Estimation of the effective amount entering the body for drugs subject to enterohepatic recirculations

Med. Eng. Phys. Vol. 17, No. 5, pp. 172-l 76, 1995 Elsevier Science Ltd for BES Printed in Great Britain 1350~4x%3/95 $10.00 + 0.00

Estimation of the effective amount entering body for drugs subject to enterohepatic recirculations Y. Plusquellec*tf

the

and G. Houint$

*UFR de MathCmatiques, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France; fUnit de Pharmacocinktique clinique, H6pital Purpan, Place du Docteur Baylac, 31059 Toulouse, France; +WINET, Centre Interdisciplinaire d’Etudes Pharmacocin&iques de Toulouse, C.H.U. Purpan, Place du Docteur Baylac, 31059 Toulouse, France Received

November

1993, accepted

March

1994

ABSTRACT A pharmacokinetic model is used to take into account multiple recirculations of drug occurring at various times after gall bladder emptying. If a dose D is initially administered, due to recirculation, an effective amount A* reaches the body. This value A* is expressed as a function of D and the model parameters, after oral administration or intravenous injection. Using areas under curoes in two different situations, the reabsorption rate may be identified.

Keywords:

Enterohepatic

circulation,

reabsorption,

Med. Eng. Phys., 1995, Vol. 17, 172-176,

compartmental

xBi

4

cumulated amount of drug in B at time T, A” effective amount of drug reaching the body AUC area under amount curve in compartment 1 after administration of a dose D to a bileduct cannulated subject AUC” area amount curve in compartment 1 after administration of a dose D when a great number of recirculations occur B storage compartment including the gall bladder Cl drug clearance D administered dose of drug the fraction of the cumulated amount 43 in B which is released into B the fraction of the cumulated amount FG of drug reaching G which is released into the central compartment G gastro-intestinal tract compartment amount of drug available for absorption G(t) from compartment G at time t time t emptying T, , T2, . . ., T,, . . . times of gall-bladder time-delay TL be addressed Paul Sabatier,

area under

curve (AUC)

April

NOTATIONS

All correspondence should Mathimatiques, UniversitC 31062 Toulouse, France

model,

to Y. Plusquellec, UFR de 118 route de Narbonne,

X,(Q

amount of drug passing from compartment 1 to compartment B during time T,, to time Ti amount of drug in compartment i at time t

INTRODUCTION

When looking for the calculation of the reabsorbed fraction of drugs undergoing enterohepatic circulation, Tse et al.’ have shown that the total cumulative effective amount A* to which the body has been exposed was linked to the initially administered dose D by the relation: A* =

D

1 - F,F,

where Fb is the fraction of drug in the body which is excreted in the bile and F, the fraction of Fb which is subsequently reabsorbed from the gut. As 0 < (1 - FaFb) < 1, A* is greater than D. This study was non-compartmental as well as the study made later by Kurita2 who has proved that the Tse et al. result must be corrected by replacing F,F, by RFaFb with R=

1-E 1 - F,(l-E)

where E represents

the hepatic

extraction

ratio.

It is our purpose to calculate A* for the compartmental model we have recently published’, taking into account a sequence of enterohepatic recirculations present after biliary emptying occurring at various times T,, . . ., T,z, . . . . A* is the cumulative amount of drug which is absorbed first after drug administration and then after each recirculation, so that, as a function of time, the amount A* reaching the central compartment may be greater than D and we have shown that the difference A* - D is proportional to the area under amount curve AUC* in plasma. Then, identification of some model parameters may be performed by a comparison between this area AUC* and the area AUC which would be obtained after administration of the same dose D in a bileduct cannulated subject so that biliary recycling is interrupted.

(1 - &)A, stays in B until time 7;. Then a fraction I;(; of the amount &A1 of the drug reaching G is reabsorbed (coefficient 12,) and again enters compartment 1. We assume that all the remaining fraction (1 - &)&A, spontaneously goes into B in a first-pass transfer process and remains there until time 7;. From time T, to time Z:L, the drug follows the same pathway as previously. We denote hy G(1) the amount of drug available for absorptlon from G at time t. It may happen that some of the drug available for absorption is still present in compartment G when B next empties: these amounts will be taken into account in equations (3), (4) and (11). At time 7& the cumulated quantity A, of drug in B consists of three additive factors: (1 - F,,)F13A, coming from compartment G (1 - &)A, staying here from time I’, X,, the amount of drug passing from compartment 1 to B during time ?; to time T2

l l

THEORETICAL SECTION: THE PHARMACOKINETIC OF RECIRCULATION

l

MODEL

Thus

We have published” a model used to depict the enterohepatic circulation of drugs. This model extended that of Pedersen and Miller4 which was completed by Shepard et al.” for several cycles of equal duration. We have generalized the model by taking into account recirculations occurring at various times 7;, 7& . . ., T,,, . . . instead of at regular intervals, and by adding possible elimination. This model is displayed in F&YP 1 in which G represents the gastro-intestinal tract while B is a storage compartment. Intravenous

A, = (1 - &&)A,

+ XlS2

At time 7:L, a fraction & of A, is spontaneously released into compartment G and is added to the not yet absorbed amount G( T2) to obtain the total amount [ G( T2) + &A21 available in G either for absorption (a part F,;) or for spontaneous transfer back to B (a part 1 - 4;). The remaining part (1 - &)A2 remains in B until time T3. Similarly, during the jth recirculation, from time T, to 7;+,, a fraction & of the accumulated amount A, of drug in B is released into compartment G, the remaining part (1 - &J Aj stays in B until time ‘I;+ I. A fraction fi;(, of the amount [ G( 7;) + &A,] 1s reabsorbed and undergoes a new cycle while the remaining part (1 - fi;) [ G( Ti) + &A,] spontaneously goes into B. Elimination takes place from compartment 1 (k,,, coefficient) and compartment 2 (L,, coefficient) during the whole period. Thus:

injection

A bolus dose D is intravenously injected in the central compartment 1 at time 0. From T,, = 0 to time T,, the drug present in compartment 1 is transferred to a peripheral compartment 2 (k,, and &, coefficients) and to a storage compartment B including the gall bladder (iz,, coefficient) following first order kinetics. At time 7;, a fraction & of the cumulated amount A, of drug in B (only the amount X,, going from compartment 1 to compartment B during time 0 to time 7;) is spontaneously released into the absorption compartment G (the gastro-intestinal tract). The remaining part

14, = x,,,

(1)

‘4, = x,,, + ( 1 - J&&)il,(1 - F,;)G(7;..,)

, +

forg?

2

(2)

with:

ORAL

INTRAVENOUS

tI

f

klz

kA G

2

1

kz 1

I,

kl o

k2 o

\I ,f

Figure 1 taneously

-

kls B

The pharmacokinetic model for multiple enterohepatir rerirculations”. released into compartment G at various times 7;. 71,. ..,, ‘r,,, _.

B is a storage

compartment

from

which

the drug

is spou-

173

EJective amount

G(T,)

in enterohepatic

recirculation:

Y. Plusqueller

and G. Houin

= 0

(3)

G( 7;) = FG[G( 5’~,)

+ F,A,-,]

emkA(Tj-q-1) for

j

x

1 2,

(4)

where X,, is the amount of drug passing from compartment 1 to B during time Tj~, to q.. Oral

Al = X,, + (1 - &)D.

At time T,, a fraction FB of Al is spontaneously released into compartment G and is added to the not yet absorbed amount G(T,) = FCLI e-k~(7’~-7iA) to get a total amount [ G( TI) + FBAl] available in compartment G either for absorption (a part F,;) or for spontaneous transfer back to B (a part 1 - F,:). The remaining part (1 - FB)A, stays in B until time 7;. The drug then follows the same pathway as previously described by equations (2) and (4). The input

function

the first

compartment

in compartment

I

The input

recirculation

injection,

and

function

in

for oral administration. For j= 2, . . ., n, whatever is given by equation (2) 7; with Xej = klB &tWt

the route

dt

of intake,

also G(T,)

= 0

Aj

(10)

-I-,

for intravenous injection, G( Tl) = &Depk~x(T1p7‘1.)

and (11)

Due to recirculations, the body has been exposed not only to the dose D but to a certain amount denoted by A*, which takes into account the successive inputs of reabsorbed drug in compartment 1. Intravenous

injection

At time t = 0, the amount D of drug enters compartment 1. At the first recirculation, the amount of drug entering compartment 1 is: 7: 7‘ ’ e,(t)dt= F,[G(T,) + FRA1]kA ’ e-kA(/- I‘, ‘dt I 7’I 1, 7‘ = F,J G( 7;) + &Al]

injection,

and (6)

Similarly, at the nth recirculation, the amount of drug entering compartment 1, due to this recirculation is:

F,[GtT,J + &A,,1 - G(T,,+,) Thus, the effective amount after n recirculations is: A*=D+&;F;,fAz+

occurs

The input

amounts

If X, (t) is the amount ment 1, then:

func+

(7)

in the storage

F,:G(T,)

(12)

A* reaching

(&;-I)2 r=l

1 is:

e,(t) = U( t - 7;) F,; [ G( 7;) + FBAj] kAe-k/ic’- V

The accumulated compartment B

1

= &;[G( 7;) + FBA,] - G( 7;)

TI,) F,; D kA epk~~(1--7i)

When the ith recirculation tion eI( t) in compartment

1 - epk.,(“i-“‘l) t

(5)

for oral administration, with s(t) = Dirac impulse delta function and U(t) = unit step function of Heaviside:

174

7‘ ’ X,(t)

+ X,, with X,, = k,,

A, = (I-F,)D

1 is:

D&t) for intravenous U( t -

(8)

CALCULATION OF THE EFFECTIVE DOSE REACHING THE COMPARTMENT 1

Thus

&fire

for intravenous

dt

for oral administration and G( T,), for j Z 2, is given by equation (4) for all routes of administration.

(1 - &)D coming from G X,, the amount of drug passing from compartment 1 to B during time TL to time TI.

l

4 = h, I 0 ’ XI(t)

(9)

administration

The drug is orally administered into compartment G and stays there during a time delay TL. From time TL to time T,, a fraction F,; of the dose D is absorbed and reaches compartment 1 but it may not be totally absorbed at time TI when the first emptying of B occurs. The amount (1 - &)D comes into the storage compartment B. At time T,, the cumulated amount A, of drug in B is composed of two additive factors: l

7

G(7)) FI

-

G(T,,+,)

As G( T,) = 0 in the case (from equation A* = D + E;,,& c A, + (fi; - 1) i Fl kl -

of drug at time t in compartAdding

the body

G(7;,+,)

part by part the relations:

(13) 3) we get:

G(T,)

(14)

A,=

(1 - &;&)A,

+ X,,, + (1 - &;)G(T,)

i=i

a-1

(1 - 4;:) 2 G(7)) + (1 - t”;,)U = -t;;:&

A ,,+ I

c

A; + c

FI

i-l

X,3, -t- (1-F;;)

2

G( 7:)

From

equations

/=I

From equations ( 14) and (15), we derive: ,I+ I A* = 11 + c &, - A,,+, - G( T,,,,) r-1 with

(21)

,=I (20) and (21) we derive: ,rt 1

(16) Thus

Fl

When infinity ‘/ Ni-I

the

recirculation

number

increases

to

-* X,(t)dl-

X,(t)dt=AUC* i0 I 0 where AUC” denotes the area under amount curve in compartment 1, following an infinite number of recirculations.. Furthermore, lim G(7’,, i ,) = 0 and lim A,,.] = 0 ,,- i T ,I- +-L because the recycled amount vanishes number of cycles is great. Thus A* = 11 + L,,, AUC”

when the

(17) The second member of equation (17) is homogenous to a mass, as well as I) and A* because k,,S is expressed in time-’ and AIJC* in mass X time.

Then equation (16) is unchanged and gives the rest& equation ( 17) which still holds. Thus, for oral administration as well as for intravenous injection: 24” = 1) + k,,, AUC* Example

We have published~ a pharn~acokinetic study of a new drug IN0 CZ 2628; starting from a first investigation in rats, before utilization in human and we have shown that this compound exhibits enterohepatic recirculations. From data and results published”, we get for subjects E, F, G, H, I of this study, after oral administration of D = 300 mg of drug the results shown in Td,& 1. Table

Oral

1

Rrscdts

fbr subjects

E, F, G. Ii, I

admi~s~ation

Before the first recirculation, the amouIlt of drug coming from G which enters comp~l.tment 1 is:

=: fii;:o(l

- p~k.~c7’~r7i.1) = &;I] -

G(y’,)

(18)

The amounts of drug due to recirculations are expressed as in the previous case, so that equation ( 13) becomes:

(z$;--1) iG(?;! i:l’

+ &G(7;)

- G(T,,+,)

(19)

PRACTICAL The ratio

SECTION

A*/D

If the same dose D is administered to a bile-duct cannulated subject in which biliary recycling is interrupted, we get an area under amount curve in compartment 1 denoted by AUC. Assuming constant drug clearance: A” -. = AUC,” I) AUC Thus:

(4, - If 2

r-1

G(7;) - G(T,+,)

(20)

-A* = AUC” -._D AUC

- II+ - -

kiBAUC* __ D

AUC” = 1 + klB ____ D

Then

175

A* -=-= AUC” D AUC AUC =

D X AUC” D -I- k,, AUC”

As RIB is a positive number, we find again the result: the AUC is increased when enterohepatic recirculations are present (AUC* > AK). Furthermore, if k,, is small, AUC* and AUC approach each other closely, in agreement with lack of recirculation. As an area under curve may be experimentally determined, even from a non-compartmental analysis, equation (22) gives a way to identify the klB coefficient which characterizes the recirculation. On the other hand, if km is known (for example, a gall bladder ultra-sound allows one to measure the amount of drug coming from the central compartment to the gall bladder and to deduce k,,), the drug clearance CI may be obtained from equation (22): AUC” ‘1, = I)AvC& ,,jl

- AUC x AUC = &

CONCLUSION From a general model used to depict the enterohepatic recirculation phenomenon, we have shown that the effective amount A* to which the body has been exposed is greater than the administered dose D. Furthermore, the relation between A* and D allows one to identify an important parameter of the model by a comparison of areas under curves when enterohepatic recirculation is present or not. ACKNOWLJ2DGEMENT The authors thank Madame M.C. Aragon efficient secretarial assistance.

for her

FENCES (I - ~~

-gg)

(23)

Then Cl = kl,,

2.4 and

AUC” A” - D AUC” - AUC = AUC” - AUC .

(24)

1. TX FLS, Ballard F, Skinn J. Est,imating the fraction reabsorbed in drugs undergoing enterohepatic circulation. J Phamacokin Biopham 1982; lO(4): 455-61, 2. Kurita M. Necessary modification of formulae estimating the reabsorption rate for drugs subject to enterohepatic recirculation. Jinhamacokin Biopharm 1991; 19(4): 469-72. 3. Plusquellec Y, Houin G. Drug recirculation model with multiple cycles occurring at unequal time intervals. J B~o~~~ Eng 1992; 14: 521-6. 4. Pedersen PV, Miller R. Pharmacokinetics and bioavailability of Cimetidine in humans. J Pham Sci 1980; 69: 394-8.

Example From Table 1 data of Tse et aZ.r related Temazepan in rats, we get: AUC” Thus

176

= 6.5;

AUC = 2.7

to t4C

for D = 0.5 mg

5. Shepard TA, Reuning RH, Aarons LJ. Estimation of area under the curve for drug subjects to enterohepatic cycling. J Pharmacokin Biopham 1985; 13(6): 589-608. 6. Plusquellec Y, Trenque T, Barre J. Tillement JP, De Biasi J, Houin G. Application of a pharmacokinetic model with multiple enterohepatic cycles to a new inotropic drug after infusion and oral administratior~. f Pham Sci 1992; 81(10): 1020-3.