koo can be dropped, b y L e m m a 4. The same p r o o f also establishes the unimodality of the impulse response in c o m p a r t m e n t b of structures Q and R, and in c o m p a r t m e n t b of structure T in the special case w h e r e there are just two bridging CM structures which are both single c o m p a r t m e n t s . F u r t h e r m o r e , since the connections to c o m p a r t m e n t b in Q and R are irreversible, the remainder of each of these s y s t e m s (up through c o m p a r t m e n t b - 1) b e h a v e s independently, with at m o s t b - 2 sign changes in the coefficient s e q u e n c e r b - l , t , . . . , Fb-l,b-l. The same p r o o f t h e r e f o r e establishes the unimodality in compartment b - 1 of Q, and in c o m p a r t m e n t s b - 1 and b - 2 of R. Since the remainder of structures Q and R is in each case a CM structure, these structures are UM. Structure S is a C M structure, e x c e p t for c o m p a r t m e n t b. F r o m KR = RA we have

- kboroj - kb,o+, r . + , , j + ( k o b - ,~j)rbj = O,

where the (a substructure connectivity, equation (27)

(27)

+ 1)-th c o m p a r t m e n t is the first c o m p a r t m e n t of the C M denoted b y the circle in Figure 7 (S). A s s u m i n g strong equation (10) holds for raj and ra+la, and from this and we get

[kb.4,o-,(,~j) + kb.o+,,/,o (Aj)] r,j rbj =

kob -- Aj

(28)

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--III

629

assuming kob# hi. The n u m e r a t o r of equation (28) is a polynomial of degree a in hi, hence rbj(h) has a zeros and one pole, and the s e q u e n c e rb,,. •., rbb has at most a + 1 sign changes. The minimum path from the origin to c o m p a r t m e n t b is of length a, and t h e r e f o r e Yb has at most two extrema, b y L e m m a 2. As already noted, there cannot be an e v e n n u m b e r of extrema, and t h e r e f o r e there is but one. The a s s u m p t i o n of strong connectivity and the restriction on kob are r e m o v e d b y L e m m a 4. F o r structure T, if c o m p a r t m e n t b is r e m o v e d , then each C M substructure lies in a CM primary c o m p o n e n t , so structure T with compartment b r e m o v e d is U M . W e have already s h o w n that Yb is unimodal in the special case w h e r e there are just two bridging C M structures which are both single c o m p a r t m e n t s . W e n o w extend this, first to the case of more than two single-compartment bridges and then to the general structure T. The r e s p o n s e in c o m p a r t m e n t b is clearly the sum of the unimodal r e s p o n s e s that w o u l d be obtained b y deleting all but one bridge, doing this for each bridge in turn. Intuition suggests that if no two of these unimodal r e s p o n s e s can c o m b i n e to p r o d u c e a bimodal r e s p o n s e , then the addition of a third will not help. B e f o r e we can c o m p l e t e our p r o o f for structures T and T' we need to establish this assertion in the form of a theorem.

5. A Theorem on the Unimodality o f Linear Combinations o f Unimodal or M o n o t o n i c Functions. W e begin b y observing that certain pairs of m o n o t o n i c functions (one increasing, one decreasing) have the p r o p e r t y that all their linear combinations are m o n o t o n i c or unimodal. For example, e -~ and 1 - e k, have this p r o p e r t y if k > 1. On the other hand, if k < 1, then there are linear combinations of these two functions, e.g. e - ' + 1 - e -k`, which have a minimum and are t h e r e f o r e not unimodal or monotonic. To determine w h e t h e r a function f is unimodal or m o n o t o n i c we consider three points fi = f(ti), i = 1, 2, 3; t~ < t2 < t3. The function f is unimodal or m o n o t o n i c only if f2 -> rain {f~, f3}. LEMMA 5. L e t f and g be f u n c t i o n s whose values at t~, t2 and t3 (t~ < t2 < t~) are f~, f2, f3 and gl, g2, g3 respectively, and let ]:2 >-min{f~, f3} and gl>-g2>-g3. Then f 2 + a g z > - m i n { f ~ + a g l , f3+ag3} f o r all a if and only if

g~(f2 - f3) + g2(f3 - fl) + gJ(fl - f2) >- O. Proof.

(29)

If g~ = g3, then g~ = g 2 = g~ and the lemma is trivially true.

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C.D. THRON

Assume, then, that g~ > g~; then equation (29) follows directly f r o m [f3-f,]

f~ + [ ~ J g ~

> -

[f3-f, ] f, + [g _ - - ~ Jgl.

(30)

Conversely, we get equation (30) f r o m equation (29); and f r o m equation (30), letting ao = (f3 - f,)/(g~ - g3), we get (f2 + otg2) - ( f , + a g o >-- ( ao - tr )(g, - g2) ~ 0 (a < SoL and (since f3 +

(31a)

Sog3 = f , + aog,)

(f2 + ag2) - (f3 + sg3) -> ( a - ao)(g2 - g~) -> 0

(a > ao).

(31b)

This completes the p r o o f of L e m m a 5. We continue with a l e m m a on linear c o m b i n a t i o n s of three functions. LEMMA 6.

L e t f a n d g be f u n c t i o n s as in L e m m a 5, a n d let h be a n o t h e r f u n c t i o n like g, Le. hi >- h2 >~ h3. T h e n if f2 -t- s g 2 >~ rain{f, + s g l , f3 + sg3} f o r all a, a n d f2 + oth2 >-- m i n { f l + s h , , f3 + trh3} f o r all s , it f o l l o w s t h a t f2 + agz + flh2 >- m i n { f l + agl + Oh j, f3 + ag3 + Oh3} f o r all a, fl s u c h t h a t a0 >0.

For g we have e q u a t i o n (29) and, similarly for h, introducing an arbitrary positive factor 3' = B/a, we have Proof.

y[h,(f2 - f3) + hz(f3 - f , ) + h3(f, - f2)] >~ O.

(32)

Adding equations (29) and (32) we get (g, + yh,)(f2 - f3) + (g2 + yh2)(f3 - f,) + (g3 + yh3)(f, - f2) -> 0,

(33)

from which L e m m a 6 follows by L e m m a 5. We are now ready for the following theorem. THEOREM 2.

I f a f u n c t i o n y is the s u m o f three o r m o r e u n i m o d a l or m o n o t o n i c f u n c t i o n s , a n d if y has a m i n i m u m , then there is a f u n c t i o n z

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--III

631

which has a m i n i m u m and is a linear combination of lust two of the unimodal or m o n o t o n i c s u m m a n d s o f y, with positive coe~icients. Proof. C h o o s e times tl < t 2 < t3 (or t l > t2> t3) such that the corresponding values y~ > Y2 < Y3. S u b s t r a c t from y any summand, or any partial sum of s u m m a n d s , which is greater at t2 than at both t~ and t3. The resulting function u is the sum of functions, each of w h o s e value at t2 is intermediate b e t w e e n its values at tl and t3, and u~ > u2 < u3. As three functions f, g, and h, consider two of the individual s u m m a n d s of u, together with the sum of the rest. Clearly, we cannot have f, g and h all increasing or all decreasing if u~ > u2 < u3. B y assigning the labels f, g and h appropriately, and b y interchanging the labels t~ and t3 if necessary, we can assure that g~ >- g2 ~ g3, hj --- h2-> h~ and f~ -< f2 ~< f3. F r o m Ul > u2 < u3 we have f2 + gz + h2 < min{fl + gl + h~, f3 + g3 + h3}. F r o m the negative c o n v e r s e of L e m m a 6 it then follows that there is s o m e a such that either f2 + ag2 < min{f~ + ag~, f3 + ag3} or f2 + ah2 < min{fl + ah~, [3 + ah3}. Neither of these conditions can hold for a < 0, b e c a u s e f2 >-f~ and g2 -< gl, and t h e r e f o r e fz + ag2 -> f~ + o~gl for o~ < 0; the same applies for f + ah. T h e r e f o r e , a > 0, and we have a new function v = f + ag or f + ah, which is the sum of functions e a c h of w h o s e value at t2 is intermediate b e t w e e n its values at tl and t3, and v~ > v2 < v3. If v has m o r e than two of the original s u m m a n d s , then the foregoing can be repeated, until one arrives at a linear c o m b i n a t i o n z of just t w o of the original unimodal or m o n o t o n i c s u m m a n d s with positive coefficients, and z2 < min{z~, z3}. This p r o v e s the theorem. 6. Completion o f the P r o o f that Structures T and T' are UM. If there is an N U M T structure with three or more single-compartment bridges, it follows from T h e o r e m 2 that some linear combination (with positive coefficients) of the contributions to Yb from just two of the bridges has a minimum. If we eliminate all the other bridges (redirecting the c o r r e s p o n d ing transfers f r o m c o m p a r t m e n t a to the outside world) and let the outflows from these t w o c o m p a r t m e n t s be suitably partitioned b e t w e e n b and the outside world, we than have a T structure with t w o single-compartment bridges and a non-unimodal, n o n - m o n o t o n i c r e s p o n s e in c o m p a r t m e n t b. But this has already b e e n s h o w n to be impossible; hence, T structures with any n u m b e r of single-compartment bridges m u s t be U M . If a bridge c o m p a r t m e n t in a T structure is the first c o m p a r t m e n t of a larger C M structure, then its impulse r e s p o n s e to an injection directly into it is the sum of exponential terms with positive coefficients (Hearon, 1979). Its impulse r e s p o n s e in the larger T structure is the convolution of this sum of exponentials on its influx from c o m p a r t m e n t a. Its efflux to

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C . D . THRON

compartment b is therefore equivalent to a sum of effluxes from a set of single-compartment bridges and since T structures with single-compartment bridges are U M , it follows that all T structures are U M . Any structure of the form T' (Figure 7) can be obtained b y reduction of some T structure b y Principle 2. It follows, since structure T is U M , that structure T' cannot be N U M and is t h e r e f o r e U M . We have therefore established a more c o m p r e h e n s i v e version of Theorem 1 : THEOREM 3. The set of U M structures comprises precisely: (a) those structures whose every primary component is either a C M structure or a structure of type P, Q, R, S, T or T' (Figure 7); and (b) structures formed from (a) by addition of 'tails' (straight or branched) and/or closed compartments. Note that a general U M structure m a y be s o m e w h a t more complicated than m a y at first appear f r o m the p r i m a r y c o m p o n e n t s , b e c a u s e of partial overlapping of the primary c o m p o n e n t s (cf. Figure 8). N o t e also that paths converging at a closed c o m p a r t m e n t m a y arise from different primary components.

7, Simultaneous Injection Into Two or More Compartments of a M a m millary System. In some practical cases it m a y not be possible to model the site of injection as a single c o m p a r t m e n t , and all injections must be modelled as simultaneous injections into two or more c o m p a r t m e n t s . The following example d e m o n s t r a t e s that the unimodality p r o p e r t y does not necessarily carry over to such cases.

.

.

.

.

.

.

.

I Figure 8. Three partially overlapping primary components of structure Q with terminal compartments b, b' and b" respectively. The first two primary components are each outlined by dashed lines; the third is the same as the second, except that compartment b" replaces compartment b'.

PEAK DRUG LEVELS IN LINEAR PHARMACOKINETIC SYSTEMS--III

633

Suppose simultaneous impulse injections are made into peripheral compartments 2 and 4 of a 4-compartment mammillary system. Suppose next that compartment 2 equilibrates rapidly with compartment 1 (the central compartment), causing the level in compartment 1 to rise. Next, these two compartments equilibrate with compartment 3, and this causes the level in compartment 1 to decline. Next, all three compartments equilibrate very slowly with compartment 4, and this causes the level in compartment 1 to rise again. Finally, the level in compartment 1 declines as a result of excretion. Numerical integration of such a model, with

K

=

0.262799 I _ 0.2 0.055556 0.007143

-0.8 0.8 0 0

-0.044444 0 0.044444 0

-

0'0028571 0 0 0.002857

and initial injections of 0.25 and 0.75 into compartments 2 and 4, respectively, showed a bimodal response in compartment 1 with peaks at 3.4 and 890 time units. The large relative difference in these times seems to be characteristic of this model, because the bimodality depends upon large differences in the equilibration rates of the various compartments. 8. Discussion. In physiological and pharmacological applications, the model of Figure 1 might fit some cases where the injected substance passes to the brain (compartment 4) by two pathways (cerebral capillaries and cerebrospinal fluid), or where an intramuscular or subcutaneous injection is absorbed by two pathways (capillaries and lymphatics). The model of Figure 2, on the other hand, may fit some cases where the substance injected into compartment 1 is an inactive precursor which is metabolized to an active form (compartment 3), compartments 2 and 4 being distribution volumes for the precursor and the active substance respectively. This model might also fit some cases of intramuscular or subcutaneous injection, where compartment 2 represents local binding or intracellular sequestration. In fact, Greenblatt and Koch-Weser (1976, Figure 1B) have published for intramuscular absorption of quinidine a curve very similar in form to curve A in Figure 2. Although these authors interpreted their curve as 'non-linear' absorption, they did not study dose-dependence, and there seems to be no reason why their results could not be fitted by a linear model such as that of Figure 2. The model of Figure 2 may also fit certain pulse-labeling experiments, where compartment 2 represents either a distribution volume of the injected tracer substance or a metabolic precursor from which the

634

C.D. THRON

substance in compartment 1 is formed reversibly. In such cases, labeling would appear in compartment 3 with a bimodal time-course. The examples of Figures 1 and 2 demonstrate that the bimodal impulse responses of multicompartment systems can be such that a slower (time-expanded) rectangular injection can sometimes produce a higher peak drug level in one compartment. As in the other examples of this phenomenon (Thron, 1981b), however, the magnitude of this paradoxical output peak effect seems to be generally extremely small. I am indebted to Professors R. Z. Norman, J. L. Snell and R. E. Williamson of the Department of Mathematics, to Professor I. H. Thomae of the Thayer School of Engineering, and to C. P. Thron for helpful discussions on various aspects of this work. LITERATURE Bellman, R. 1970. Introduction to Matrix Analysis, 2nd ed. New York: McGraw Hill Book Company. Browne, E. T. 1930. "On the Separation Property of the Roots of the Secular Equation," Am. J. Math. 52, 843-850. Churchill, R. V. 1972. Operational Mathematics, 3rd ed. New York: McGraw-Hill Book Company. Coddington, E. A. and N. Levinson. 1955. Theory of Ordinary Differential Equations. New York: McGraw-Hill Book Company. Franklin, P. 1940. A Treatise on Advanced Calculus. New York: John Wiley & Sons. Gnedenko, B. U. and A. N. Kolmogorov. 1968. Limit Distributions f o r Sums o.f Independent Random Variables. Translated and revised by K. L. Chung, with appendices by J. L. Dobb and P. L. Hsu. Reading, Massachusetts: Addision-Wesley Publishing Company. Greenblatt, D. J. and J. K o c h - W e s e r . 1976. "Intramuscular Injection of Drugs," New Engl. J. Med. 295, 542-546. Harary, F., R. Z. Norman and D. Cartwright. 1965. Structured Models: An Introduction to the Theory o[ Directed Graphs. New York: John Wiley & Sons. Hearon, J. Z. 1963. "Theorems on Linear Systems." Ann. N. Y. Acad. Sci. 108, 36-68. --. 1979. "A Monotonicity Theorem for Compartmental Systems." Math. Biosci. 46, 293-300. --.and W. P. London. 1972. "Path Lengths and Initial Derivatives in Arbitrary and Hessenberg Compartmental Systems." Math. Biosci. 14, 121-134. Himmelblau, D. M. 1970. Process Analysis by Statistical Methods. New York: John Wiley & Sons. Karlin, S. 1964. "Total Positivity, Absorption Probabilities and Applications." Trans. Am. Math. Soc. 111, 33-107. --. 1968. Total Positivity, Vol. 1. Stanford: Stanford University Press. Keilson, J. 1979, Markov Chain Models--Rarity and Exponentiality. New York: SpringerVerlag. Schoenfeld, R. L. 1963. "Linear Network Theory and Tracer Analysis." Ann. N.Y. Acad. Sci. 108, 69-91. Sheppard, C. W. 1971. "Stochastic Models of Tracer Experiments in the Circulation. III. The Lumped Catenary System." J. theor. Biol. 33, 491-515. and A. S. Householder. 1981. "The Mathematical Basis of the Inter-

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pretation of Tracer Experiments in Closed Steady-State Systems." J. appl. Phys. 22, 510-520. Thron, C. D. 1972. "Structure and Kinetic Behavior of Linear Multicompartment Systems." Bull. math. Biophys. 34, 277-291. --. 1980. "Linear Pharmacokinetic Systems." Fed. Proc. 39, 2443-2449. 1981a. " P e a k Drug Levels in Linear Pharmacokinetic S y s t e m s - - I . The Effect of Rate of Injection." Bull. math. Biol. 43, 693-703. 1981b. " P e a k Drug Levels in Linear Pharmacokinetic S y s t e m s - - I I . Conditions for a Paradoxical Injection Rate Effect with Rectangular Input." Bull. math. Biol. 43, 705-715. RECEIVED 5-30-81 REVISED 7 - 1 0 - 8 1

NOTE ADDED IN PROOF Keilson (1979) discusses several aspects of unimodality, including 'strong unimodality', which insures that the convolution with any unimodal function is unimodal. In the present study, the unimodality of 'tails' added to a UM structure follows directly from the strong unimodality of an individual compartment. The impulse response of the last compartment of a catenary system, or more generally an upper Hessenberg system (Hearon and London, 1972), is strongly unimodal.