Assessment of rate and extent of drug absorption

Pharnuw. Ther. Vol. 14. pp. 123 to 160. 1981 Printed in Great Britain. All rights rc~,er~ed

0163-7258 81 [}211123-38'~19 ( ~ 0 Copyright q~ 1981 Pcrg~mlon Pre,,~ Lid

Specialist Subject Editors: MALCOLMROWLANDand G. TUCKER

ASSESSMENT

OF

RATE AND EXTENT ABSORPTION

OF

DRUG

DAVID CUTLER* The School of Pharmacy, University of London, Brunswick Square. London W C I, England

1. INTRODUCTION The time course of the concentration of a drug at its site of action is determined by a variety of processes which can be classified as either processes of disposition or processes of absorption. The amount of drug which reaches the site of action is determined in part by the extent of absorption and, the rate of delivery of drug to its site of action is determined in part by the rate of absorption of the drug. While the disposition processes may be decisive in determining the availability of a drug at its site of action, for a given drug these processes are not normally amenable to modification. Usually, the only means of controlling the access of a drug to its site of action is through control of the absorption process by a choice of formulation or route of administration. Furthermore, unintended variations in the activity of a drug-product can arise through variations in the absorption process. An appreciation of the role of the absorption process in the overall activity of a drug product, and the development of procedures to control absorption, require methods of quantifying the absorption process. Only in special cases can an absorption process be directly observed; e.g. in intestinal loop preparations in animals. In human studies experimental data relating to absorption usually consist of some measure of the consequence of the absorption process, such as the plasma concentration of the administered drug or its urinary excretion rate. Mathematical analysis of the data is needed to deduce the characteristics of the absorption process. Methods which can be used for this purpose are reviewed here. The primary aim of investigations of the kinetics of absorption is to obtain a description of the absorption process which is independent of the disposition processes. In some cases, as in considering bioequivalence of different formulations, the information sought is less than a complete description of the absorption process; in other cases--e.g, when the absorption mechanism is of interest--information on the time and dose dependence of the absorption process is the ideal.

2. DEFINITION OF THE TERM "ABSORPTION" Perhaps the most obvious interpretation of the term "absorption" is the disappearance of the drug from its site of administration. Another interpretation is that this term refers to the appearance of the drug in the general circulation. In some cases there may be a considerable difference between these interpretations. For example, following oral administration, drug which leaves the gut lumen must cross the intestinal wall and pass through the liver before reaching the general circulation. If drug is eliminated during passage across the intestinal wall or through the liver, the extent of absorption, interpreted as the amount of drug reaching the general circulation, will be less than the extent of absorption interpreted as the amount of administered drug which disappears from the administration site. If the drug is delayed by interactions with the tissues of the gut wall or the liver, the rate of absorption will also differ for these alternative interpretations. *Present address: Department of Pharmacy,Universityof Sydney,Sydney,2006, Australia. 123 J.P.T. 14 2

A

124

DAVID CUTLER

In the present context interest centres on absorption as the first step of the process which delivers the drug (or possibly an active metabolite) to its site of action. With specific exceptions (e.g. when the drug is administered for a local effect, or when an active metabolite is formed before the administered drug reaches the general circulation) the drug must reach the general circulation before it can exert its pharmacological effect. With this in mind the appropriate interpretation of the term "absorption" is the process of delivery of drug to the general circulation. Provided there is no significant first-pass effect this interpretation remains appropriate when the activity of the drug is due, partly or wholly, to the formation of an active metabolite; delivery of the drug to the general circulation is the first step in the sequence of processes leading to the arrival of the metabolite at its site of action. Another reason for adopting this interpretation is that most methods of calculation provide an estimate of the rate or extent of delivery of drug into the general circulation, a consequence of the assumptions made and the usual conditions of the study. The term "bioavailability" is widely used to refer to the rate and extent of absorption; however, many workers use the term to refer only to the extent of absorption. In view of this ambiguity, the more explicit terms rate and extent of absorption will be used throughout this review.

3. A S S E S S M E N T O F T H E E X T E N T O F A B S O R P T I O N When a complete description of the rate of absorption is obtained over the entire absorption period the extent of absorption can be calculated as the integral with respect to time of the absorption rate. Assessment of the rate of absorption is discussed in a following section. This section is concerned with methods of assessing the extent of absorption which do not involve a determination of the absorption rate. These methods rely on fewer assumptions than methods for assessing the absorption rate and are easier to apply to experimental data. In some cases (e.g. in calculating the extent of absorption of a drug from a formulation relative to that from another formulation of the same drug) calculations concerning the extent of absorption can be performed when rate calculations are not possible. However, it should always be borne in mind that determination of the extent of absorption provides incomplete information on the absorption process. For example, drug products which are absorbed to a similar extent will show,markedly different plasma concentrations of the drug when the absorption rates from the two formulations differ widely. In assessing the extent of absorption the point of interest is usually the fraction of the administered dose which is absorbed (i.e. which reaches the general circulation). We denote this fraction by F with the administered dose denoted by D. The total amount of drug reaching the general circulation is therefore FD. The equations which follow will be presented in terms of F rather than the total amount of drug absorbed. I n some cases, depending on the data available, the fraction F cannot be evaluated but relative values of F can be determined. For example, if Fx denotes the fraction of a test dose x which is absorbed, and F~ denotes the fraction of a reference or standard dose of the same drug which is absorbed, the aim is to calculate the ratio Fx/F~. Such calculations are appropriate when a new formulation is compared with an existing one. The equations on which calculations of the extent of absorption are normally based can be derived in two ways, which rely on different assumptions. One derivation is based on mass balance considerations, the other on the assumption that the disposition of the drug is by means of linear processes. Both derivations will be presented, to demonstrate the range of application of the derived equations and the assumptions underlying these applications. Both these approaches can be said to be "model independent" in the sense that they do not require explicit models of the disposition of the drug.

Assessment of rate and extent of drug absorption

125

3.1. DERIVATION OF BASIC EQUATIONS 3.1.1. Mass Balance Derivation The mass balance methods are based on the principle that all of the drug which enters the body is ultimately eliminated, either by excretion of the parent drug or by metabolic processes. When elimination is first order with respect to the plasma concentration of the drug, C, the elimination rate, dA~t/dt, is proportional to C and can be written as dAeddt = CL.C

(1)

CL is the total clearance, the sum of the clearances due to the individual elimination processes (e.g. clearance due to renal excretion of the unchanged drug + clearance due to hepatic metabolism). In general, CL may be a function of time or of plasma concentration, but for the present it will be assumed that CLis a constant. Ae~ is the cumulative amount of drug eliminated up to time t. We take time zero as the time at which the drug is first administered and write A~t(O) = 0 to indicate that at this time no drug has yet been eliminated. We write Aet(~) to denote the total amount of drug eliminated up to time infinity (time infinity is taken as some time after which no further elimination occurs; its exact specification is unimportant). Integrating eqn(1) from time zero to time infinity gives Aet(oo) = CL

~0°° C(t) dt

(2)

when CL is a constant. To simplify the notation we write AUC =

C(tl dt

(AUC is the area under the plasma concentration-time curve) and eqn(2) becomes Aez(cc) = CL.AUC.

(31

According to the mass balance principle A~t(~) is equal to the amount of drug absorbed. Thus, F.D = CL.AUC or

F = CL,AUC/D.

(4)

Note that CL.AUC refers to the total amount of drug eliminated from the plasma. In practice the plasma drug concentration is usually measured in a venous blood sample. This will include no contribution from drug which is eliminated before reaching the general circulation (e.g. on the first pass through the liver); consequently CL.AUC can be interpreted as the total amount of drug eliminated from the general circulation. Identifying this quantity with F.D therefore defines F as the fraction of the administered dose D which reaches the general circulation. Equation (4) has been derived with C in eqn(1) representing the plasma concentration of the drug. Corresponding equations can be written in terms of the whole-blood and free-plasma concentrations of the drug. In these cases the meaning of CL, and its numerical value, will differ from that in eqn(1). However, provided that the.same interpretation is placed on the concentration throughout, the applications discussed in the following sections apply equally well to whole-blood or free-plasma concentrations, provided that the elimination rate can be regarded as being proportional to the measured concentration. It can be seen from the derivation of eqn(4) that no assumption has been made concerning the distribution processes. It follows that eqn(4) remains valid for linear or nonlinear distribution processes. However, nonlinear binding of the drug to plasma

126

DAVID CUTLER

proteins may invalidate eqn(4) since the assumption that CLis a constant may not apply. This case can be treated--at least in theory--by regarding CL as a function of time, applying the methods discussed in Section 3.4. Equation (4) provides the basis for methods for calculating the extent of absorption using plasma drug concentration-time data. It shows that the fraction absorbed F is proportional to the dose-adjusted area AUC/D. The use of this equation requires a value for CL. Assuming CLto be invariant between studies its value may be found by conducting a separate study with an intravenous dose of the drug. Since the drug is introduced directly into the general circulation F = 1 for an intravenous dose. Using the subscript it" in eqn(4) to denote quantities appropriate to the intravenous study, 1 = CLivAUCi~,/Div or

CLi~, = Div/AUCiv.

(5) Using the subscript x to refer explicitly to a particular extravascular test dose study, eqn(4) becomes Fx = CLxAUCx/Dx.

(6)

Combining eqn(5) and eqn(6) gives CLx AUCx Dit, Fx = CLi,, Dx AUCit,'

(7)

With the assumption that the clearance is invariant between studies, CLx = CL,. and eqn(7} becomes Fx = AUC~ Div D~ AUCiv"

(8)

Thus, F~ is given by the ratio of the dose-adjusted area for the test dose study (AUC~/D~) to the dose-adjusted area for the intravenous dose study (AUC,,/Di,,). The corresponding equation for the relative extent of absorption can be derived in a similar manner. For the standard extravascular dose study, indicated by the subscript s, an equation of the same form as eqn(6) arises. F~ = CL~AUCs/D~.

(9)

Combining eqn(6) and eqn(9) Fx Fs

-

CL~ AUCx D~ CLs D,, AUC~"

(10)

Again assuming that the clearance is the same for both studies, CLx = CL~, the following equation is obtained which shows that the relative extent of absorption is also the ratio of dose-adjusted areas. Fx _ AUCx Ds F~ D~ AUC~"

(11)

It is generally thought that variations in clearance for a given individual are less than those between individuals (but see Wagner and Ayers, 1977). For this reason, the two studies required for the use of eqn(8) or eqn(ll) are conducted, with a suitable interval between studies, in the same subject. Extension of eqns(8) and (11) to allow data on the urinary excretion rate of the unchanged drug to be used is straightforward. If Ae is the cumulative amount of drug excreted unchanged in the urine up to time t, the urinary excretion rate can be expressed

Assessment of rate and extent of drug absorption

127

as

dAe/dt = CLRC.

(12)

where CLg is the renal clearance of the drug. Taking CLR tO be a constant, integration of eqn(12) gives A~(~) = CLRAUC.

(13)

Ae(~) is the total amount of drug ultimately eliminated in the urine unchanged i.e. the total urinary recovery of unchanged drug. In deriving eqn(13) we have taken A~ {0), the cumulative amount of unchanged drug excreted up to time zero (when the dose is administered) as zero. Combining eqn(13) with eqn(14) gives CL Ae(oo) CLI~ D

F-

(14)

which shows that F is proportional to the total urinary recovery of unchanged drug. The ratio.~ = CLR/CLis the fraction of drug which reaches the general circulation which is excreted unchanged in the urine. Introducing f~ into eqn(14) gives the alternative form

F = A~(%)/(f~D).

(15)

Following the procedure used to derive eqn(7), with the subscript x again denoting the extravascular test dose study, and the subscript iv denoting the intravenous study, eqn( 141 gives

CLx CLR.i,, A~,~(oo) Di~ CLR.x CLi,~ D,, A~.iv(oc)"

Fx =

(16)

The alternative equation, eqn (15}. gives

Fx - f~'' A~'x(°°) D~ fe.,, Dx A,.iv(~}"

(17)

With the assumption that f~ is the same for both studies, fe,x = f~.i~, eqn (17) gives

Fx = A~.,,(~) D~ Dx A~.i~(oo)"

(18)

Thus, F~ is given by the ratio of the dose-adjusted urinary recovery for the extravascular test dose (A~.x(~z)/D,,)to the dose-adjusted urinary recovery for the intravenous dose study (A~.~.(7:)/D,.). An equation which expresses the relative extent of absorption in terms of urinary recovery data can be obtained from eqn(15). Writing the subscript x for the test dose study and the subscript s for the standard dose study, eqn(15) gives

F~ f~

L.~ A.,~I~:t -

f~.x

Dx

D~ A~.Azc)'

(19t

With the assumption that fe.x = f,., eqn[19) becomes

Fx F~

-

A~,,,(~) D~ Dx A~.~,(~)"

(20)

3.1.2. Linear Systems Derication The equations for the extent and relative extent of absorption derived in the previous section can also be derived on the basis of the assumption that the disposition of the drug is by means of linear processes; i,e. that the body behaves as a linear system. The main advantage of this approach is that it demonstrates how a wider range of data may

128

DAVID CUTLER

be employed in estimating the extent of absorption than is apparent from the mass balance derivation. In regarding the body as a linear system, with respect to drug disposition, we are concerned with the response of the system to a particular drug input. By the response is meant any quantifiable consequence of the presence of the drug. Examples are the plasma concentration of the drug or of a metabolite of the drug, the urinary excretion rate of the drug or of a metabolite, or the intensity of the pharmacological response. The input to the system is the delivery of the drug to a particular point, e.g. to a point in the general circulation. The system is said to be linear if the superposition principle applies. If GI is the response to the input R~, and G2 is the response to the input Rz, the system is said to be linear if the response to the input (RI + R2) is (G~ + G2). In the Appendix an equation is derived which relates the response G(t) to an input at the rate R(t) and the response G~(t) to a unit impulse input for a linear system. Equation (A5) of the Appendix is

;o

G(t) dt =

fo

R(t) dt

fo

Go(t) dt.

(21)

The exact meaning of the input rate R(t) depends on the unit impulse input used to produce the response G6(t). R(t) is the rate of delivery of drug to the particular point, say L, at which the unit impulse input is applied. For example, if Lwere a point in the general circulation (an appropriate interpretation if G~(t) is the response to an intravenous bolus dose) then R(t) is the rate of delivery of drug to the general circulation. If the unit impulse input were introduced into the portal vein, then R(t) could be interpreted as the rate of delivery of drug to the portal vein after oral dosing. An oral solution dose may be regarded as an impulse input into the gut fluids. In this case R(t) could be interpreted as the in vivo release rate from an orally administered formulation. Whatever the particular interpretation of the input rate R(t) we can write

ff

R(t) dt = FD.

(22)

where D is the administered extravascular dose and F is the fraction of the administered dose which reaches the point L. Corresponding to the examples discussed above F could refer to the fraction of the administered dose which reaches the general circulation, the portal vein or the gut fluids, depending on the unit impulse input used to produce the response Go(t). Combining eqn(21) and eqn(22) and rearranging gives

fo ~ G(t) dt F -

(23)

D

fo

G~(t) dt

r

Equation (8) can be regarded as a special case of eqn(23) when Ga(t) is the plasma concentration of the drug following a unit dose rapid intravenous injection and G(t) is the plasma concentration of the drug following the input at rate R(t). In this case

fo ~ G~(t) dt = AUCi,./D.. and

fo ~ G(t) dt = AUCx for a particular test dose x: with Fx in place of F, eqn(8) follows from eqn(23).

Assessment of rate and extenl of drug absorption

129

Equation (18) is another special case of eqn(23) which follows when the response is interpreted as the urinary excretion rate of the unchanged drug. With G~(t) as the urinary excretion rate of the unchanged drug following a unit dose rapid intravenous injection and G(t) as the urinary excretion rate of the unchanged drug following the test dose x,

G~(t) dt = Ae, iv(oo)/Div and

fo ~ G(t) dt = Ae, x(OO ) and eqn(18) follows from eqn(23). Another application of eqn(23) follows when the response is interpreted as the plasma concentration of a metabolite, with G~(t) the response after a unit dose rapid intravenous injection of the parent drug, and G(t) the response after the input R(t). With the subscript m denoting the metabolite, so that AUC.,.~ and AUC,..x are the areas under the plasma concentration-time curves for the metabolite after an intravenous dose and an extravascular test dose x of the parent drug,

fo °~ G~(t) dt = AUC,..~v/Div and

fo ~ G(t) dt = AUCm.~. Equation (23) gives Fx = AUC,.,~ Div D~ AUC,,, iv'

(24)

An important limitation of eqn(24), and similar equations involving metabolite data, is that the metabolite which is measured must not be formed in appreciable quantities at the site of administration, or by a first-pass effect. More generally, the response which is measured must be attributable solely to the input represented by the rate R(t) without a contribution from other sources. After oral administration, when a significant first-pass effect operates, AUC.,,x in eqn(24) will include a contribution from unchanged drug which survives the first pass through the liver (which is accounted for by the theory from which eqn(24) is developed) and a contribution which arises due to metabolite formed during the first pass through the liver (not accounted for by the theory leading to eqn(24)}. To use metabolite data in this case would require a means of estimating the contribution to AUC,,,x of metabolite formed during the first pass. If f,. is the fraction of the administered dose which is converted to the measured metabolite during the first pass, this contribution isf,.D/CL(m) where CL(m) is the total clearance of the metabolite, which result can be obtained from eqn(4) after appropriate substitutions. In some cases there may be uncertainty about the specificity of the assay method used to determine the concentrations which serve as the raw data in estimating the extent of absorption, which would appear to cast doubt on the estimates obtained. However. in applying eqn(23) it is permissible to interpret the response as the outcome of an assay procedure without regard to its specificity. The important property of the response is that it should be linearly related to the input. A limitation of the use of a nonspecific assay procedure arises when there is a significant first-pass effect to produce a metabolite which contributes to the assay value. This introduces an error for the reasons discussed in the preceding paragraph. •

130

DAVID CUTLER

Equation (23) can also be applied when a unit impulse input other than an intravenous bolus injection is used. If G6(t) is the plasma concentration of the drug following an oral solution of the drug containing a unit dose (interpreted as a unit impulse input into the gut fluids) and G(t) is the plasma concentration of the drug following an oral formulation of dose D, eqn(23) gives F as the fraction of the administered drug which is released to the gut fluids. Metabolite data can also be used, in this case without the need to consider the possibility of a first-pass effect, since the measured response can be attributed solely to the release of drug from the dosage form. A first-pass effect occurring subsequent to the release of drug from the dosage form is accounted for by the unit impulse response G6(t). Similarly, data from a nonspecific assay can be used in this case without concern for the possibility of error due to a first-pass effect. In principle eqn(23) can also be used to determine the extent of absorption based on pharmacological data. However, it is generally to be expected that a measure of the intensity of a pharmacological effect will not be linearly related to the drug input. A procedure which overcomes this limitation has been developed by Smolen (1976a,b). The connection between the linear systems description of disposition and Smolen's method has been shown by Cutler (1978a). 3.2. ASSESSMENT OF THE EXTENT OF ABSORPTION DURING MULTIPLE DOSING The equations of the previous section are suitable for estimating the extent of absorption following a single dose. In this section the corresponding equations will be derived which are suitable for the estimation of the extent of absorption during the steady-state or plateau period of a multiple dosing regimen. The required equations can be developed on the basis of the mass balance principle. Assuming the drug to be eliminated by first-order processes, the elimination rate is given by eqn(1). With z denoting the time interval between successive doses, integration of eqn(1) from time t = nz to time t -- [n + 1]~ (i.e., over a single dosing interval) gives, with CL constant, Ael([n + 1] r) -- Aet(nz) = CL

f

[n+ lJr

C(t) dt

• t?l T

= CL. AUC,.

(25)

where in + 11~

AUC. =

C(t) dt ,in1:

is the area under the plasma concentration-time curve over the period of one dosing interval. The left-hand side of eqn(25) is the amount of drug eliminated during the dosage interval. If the steady-state has been reached the amount of drug entering the system (i.e. the amount absorbed) during the dosage interval, which we write as (FD).. must equal the amount eliminated. Using eqn(25) we have (FD), = CL.AUC,.

(26)

If the dose administered at the beginning of a dose interval is completely absorbed by the end of that interva.l, we have (FD), = FD where D is the administered dose and F is the fraction absorbed--that F and D are the same for different dose intervals is implicit in the assumption that the steady-state has been attained. However, it is possible that absorption of a dose continues after the end of the dose interval during which it is administered. Suppose that, for a typical dose, the fraction F1 is absorbed during the dose interval following its administration, the fraction F2 is absorbed in the following interval, and so on. Then, F = F~ + F2 + .... Assuming that each dose behaves in the same way the fractions F2 . . . . which do not contribute to the amount absorbed during the interval following administration will be made up by identical contributions from

Assessment of rate and cxtent of drug absorption

131

doses administered in previous intervals. Thus we again have (FD)n = FD with F denoting the fraction of the administered dose which is ultimately absorbed, either during the interval in which the dose is administered, or in subsequent intervals. Substituting for (FD)~ in eqn(26) gives, after rearranging, F = CL.AUCn/D.

(27)

This equation has the same form and the same applications as eqn(4). For example, with AUC.,x and AUC,.s denoting the area under the plasma concentration-time curve over a single dose interval at the steady-state, for test dose Dx and standard dose D~, the relative extent of absorption is given by Fx/F.~ = AUCn.xDs/(AUCn..,Dx)

when CLis constant between studies. Comparison of a test extravascular dose administered according to a multiple dose regimen with an intravenous dose gives an estimate of the absolute extent of absorption. If the intravenous dose is given by a multiple dose regimen, eqn(27) gives Fx = AUCn.xDiv/(AUCn.ivDx) where the subscript it, denotes the intravenous dose. When a single intravenous dose is used as the reference, combination of eqn(27) and eqn(5), with the assumption that C L ( = CL.,) is constant between studies, gives Fx = A UC,,.xDi,,/(A UCivDx)

where A UC., is the area under the plasma concentration-time curve after a single intravenous dose Die. Extension of eqn(27) to allow the use of urinary excretion data is straightforward. Integration of eqn(12) over a single dosage interval, assuming renal clearance CLR to be constant, gives Ae.n = CLRAUC,; Ae.n is the amount of drug excreted during a single dose interval. Equation (27) becomes F = CL.Ae,n/(CLRD)

which has the same form and the same applications as eqn(14). The derivation of eqn(27) has been based on an application of the mass balance principle over a single dose interval. However, the derivation applies equally well over a number of dosage intervals; furthermore, with a suitable choice of intervals over which AUCn is evaluated, the dose intervals need not be the same. The derivation applies over any interval, say r', which is such that the plasma concentration profile is reproduced over successive intervals z'. For example, AUCn in eqn(27) could be interpreted as the area under the plasma concentration-time curve over a single day, with unequal dose intervals during a day, but the same daily regimen for each day. A major practical drawback of this approach is the need to continue dosing until the steady-state is reached, which is time-consuming and involves prolonged exposure of the subject to the drug. Kwan et al. (1975) have presented a method for evaluating the extent of absorption at nonsteady-states during a multiple dosing regimen (i.e. before the steady-state is reached). The extent of absorption of a test dose x relative to that of a standard dose s is given by Fx _ [AUCm + i

-LA--ffC

"] 1 - e-i~.~ e-m;"'J 1 - - e - m , . z ~

when ! doses of the standard formulation are administered, followed by m doses of the test formulation, all at intervals z; AUCI is the area under the plasma concentration-time curve over interval ! and AUCm+t is the area under the plasma concentration-time curve in the interval m + 1. It is assumed that after each dose is administered the plasma concentration reaches a log-linear phase, characterized by the exponential coefficient 2z.

132

DAVID CUTLER

during which the plasma concentration is proportional to exp(- )~zt), before the next dose is administered. (The equations presented by Kwan et al. (1975) are in terms of mean concentrations over a dosage interval, equal to AUC./r for the nth dosage interval.) Thus, F~/Fs can be estimated in a single study from area estimates and an estimate of the exponential coefficient 2z. Further developments of this approach, to allow for non-equal dosing intervals and doses, are discussed by Yeh and Kwan (1976). 3.3. ASSESSMENTOF THE EXTENT OF ABSORPTION WHEN CLEARANCE VARIES BETWEEN STUDIES

The equations for estimating the extent of absorption presented in the preceding sections rely on the assumption that there is no quantitatively significant change in the elimination processes between the test dose study and the reference dose study. Equations (8) and (ll) assume that the total clearance CLis a constant, equations (18) and (20) assume that the fraction of the absorbed drug which is excreted in the urine unchanged is a constant; i.e. that the ratio of renal to total clearance does not change between studies. Conducting both studies in the same subject goes some way towards satisfying this requirement but intra-subject variations may still occur to a significant degree in some cases. Kwan and Till (1973) have suggested an approach to assessing the extent of absorption based on estimating (by measurement and/or assumption) the changes in clearance which do occur. Equation (8) follows from eqn(7) with the assumption CLx = CL.,, which is simply a means of estimating the unknown C L . / C L . , . The Kwan-Till approach aims to improve the estimate of Fx by obtaining a better approximation for CLx/CL~,.. In particular cases knowledge of the mechanism of elimination of the drug might indicate a suitable approach. When an appreciable amount of-drug is eliminated by urinary excretion, and in the absence of evidence to indicate other approximations, the following procedure is likely to be appropriate. It is based on the assumption that changes in total clearance can be attributed largely to changes in renal clearance. The total clearance in both studies can be expressed as the sum of a i'enal component (CLR,~ or CLR..,) and a non-renal component (CLNR.x or CLNR.~..): the non-renal component is the sum of the clearances due to all routes of elimination (not necessarily known) other than renal excretion. Thus. CLx = CLNR.:, + CLR.:,

and CLi,,

= CLNR.it, + C L R . i v

Assuming that nonrenal clearance is the same for both studies. CLNR.~, = CLNR.~,.. combination of the last two equations gives CLx = CLR.x - C L R . i , . + CLi,, = CLR.~, - CLR.~,, + D~,./AUCI,,.

(28)

using eqn(5). Introducing eqn(28) into eqn(6),

Fx =

Di~ ] AUCx CLR.x -- CLn... + AUCI,,J Dx

(29)

Comparison of eqn(29) with eqn(8) shows that the term CLR.x - CLR.It, amounts to a correction for the change in renal clearance. In addition to the plasma concentrations needed to estimate AUCx and AUC,, eqn(29) requires simultaneous measurement of the renal clearance in both the test dose and intravenous dose studies. An equation corresponding to eqn(29) for use with urinary excretion data is easily derived from eqn(29). With A,,.x(oo) -- CLR.~,AUCx

Assessment of rate and extent of drug absorption

133

and A,,.i,,(~ ) = CLR,i,,A U Ci, eqn(29) gives

Fx -

CL.,~[CL.,x_-CL.,,. I CL,.,~

_D~ ] A,.x(~) + A,.i~(oo)J Dx

(30)

Equations (29) and (30) are exact, subject to the assumptions made. Corresponding equations can be derived for the relative extent of absorption, Fx/Fs, but require an additional approximation to be of practical use. In place of eqn(29) we obtain

o, ]AuC - L

F,

' + AUC,J

Ox '

(31)

'Fhis equation cannot be used in this form to estimate F,,/F~ due to the appearance of the unknown F~ on the right-hand side. An approximate equation (cf. the approximation suggested by Kwan and Till (1973)) follows on taking F~ = I on the right-hand side of eqn(31) to give

~

-

CLR.x - CL..~ + ~

o, ],uc. ~

(32)

Comparison of eqn(32) with the exact eqn(31) indicates that the error in this approximation will be small when F~ is close to one, and when the change in renal clearance is small. The error can be expressed by the term

Eo = [ (Fx/Fs)eqn(32) - (Fx/Fs)eqnt31)]/(Fx/Fsleqn(31) = (I - Fs)(CLR,, - CLs.:,)/CL,,.

(33)

on substituting the expressions for Fx/Fs from eqns(31) and (32). This error can be compared with the error arising, under the same conditions, through the use of eqn(11) which ignores the possibility of a change in renal clearance. The corresponding error is given by

E',~ = [ (Fx/Fs)eq n(xI) -- (Fx/Fs)eq n(31)] / ( F ~ / F s ) ~ q . ( 3 x) = (CLR.~ - CLR,x)/CLx.

(34)

on substituting the expressions for F:,/F~ from eqns(ll) and (31). Since 0 < F~ ~< 1, IE. I < I E'.I for all values of Fs; the approximation of eqn(32) leads to a better estimate than the uncorrected area ratio of eqn(ll), whatever the value of F, when the non-renal clearance is constant. However, if a change in renal clearance between studies is accompanied by a compensating change in non-renal clearance, eqn(11) might yield the better estimate. Corresponding equations for use with urinary excretion data are readily obtained: with

A~,,,( ~ ) = CLR,xA UC:, and A~.~(zc) = C~.~AUC~ eqn(31) gives

Fx [CLR.x - CLRs CLR.sD,] Ze,x(oO ) F---~= _ -f~ " + Aem~(~)] CLa.xD,,"

(35)

As with eqn (31) this equation is exact (subject to the assumptions made) but is not of practical use since the unknown F~ appears on the right-hand side. Approximating F~ = 1 on the right-hand side ofeqn (35) (Kwan and Till, 1973) gives

F:,

-

[

CLR,sD,] Ae,x(OO) CLg,,, - CLR,, + Ae,,(~)J ~ . ~ f f ~ '

(36)

134

DAVID CUTLER

The error in this approximation can be expressed by the term E,, = E (Fx/F,)~q,05) - (Fx/F~)~q.t36)]/(Fx/F~)~q,(35) = (1 - F,)(CLR,~ - CLR.,3/CLx.

(37)

using eqn(35) and eqn(36), indicating that the error is small when F~ is close to one and when the change in renal clearance between studies is small. The error arising, under the same conditions, from the use of eqn(20), which assumes no change in the fraction of absorbed drug which is excreted in the urine, can be expressed by E i, = [ (Fx/F~)¢q.,i2o)

-

-

(Fx/F~)~,q,,~3s)]/(F,,/F.O~.q,,~35 ~

(38)

= (CLNR.jCLa.~)(CLR.~ - CLR.x)/CL~.

using eqn(20) and .eqn(35). From eqn(38) we note that E'e approaches zero when nonrenal clearance (CLNR., = CLNR.,,) approaches zero. which is not the case for E,,, given by eqn(37). Conversely, eqn(37) shows that Ee approaches zero when F~ approaches one, whereas F~ does not influence E~.. Therefore, which of E,, or El, is the smaller depends on F, and the ratio CLNR.s/CLR.,. Comparing eqn(37) with eqn(38) indicates that IE,.F < IE;,[ (the error in cqn(36) is less than that in eqn(20)) when 1 - F~ < CLNR.jCLR.~

Hwang and Kwan (1980) express the last relationship in the form f..., < ½11 + A,,..~( zc )/D.O.

(39)

which is of greater practical convenience. Given a value of A,,..dm) following the dose D,, and an estimate off,.,, the inequality (39), if satisfied, indicates that eqn(36) involves the smaller error; otherwise, eqn(20) involves the smaller error. Thb method described by Lalka and Feldman (1974) for the estimation of the absolute extent of absorption without the use of an intravenous reference dose is based on the assumption that non-renal clearance is unaltered between studies, but that changes in renal clearance occur, or can be induced, between studies. From the relationships C L = CLR + CLNR = F D / A U C we obtain CLNR = F D A U C

-

(40)

C L R.

If two studies are conducted under conditions of different renal clearance, with clearances AUG2 will in general differ. If the same test dose is given in both studies, assuming that F is unaltered and that the non-renal clearance is unaltered, eqn(40t gives CLR.I and CL~.2, the resulting areas AUC1 and

FD AUC1 - CLR.1

=

FD

AUG2

-

CLR.2

which rearranges to give F = (CLR.1 - CLR.2) F AUC1AUC2

]5

]

LA~2 7 AUC~J'

(41)

The main difficulty in applying eqn(41) is likely to be the error in F arising due to the errors in renal clearance and AUC estimates. Modest errors in these estimates may lead to very large errors in the differences CLR.1 -- CLR.2 and AUC2 - AUC1 with a correspondingly large error in F.

Assessment of rate and extent of drug absorption

135

3.4. ASSESSMENT OF THE EXTENT OF ABSORPTION WHEN CLEARANCE VARIES DURING A STUDY, DUE TO TIME-DEPENDENT OR NONLINEAR PHENOMENA

The methods described in the previous section apply when the clearance varies between studies but assume that the clearance is constant for the duration of a single study. When variation in clearance occurs during a study, eqn(1) can be written

dAet/dt = CL(t)C(t).

(42)

where CL(t) is written to emphasize the time dependence. This time dependence can arise in various ways. For example, time dependent renal clearance can arise due to variation in urine flow rate or urine pH ; time dependent hepatic clearance can arise due to hepatic enzyme induction. Nonlinear elimination processes can also be regarded as a cause of time dependent clearance. For example, if a metabolic process follows Michaelis-Menten kinetics, the elimination rate for this process takes the form VmC(t)/(Km + C(t)) where V,. and Km are constants. The elimination rate may also be expressed as CL,.(t)C(t) where CL~,(t) is the clearance associated with the formation of the metabolite. Equating these expressions gives

CL,.(t) = Vm/(Km + C(t) ).

(43)

The time dependence of the clearance arises due to the time dependence of the concentration C(t). In some cases it is convenient to recognise explicitly the concentration dependence but in other cases a nonlinear process can be adequately described through its time dependence. Integrating eqn (42) from t = 0 to oo, with Ae~(0) = 0, gives

Aet(~) =

CL(t)C(t) dt

for the total amount of drug eliminated. By the mass balance principle this quantity equals the amount of drug absorbed, FD. Thus

F = ~

CL(t)C(t) dt.

(44)

When CL(t) is "known", by measurement and/or assumption, the integral in eqn(44) can be evaluated (numerically if necessary) to give an estimate of F. Expressing the total clearance as the sum of a renal and non-renal component,

CL(t) = CLm~(t) + CLR(t) eqn(44) becomes

lifo

F = ~

CLNR(t)C(t) dt +

fo

]

CLR(t)C(t) dt .

(45)

If renal clearance is measured as a function of time the second integral on the right-hand side of eqn(45) could be evaluated numerically. A simpler procedure is to base the estimate of this term on the urinary recovery of the unchanged drug. When renal clearance is a function of time the urinary excretion rate of unchanged drug isgiven by

dAe/dt = CLR(t)C(t). Integrating from t = 0 to oo, with Ae(0) = 0, gives

ff

CLR(t)C(t)dt = Ae(oo).

136

DAVID CUTLER

Equation (45) becomes

'[fo

]

F = D

CLNR(t)C(t)dt + Ae(oc) .

(46)

Since non-renal clearance is generally not directly measurable some assumptions are required to enable evaluation of the integral in eqn (46). If CLN~(t) is assumed to be constant,

fo °° CLNa(t)C(t) dt = CLsRAUC and eqn (46) becomes F = ~ 1 [CLNRAUC + Ae(~)].

(47)

In this equation CLNR is the only unknown. It may be estimated, assuming non-renal clearance to be unaltered between studies, with the aid of an intravenous dose. With the subscript ic in eqn(47) to indicate the intravenous dose, and noting that F,, = 1, eqn(47) gives, after rearranging CLNR,Iv

=

(Dit, - A~..,(cc))/AUC.,.

(48)

With the subscript x to denote a particular extravascular test dose, and the assumption that CLNR.x = CLNR,iv, combination of eqns (47) and (48) gives

Fx =

[Div

--

Ae. iv(oO)] A ~ . .

+ Ae.x(oc) .

(49)

This equation, given by Oie and Jung (1979), requires for its application measurements of plasma concentrations (to estimate AUCx and AUCIt,) and total urinary recoveries (A~.x(~) and A~..,(oo)). The major assumption on which eqn(49) is based is that the non-renal clearance is constant. The time dependence of renal clearance could account for such effects as variable urine flow rate and variable urinary pH ; it could also account for nonlinear processes such as capacity limited tubular secretion or nonlinear protein binding (the latter insofar as it influences renal excretion). In principle, eqn(46) could be used to calculate F when CLNR varies with time. It might be possible to express the time dependence of CLNR(t) with the aid of a theoretical model describing the kinetics of a proposed change in clearance, e.g. due to hepatic enzyme induction or a change in hepatic blood flow. If the time dependence of non-renal clearance can be attributed to a nonlinear metabolic process, the time dependence can be expressed in terms of the concentration dependence of the metabolic process. If non-renal elimination is by a Michaelis-Menten process, in parallel with a first order process, CLNR may be written in the form

CLNR(t) = V,,/(Km + C(t)) + CL1. The constants Vm and K,, characterize the Michaelis-Menten process and CL~ is the clearance due to the linear process. Equation (46) becomes

F = ~

I'm

K m + C(t) dt + CL1AUC + Ae(oo) .

(50)

When Vm, K s and CL~ are known the use of this equation is straightforward. The integral could be estimated by evaluating the term C(t)/[Km + C(t)] at each data point before applying one of the methods of area estimation discussed in the following section. The major difficulty is in estimating the required parameters. An approach used by Martis a n d L e v y (1973) and by Jusko et al. (1976) estimates these parameters from an intravenous study. This approach assumes a model for the dispo-

Assessment of rate and extent of drug absorption

137

sition of the drug to obtain these estimates. If a one-compartment model, with volume of distribution E is assumed, the equation

V dCi____~= _ _ V,.Ci~ dt K m + C~

(CLt + CLR)Civ

with C~v = D/V at t = 0, describes the disposition of the drug following an intravenous dose D, with parallel Michaelis-Menten and first-order elimination. A nonlinear curve fitting procedure was used by Martis and Levy (1973) and by Jusko et al. (1976) to estimate the parameters. This procedure differs from the methods discussed previously in requiring a specific model for drug disposition. An alternative model-independent method using a constant rate intravenous infusion could be used to provide these estimates. When a steady:state concentration (C~) is reached, mass balance considerations give Ro -

Vm C ss

K m + C~

+ (CL~ + CLR)C,,.

Different infusion rates Ro will give rise to different steady-state concentrations C~s. Nonlinear curve fitting of the resulting (Ro,Cs~) data, with independent assessment of CLR, gives the required parameter estimates.

4. M E T H O D S OF AREA ESTIMATION Assessment of the extent of absorption usually involves the estimation of an area, such as the area under a plasma concentration-time curve. The area is given by an integral AUC =

;?

C(t) dr.

(51)

with data C(t 3 available only at particular times ti (i -- 1. . . . . m). Evaluation of this integral involves tWO steps. The first step is to estimate the area AUC(0, tin) =

f?

C(t) dt.

(52)

where tm is the last data point. The second step is to estimate AUC(t=, ~ ) =

C(t) dt

(53)

which is the area lying outside the time range spanned by the data. Estimation of AUC(tm, oc) inevitably involves some form of extrapolation; it is always necessary, if only approximately, to show. that the contribution of the extrapolated area to the total area is negligible. In some cases, as in eqn(50), the function to be integrated is not directly measured but can be constructed from the data; the integration in eqn(50) could be performed with the methods discussed here, with the interpretation C(ti) for C(t~)/[K= + C(ti)], although special attention to the extrapolation term might be required. Estimation of areas as required in the Wagner-Nelson and Loo-Riegelman methods, discussed in Section 5.2, can be performed by the methods appropriate for eqn(52).

4.1. ESTIMATIONOF AUC(O, tm) The simplest and most widely used method for estimating the area over the time range spanned by the data is the trapezoidal method. A linear interpolation is used between successive data points t~ and t~÷ 1 and the area under the curve between these data points,

138

DAV]O CUTLER

AUC(ti, t~+ 1), is given by the area of the trapezium so formed: AUC(ti, ti+ 1) = ½[C(ti+ 1) -~- C(ti)](ti+ 1 - ti).

(54)

The area AUC(0, t,,) is obtained by adding the areas of the individual segments. m--1

tin)

AUG(0,

---- 2

AUC(ti,

ti+ 1)

(55)

i=1

where tl = 0. Equations (54) and (55) can be combined to give

AUC(0, t,.) = ~

T,C(t,)

(56)

i=1

where Zl -- t2 -

tl,

T/ - - t i + 1 --

ti_l,

for

i=2 ..... m-1

T m = tm -- tin-1"

The ability of the trapezium method to provide a good estimate of AUC(0,t,,) clearly depends on the error introduced by the linear approximation to the data between successive data points. If the function to be approximated shows marked curvature, and if the intervals between data points are long, significant errors can be expected. For the portion of a curve increasing to a maximum the trapezoidal method will under-estimate the area and for the portion of a curve which decreases in an approximately exponential manner (the entire curve for an intravenous bolus dose) the method will over-estimate the area. Studies of the errors associated with trapezoidal integration have been conducted by Chiou (1978) and by Yeh and Kwan (1978). Chiou (1978) has shown that for a curve decreasing mono-exponentiaily the error is determined by the ratio of the interval between data points to the half-life of the exponential function. When this ratio is 0.5 the error is about 10,, when the ratio is 2 the error is about 16"o, and when the ratio is 4 the error is about 57?0. Chiou (1978) also showed that large errors can arise in area estimates during the absorption phase following extravascular administration; in two examples errors around -35'30 were observed. Chiou (1978) and Yeh and Kwan (1978) investigated the log-trapezoidal method, which employs an exPonential rather than a linear approximation over a data interval. The resulting estimate for the area under the curve between successive data points ti and t.:+ 1 is AUC(ti,ti+ 1) = [C(ti+ 1) - C(ti) ] (ti+ 1 - ti)/ln [C(tl)/C(ti+ 1)].

This method was shown to be better than alternative methods examined for a function which declines mono-exponentially. For an ascending curve in the vicinity of maximum the log-trapezoidal method gives very large errors and its use should be avoided in these situations. Yeh and Kwan (1978) have also investigated the use of cubic polynomials (Lagrange and cubic spline) for interpolating between data points. The Lagrange method fits a cubic polynomial over the interval (ti, ti+ 1) using also the data at ti-1 and ti+ 2; a quadratic is used for the first and last intervals. The cubic spline method fits a cubic polynomial over each data interval with the constraint that at each data point the fitting function and its first two derivatives are continuous. This property ensures a smooth transition between successive data intervals in contrast to the Lagrange and trapezoidal methods. For both the Lagrange and cubic spline methods the estimate of the area for each interval is obtained by analytical integration of the approximating polynomial and the total area is given by the sum of the estimates for the individual intervals, as in eqn(55).

Assessment of rate and extent of drug absorption

139

The aim of the cubic polynomial methods is to obtain a better local approximation to the curvature of the function to be integrated. Using simulated error-free data Yeh and Kwan (1978) confirmed the expectation that the cubic polynomials should provide significantly better estimates for an ascending curve and in the vicinity of a maximum; for a mono-exponentially decreasing curve the log-trapezoidal method gave the best estimates. For estimating AUC(0,tm) following an extravascular dose the best method examined used a cubic spline for the period before a mono-exponential decline was apparent, and the log-trapezoidal method thereafter. However, the differences observed using error-free data were less marked with simulated data with added noise. As Yeh and Kwan (1978) point out the use of cubic polynomials for interpolation may result in spurious oscillations, leading to large integration errors, particularly when the data error is large. To detect oscillations requires evaluation of the interpolating polynomial at several points within the interval. One of the objects in using a cubic spline is to mimic, and automate, the human facility for drawing a smooth curve through a set of data points. If simplicity of calculation is given a high priority, graphical interpolation followed by application of the trapezoidal rule with a small interval is likely to give satisfactory estimates for an ascending curve, or in the neighborhood of a maximum. 4.2. ESTIMATIONOF AUC{tm, ~ ) The classical method of estimating AUC(t,,, oc) is to assume that C(t) decreases monoexponentially after the last data point' i.e. follows an equation of the form C(t) = C(tm) e -~z"-''',

for

t > tm

(57)

where C(tm) is the value of C at time tm and 2z is a constant. Integrating this expression gives AUC(tm, ~ ) =

C(tm) e -~'~"-'"~ dt

(58)

= C(t,)/2z.

In this calculation it is necessary that the mono-exponential decline expressed by eqn(57) be followed for a sufficient time before the last data point to enable estimation of 2z (e.g., by determining the slope of the terminal portion of a plot of in C vs t). This will occur when absorption is sufficiently rapid; the mono-exponential decline of eqn(57) is expected at late times when elimination is by first-order processes. This might also be a reasonable approximation if elimination follows Michaelis-Menten kinetics since at low concentrations (C ~ Kin) the Michaelis-Menten process is approximately first-order. If absorption is sufficiently slow to be the rate limiting step in the overall absorptiondisposition process, eqn(57) may be followed, if the absorption process is first-order. In this case 2z is the absorption rate constant. In intermediate cases, when neither absorption nor elimination is rate limiting, the single exponential decline of eqn(57) is not expected. If it is found that a polyexponential function of the form C(t) = ~ Bi e -b'' i=1

adequately represents the data over the time interval (0, tin), with the parameters Bi and b~ obtained by nonlinear curve fitting, the assumption that the same expression applies for times t > tm gives AUC(tm, ~ 1 =

Bi e -O't dt ,~ i = 1

= ~. Bie-b't'/bi. i=l

JP.T

14 2

I~

140

DAVID CUTLER

Alternatively, using the poly-exponential expression to represent the data over the entire time-interval (0, oo) gives AUC(0, oo) = ~ Bi/bi. i-I

Wagner and Ayers (1977) have examined a number of alternative methods for the single exponential case; they recommend two as being preferable to the use of eqn(58). If C(t5 declines exponentially after some time t~ < t,, we have

C(t) = C(G) e -xz~'-'x~,

for

t > tx.

for

t > tx

Integrating from time t~ to time t,

AUC(tx, t) = pl[1 - e-;'z(t-tx)],

(59)

where Pl = AUC(tx, oo). AUC(G,t) can be obtained from the C(t) vs t data at each data point by the methods of Section 4.1; e.g. by the application of the trapezoidal method over the interval (G,tS. Fitting the resulting values of AUC(G,t) to the two parameter function of eqn(59) by a nonlinear curve fitting routine yields values of the parameters p~ and 2z. The parameter p~ gives the contribution to AUC of the extrapolated curve, in this case after the data point G, rather than after the last data point tin. AS an alternative to the use of a general nonlinear curve fitting procedure Wagner and Ayers (19775 also considered a modification of the accelerated convergence method of Amidon et al. (19755, which is specific for an equation of the form of eqn(59). With simulated data Wagner and Ayers (19775 found that the accelerated convergence method gave estimates of AUC(t~,,oo) which were were closer to the true values than those provided by the nonlinear curve fitting procedure. However, the accelerated convergence method can only be applied with equally spaced data points during the terminal single exponential phase. It should be appreciated that any method for estimating the area lying outside the time range spanned by the data is inherently unreliable. The assumption that the trends observed up to the last data point are followed by the data after the last data point may lead to large and unpredictable errors. Observations should be continued to plasma concentrations as low as possible so that the contribution to the overall area of the extrapolated area is as small as possible, thus reducing the proportional error due to the extrapolation term.

4.3. ESTIMATION OF Ae(fm, 005"

Methods based on total urinary recovery of unchanged drug do not appear to require numerical treatment to estimate the quantity Ae (0, oo), However, if elimination is very slow a significant amount of drug may be eliminated at urine concentrations below the detectable level. Alternatively, it may be inconvenient to continue collection of urine for long periods to estimate what is actually a minor contribution to the total amount of drug eliminated. From eqn(12), if CL~ is constant, Ae(tm, ocJ) = C L R A U C ( t m , ~ ) .

If the plasma concentration is assumed to decrease exponentially, as given by eqn(57), eqn(58) gives Ae (tm, c~) = CLRC(tm)/).z *In this section the term A~(ta,t2) is used to indicate the a m o u n t of drug excreted in the urine in the time interval (tl,t2). Elsewhere, Ae(tl) = A e (0,ti).

Assessment of rate and extent of drug absorption

141

CLRC(t.,) is the urinary excretion rate of drug at time t,.. Similarly, eqn{59) gives Ae(tx, t) = Pl(1 - e -~Z"-t'~)

for t > tx.

(60)

where

PI = A~(tx,~) Fitting the data A,,(tx,t) at various times t to the two parameter eqn(60) gives the required estimate of P~.

5. ASSESSMENT OF THE RATE OF ABSORPTION A measure of the extent of absorption gives useful but incomplete information about the absorption process. A fuller understanding requires information about the timecourse of the absorption process. In some cases, as in comparing the in vivo performance of different dosage forms of the same drug, simple measures such as the magnitude and time of occurrence of the maximum in the plasma concentration-time curve may be adequate. More information is provided by the statistical moments of the absorption process. When the mechanism of the absorption process is of interest the ideal is a representation of the time dependence of the absorption rate (or amount absorbed at various times) either in numerical form or as a mathematical function. Studies at different doses would reveal the dose dependence of the absorption kinetics. The exact meaning of the term rate of absorption depends on the nature of the data and the mathematical procedures of analysis. In the following, unless otherwise stated, the rate of absorption is the rate of delivery of the drug to the general circulation.

5.1. SIMPLEMEASURESOF ABSORPTION RATE; CraaxAND t~a~ It is generally observed, following an extravascular dose of a drug, that the plasma concentration of the drug passes through a maximum some time after administration. The magnitude of the concentration at its maximum, C.,,., and the time at which the maximum occurs, t,,,s, give some indication of the absorption rate. Roughly speaking, the earlier the maximum occurs the greater the rate. For a given extent of absorption, Cm,xincreases with increasing rate. These features may be demonstrated for a one compartment model with first-order absorption. We have tmax --'-- ln(ka/k)/(ka - k)

(61)

and

FD f k ~k/tk~-k) Cmax = ~ \ ~ /

(62)

where FD is the absorbed dose, ka is the first-order absorption rate constant, k is the first-order elimination rate constant and Vis the volume of distribution. There are several difficulties in basing firm conclusions about absorption rates from observations of Cmaxand tmax.Firstly, both parameters depend in part on the disposition of the drug; i.e, they do not refer solely to the absorption process. Secondly, while interpretation of Cmaxand tmaxis straightforward in simple cases (as in eqns (61) and (62)) this interpretation does not readily generalize. For example, it cannot be said in general that the shorter tmax the greater the absorption rate for an arbitrary absorption process; in comparing two rates (RI and R2) we may have RI > R2 at certain times and RI < R2 at other times. It is also clear that tmaxdepends on the absorption rate up to the time tma~ but is uninfluenced by the absorption rate at later times. Finally, expressions for tma~and C,,a~ in terms of some parameter describing the absorption process can only be obtained in very simple cases. It is not generally clear, for example, how tma~ and Cma~ might change in response to a change in the disposition of the drug.

[42

DAVID CUll I R

C...... and t ...... are usually estimated simply by inspection of the data. To automate this procedure, and possibly to improve the estimates obtained, a quadratic function can be fitted to the data in the vicinity of the maximum, with estimation of C ...... and t ...... by determining the maximum and time of maximum of the quadratic (Saunders and Natunen, 1973). 5.2. ASSESSMENT OF ABSORPTION RATE BY MASS BALAN(I~ METHODS

Mass balance methods for measuring the extent of absorption were introduced in Section 3.1.1 by noting that after all the administered drug has been eliminated the total amount of drug eliminated is equal to the total amount absorbed. The associated equations for estimating the absorption rate are based on an instantaneous mass balance; drug which has been absorbed up to time t must either still be present in the body, or have been eliminated. This principle is expressed by the equation A~,b~(t) = A (t) + A,,M)

(63)

where A,,h~(t)is the cumulative amount of drug absorbed up to time t. A (tl is the amount of drug (unchanged) in the body at time t and A~t(t) is the cumulative amount of drug eliminated up to-time t (by all routes, excretion and metabolism). The rate of absorption, R~,~M), is related to the cumulative amount absorbed by the equations

Aab,dt) =

R~b~(T) d T

(64)

or

R~b~(t) = dAubs~dr

(65)

which allow interconversion of the two quantities. Assuming first-order elimination the elimination rate is given by eqn(l),

dA,,/dt = CL.C(t). Assuming for the present that CLis a constant [other assumptions are considered laterl, integration of eqn(l) from time 0 to t, with A,.j = 0 at time 0, gives A,,M) = CL f~ C(T) dT.

(66)

Combining eqn(63) and eqn(66),

A,b~(t) = A(t) + CL f l C(T)dT.

(67)

Application of eqn(67) requires a means of estimating A(t), in practice based on a measure of the plasma concentration of the drug. The Wagner-Nelson and Loo-Riegelman methods (discussed in the following sections) employ the one and two compartment models respectively for this purpose. In both cases the resulting estimate for A(t) refers to the amount of drug which has reached the general circulation. The elimination term in eqn(67) refers to the cumulative amount of drug eliminated from the general circulation, Consequently, the mass balance principle requires that A,b,,(t) refer to the cumulative amount of drug which has reached the general circulation. The corresponding rate of absorption, given by eqn(65), is the rate of entry of drug into the general circulation. 5.2.1. The Wagnep~Nelson Method In this method (Wagner and Nelson, 1963) disposition is assumed to follow the onecompartment model. The amount of drug in the body is given by

A (t) = VC(t).

(68)

Assessment of rate and extcnl of drug absorption

143

where Vis the volume of distribution. Introducing this expression for A (t)into eqn(67) gives A.b~lt) =

VC(t) + C L

fo

C(T) dT.

169)

A separate intravenous study, in the same subject, may be used to determine V and CL. Following the test dose, with concentration C(t~) at the sample times tl, eqn (69) becomes A.b~(ti) = VC(ti) + C L " AUC(0, ti)

(70)

where AUC(0, ti) =

C(T) d T

may be estimated by the methods discussed in Section 4.1. Thus, values are obtained for the cumulative amount absorbed at the sample times ti. Application of eqn(70) yields absolute values for the cumulative amount absorbed. Relative values may be obtained, with further assumptions, without the need for a separate intravenous study. On dividing both sides of eqn(70) by l! and noting that the elimination rate constant for the one compartment model is k = CL/V,

A~b~(ti)/V = C(t i) + k.AUC(O,q).

(71)

If absorption is sufficiently rapid k may be estimated from the terminal phase of the plasma concentration-time curve following the test dose; if the assumptions made are valid, - k is the slope of the terminal phase of a plot of lnC vs t. Equation (71) yields values of A~b,(tl)/V; since Visa constant, a plot of these values against time has the same shape as a plot of the absolute values but is in arbitrary units when Vis unknown. While the prospect of determining the absorption profile without the need for an intravenous dose study is appealing, a number of pitfalls exist with this approach. Onecompartment disposition is followed in a minority of cases. With extravascular data alone it is not possible to determine whether disposition follows the one-compartment model. In particular, the observation that the plasma concentration-time profile after an extravascular dose has the general appearance expected for a drug following single compartment kinetics (a bi-exponential profile) can also arise when two-compartment kinetics are apparent with intravenous dosing (Ronfield and Benet, 1977); therefore, this observation is not a reliable indicator of one-compartment disposition. Wagner (1974) has shown that the Wagner-Nelson method in some cases (depending on the magnitudes of the two-compartment model parameters) gives a good approximation to the absorption profile with two compartment disposition. However, without intravenous data it is not clear how these cases can be distinguished from those in which the Wagner Nelson method yields a poor approximation. Another difficulty in applying eqn(71) arises from the assumption of rapid absorption. If absorption is sufficiently slow k cannot be determined from the terminal phase of the plasma concentration-time profile. If absorption is first-order, with rate constant k, < k the terminal slope of a In C vs t plot is - k , rather than - k . The erroneous use of the value obtained in eqn(7) would give values of A~b~(ti)/V in error by the amount (k,, - k). AUC(O, ti). The main application of eqn(71) would appear to be in studying the absorption of a drug, with well-established one-compartment kinetics, from different formulations or in different subjects. Equation (701 can be extended to allow the use of urinary excretion data. On multiplying both sides of eqn(70) by CLR. the renal clearance of the drug. and substituting

dA~/dt = CLnC(t)

144

DAVID CUTLER

and Ae(O, t) = CLR

C(T) dT

we have (72)

CLRAab~(ti)/V = (dA,,/dt L + kAe(0, h)-

where (dAe/dt),, is the urinary excretion rate at the time t~ (which has to be estimated from the A,, (0,t)data). With C L R / V constant, eqn(72) provides A,b~(t) in arbitrary units. The major difficulty in applying eqn(72) is the practical one of obtaining data at sufficiently short time intervals to adequately describe the absorption profile. The approach is therefore limited to drugs which are slowly absorbed and slowly eliminated (slowly eliminated since absorption must be rapid compared with elimination). 5.2.2. The Loo--Rie.qelman Method The Loo-Riegelman method (Loo and Riegelman, 1968) is an application of eqn(67) using the two compartment model to estimate the amount of drug in the body. A (t). If At(t) and A2(t) represent the amounts of drug in the central and peripheral compartments respectively, we have (73)

dA2/dt = kl,2Al - k2.1A2.

where k1.2 and k2.x are the intercompartmental transfer constants for the two-compartment model. Equation (73) assumes that no elimination occurs from compartment two. Taking A2 = 0 at t = 0, integration of eqn(73) gives A2(t ) = kl. 2

e :"ff

ek'~ ITAI(T) dT.

The total amount of drug in the body is given by A(t) = Al(t) + A2(t)

= At(t) +

kl.2

e -k'- "

fo

ek:.'rAl(T) dT.

Assuming that the measured plasma concentration C(t) represents the concentration at some point within the central compartment, we have At(t) = VtC(t), where V1 is the volume of distribution of the central compartment. Thus, A(t) = VIC(t) + V l k l . 2 e -k'-''

ek'~'rC(T) dT.

Combining this equation with eqn(67) gives Aab,(t) = V1C(t) + Vlkl.2 e -k'-''

e k 2 ' r C ( T ) d T + CL

C ( T ) dT.

(74)

Equation (74) expresses the cumulative amount absorbed in terms of the measured concentration C(t), the parameters of the two compartment disposition model, (V1, kl.2 and k2.1) and the clearance CL. Assuming the parameters to be constant between studies. the compartment parameters can be determined with an intravenous dose study in the same subject (see, e.g. Riegelman et al., 1968): CL can be determined by eqn(5). It remains to evaluate the integrals in eqn(74), in the original presentation Loo and Riegelman (1968) use a linear approximation for C(t) over each data interval. The resulting equation can be written A,b,,(t3 = V1C(h) + V1C2(ti) + CL

fo

C(T) d T

(75)

Assessment of rate and ¢xlent of drug absorption

145

where C2(ti) = C 2 ( t i - t ) e -k2'A'' + k.l"2 C(ti-1)(1 - e -k2'a'') k2,1

k1,2 [ C ( t i ) - C ( t i - O ] (e_k2,a, ~ + k2.1Ati - 1) + (k2.1)2

At i

(76)

with Ati = tl - ti_ 1. C(t~) is the measured concentration at time ti. Each C2(tl) uses the C2(ti_l) for the previous interval, with C2(ti_l)--0 for the first interval. Th6 integral in eqn(75) can be evaluated by the trapezoidal method. In the original presentation, Loo and Riegelman (1968) used a two-term Taylor series approximation for the final term in eqn(76); this has been shown by Boxenbaum and Kaplan (1975) to be a potential source of error, and since there is little advantage in using this approximation it should be avoided. The Loo-Riegelman equations are often presented as an expression for Aab~/Vt instead of A~b~,as in eqn(75). Since the parameter Vx is obtained from the data needed to calculate kl, 2 and k2.1 there appears to be little advantage in calculating the relative, rather than the absolute, absorption profile. In many cases it can be expected that approximations to C(t) other than the linear approximation leading to eqn(76) will result in better estimates. Wagner (1975) has proposed a cubic spline interpolation procedure prior to the application of the LooRiegelman equations, which are applied to the interpolated values with a small interval between successive times. In the derivation of eqn(75) it has been assumed that elimination occurs solely from the central compartment--the elimination rate is taken to be proportional to the measured concentration C, which is assumed to be the concentration in the central compartment. Wagner (1975) has shown that the same equations can be applied without change when elimination occurs from the peripheral compartment. value

5.2.3. Assessment of Absorption Rate with Variable Clearance The Wagner-Nelson and Loo-Riegelman methods are readily adapted to deal with variable clearance. Writing CL(t) to emphasise the time dependence of the clearance, in place of eqn(67), A,bs(t) ---- A(t) +

CL(T)C(T)dT.

(77)

When CL(t) is known this equation can be used in the same way as eqn(67), to develop equations analogous to eqns{69) and (75). For example, if the time dependence of CL is due solely to the time dependence of renal clearance, CL~(t), with non-renal clearance CLNR time independent, eqn(77) gives Aabs(t) = A(t) + CLNR

C(T) d T +

C L R ( T ) C ( T ) dT.

(78)

The last integral in this equation can be identified as the Cumulative amount of drug excreted unchanged in the urine, Ae(t). Thus Aab~(t) = A(t) + CLNR

f2

C(T) d r + Ae(t).

(79)

CLNR may be estimated following an intravenous dose, using eqn(48), assuming that non-renal clearance does not vary between studies. While theoretically straightforward two practical problems can be expected in the application of eqn(79). Conventional methods for estimating the parameters needed to calculate A(t) require constant clearance; these methods can be applied only if CLR is constant during the intravenous study to determine these parameters. The other difficulty

146

DAVID CUTI.I R

(which applies whenever urinary excretion data is used in absorption rate studies) is obtaining data on A,,(t) at sufficiently short time intervals: When the time dependence of the total clearance CL is due to a Michaelis Menten elimination process, the time dependence of CL is best expressed in terms of the concentration dependence of the Michaelis-Menten process. For example, if elimination occurs by means of a Michaelis-Menten process in parallel with linear processes, CL has the form CL(t) = V,,/[K,, + C(t)] + CL~ where V,,, K s and CL~ are constants. Equation (77) gives Aabs(t)

=

A(t) + CL~AUC(0, t) + V,, f j K,, C(T) + C(T) dT.

An equation of this form has been applied by Martis and Levy (1973) and by Jusko et al. (1976) for the one Compartment model. In this case A (t) = VClt), where Vis the volume of distribution of the compartment. The parameters V,,, K,,. and CLx were obtained by a nonlinear least-squares procedure applied to the plasma concentration data following an intravenous dose. 5.2.4. Assessment of Absorption Rate with Nonlinear Disposition Assessment of the extent of absorption based on the mass balance approach is not dependent on assumptions concerning the distribution processes. In contrast, assessment of absorption rates involves explicit assumptions to enable evaluation of the term A(t) in eqns(67) or (77). In theory, nonlinear disposition poses no particular problems in applying these equations: a model is required which accounts for the nonlinear distribution processes.. For example, if the one compartment model applies, with saturable tissue binding to a single site, an equation of the following form is expected A it) = VfC + B ......C/(Kb + C). When the parameters Vf, B ...... and K~ are known the use of this equation for A (t) in eqn(67) or eqn(77) is straightforward. The major difficulty in this approach, as with nonlinear elimination, is in estimating the required parameters. The nonlinear leastsquares procedures used by Martis and Levy (1973) and Jusko et al. (1976) could be adapted to obtain these parameter estimates using plasma concentration data following an intravenous dose. 5.3. ASSESSMENTOF ABSORPTION RATE BY DECONVOLUTIONTECHNIQUES Another approach to the assessment of absorption rates is provided by the deconvolution methods. These methods arise from considering the body as a linear system with respect to drug disposition. Application of this approach to the assessment of the extent of absorption has been discussed in Section 3.1.2. The input response relationship G(t) =

R(T)G~(t - T l d T

(80)

is derived in the Appendix (eqn(A1)). R(t)is the input rate of drug into the body and G(t) is the resulting response. G,~(t) is the response following a unit dose impulse input (such as a rapid intravenous injection). The major assumptions underlying eqn(80) are that the response is linearly related to the input and that the system is time-invariant: i.e. that the unit impulse response G,~lt) is independent of the time of administration of the unit impulse input. However. no assumptions are made concerning the detailed structure of the system, in contrast to the Wagner-Nelson and Loo-Riegelman methods, discussed in Section 5.2. which assume a compartmentalised system.

Assessment of rate and extent of drug absorption

147

Various interpretations can be given to the term response, as discussed in Section 3.1.2. The most common interpretation identifies the response as the plasma concentration of the drug. The input to the system is also open to different interpretations (Section 3.1.2). The interpretation of the previous section, with R(t) denoting the rate of entry of drug into the general circulation, arises when the unit impulse input used to define G~(t) is a rapid intravenous dose. If G~(t) is the response to a unit dose oral solution of the drug, R(t) can be interpreted as the in t~it~orelease rate from an oral formulation. As noted in Section 3.1.2, the linear systems approach may be applied with metabolite data, or data arising from a non-specific assay, provided that it can be assumed that measured metabolites are not produced in appreciable quantities before the drug reaches the general circulation (e.g. by chemical or enzymatic processes in the gut, or on the first pass through the liver, after oral administration). The integral in eqn(80) is called a convolution integral. It can be evaluated (analytically or numerically) when R(t) and Ga(t) are known. An evaluation of this kind is required to predict the response G(t) (say. the plasma concentration of the drug) for a particular input R(t) when Ga(t) is known. When eqn(80) provides the basis for an estimation of the absorption rate, G(t) and Gn(t) are known (numerically, in the form of experimental data) and R(t) is the unknown to be estimated. This procedure, the "inverse" of evaluating the convolution integral, is called deconvolution. Many methods have been proposed for numerical deconvolution. The simplest are the finite difference methods but these methods may perform badly on data with appreciable error (Gamel et al., 1973). More stable, but computationally more complex, are the least-squares methods (Cutler, 1978b, c), These methods are discussed in the following sections. Other methods (such as Fourier analysis) are discussed by Gamel et al. (1973) but appear to offer no advantages in terms of stability or simplicity over the finite difference or least-squares methods.

5.3.1. Finite D!fferenee Methods for Numerical Deconvolution A number of related methods have been proposed which fall into this class. An approximation is found for the absorption rate R(t) over the nth interval (t._ ~,t.) in terms of the approximations already derived for the previous intervals. To obtain the required approximation for R(t) eqn(80) is written as

~

4t M

G(t,) =

R(T)G~(t, - T) dT

0 lOn- 1 R(T)G~(t.

- T) dT +

fltn n-i

R(T)G~(t, - T) dT.

Rearranging,

j•/tn

R(T)G~(t.- T)dT = G(t.)-

~O/n I R ( T ) G ~ ( t . -

T)dT.

(81)

M-I

Different methods arise from eqn(81) depending on the approximation used for R(t) over each interval. One method, due to Chiou (1980) (who presented the method as arising from "empirical" arguments: it can be regarded as a deconvolution method) approximates the input rate by a train of impulse functions. With this approximation. R(t) = ~ ai6(t - il)

i=1

(82)

where m is the number of intervals, the al are constants and ti = (tl + t~_~)/2 is the mid-point of the ith interval (with to = 0). 6(t) is the unit impulse function. The constants

148

DAVID CUTLER

a~ are given by ai = J~rti R ( T ) dT. ti 1

(83)

Thus a~ is the amount of drug absorbed during the interval. Equation (82) is therefore equivalent to the approximation used by Chiou (1980). of considering the entire amount absorbed during an interval to be delivered as a pulse at the mid-point of the interval. Using eqn(82) to approximate the integrals of eqn(81) gives R(T)G~(t, -

T)dT

--- ~

aiG~(t, - {i)

i=l

and f,/"

R(T)Ga(t, -

T)dT

= a, Ga(t, - t,).

fn 1

Introducing the last two equations in eqn(81) and rearranging gives G(t,) tl n

n 1 ~ aiG~(t, - il) i=l G~(t, - i,)

(84)

For n = 1 we have al --- G(tl)/Ga(tl - tl)

where tl = tz/2.

Equation(84) expresses a, for each interval in terms of the previously calculated a._ 1. a,_ 2. . . . . al and the quantities G(t.) and Ga(t, - t~) which are obtained from inspection of the data. after calculating the mid-point times il. The cumulative amount of drug absorbed is related to the a~ by the equation

A~b~(tn) =

~ an i=1

and an approximation for the absorption rate is given by R(t) = a,/(t,-

t._l),

for

t . _ l ~< t < t..

Another method, proposed by Rescigno and Segre (1966). approximates the absorption rate by a constant over the time interval (ti-1. t~). thus. the approximation to R ( t ) has the form of a staircase function. Taking R(t)=

R~.

for

ti-~ <~ t ~ t i

the integrals in eqn(81) become f'"

R ( T ) G 6 ( t , - T) d T = R . B , . .

tn I

and ~tO~

' R(T)G~(t, -

T) d T =

n~ 1

~

RiB,.~

i=l

where J~rtl B,.i =

Go(t. -

i-i

T) dT.

(85)

Assessment of rate and extenl of drug absorption

149

Introducing these equations into eqn(81) and rearranging, (86) For n = 1, Ra = G(tl)/BI.1 = G(tl)//fo'

G~(tl -

T) dT.

Equation (86) expresses R., the constant approximation to the absorption rate over the nth interval, in terms of the approximations obtained for the previous intervals (Rg, i = 1. . . . . n - 1) and the integrals B.,~. The amount absorbed during the ith interval is approximated by the term Ri (t~ - t~_ 1). The cumulative amount absorbed up to time t. is given by Aab~(t,) =

~ Ri(ti - ti-1). i=l

The use of eqn(86) involves evaluation of the integrals B,.~ given by eqn(85). Substituting r = t, - T in the integral of eqn(85) gives Bn.i =

G,5(z) dr.

(87)

n--ti

Thus, B,.~ is the area under the Gaff) vs t curve over the interval (t, - t~_ 1,t, - t~). In the case of unequal time intervals B,.i must be evaluated for each i and each n (m(m + 1)/2 evaluations for m data points). When the time intervals are of equal length, the number of evaluations is reduced to m, for m data points. Writing t~ = iAt, where At is the constant time interval, eqn(87) gives B,.,_i =

f a

( i + l)At

G~('r) d~ = BI. -

(88)

iAt

where the term BI is introduced to indicate that this term depends on i but not on n. Changing the index in the summation in eqn(86) gives R, =

[

G(t,)-

R,_iBI i=1

]/

B'o.

(89)

Equation(89) is more efficient than eqn(86) when the time intervals are equal. Various methods can be used to evaluate the integrals of eqns(87) or (88). Rescigno and Segre (1966) estimate the integral in eqn(88) with the approximation ~(i+ l)AtG~(z) dz = AtG~([i + 1]At). d i At

Vaughan and Dennis (1978) have shown, with simulated data, that this approximation may lead to large errors. As an alternative, the integrals could be evaluated by the trapezoidal method (Section 4.1): eqn(87) gives B,.i = ½[Ga(t, - tl-a) + Ga{t, - ti)] [ti - t i - l ] .

(90)

Application of this equation, for unequal data intervals, would require an interpolation procedure to estimate G~ at the times t, - ti-~ and t, - ti. With equal time intervals. evaluation of eqn(88) by the trapezoidal method gives B~ = ½[G~(ti+a) + G~(ti)]At.

191)

150

DAVID Ctr]l.l{R

Another approach to the evaluation of B,.~ and B'~ is to fit the Ga(t) vs t data to a function which can be integrated analytically over each interval required. For example, if a poly-exponential function is employed, with N

Ga(t) =

Aj

e - ,,/t

j : 1

eqn(87) gives

.u ASo,',,

B,., --- L

j = 1

(e"'" - e "i'' ').

~.lj

With equal time intervals, eqn(88) gives

BI=

Aj

-- ej - I tlj

~_,

ajar).

The main difficulty in applying finite difference methods is that these methods are unstable in the presence of data noise (Gamel et al., 1973: Benet and Chiang, 1972j. Gamel et al. (1973) suggest the use of a smoothing procedure prior to deconvolution. While avoiding the instability problem, the smoothing procedure may distort the estimate of the input rate R(t). The main advantage of the finite difference methods is their computational simplicity which allows their use with facilities which would be inadequate for the more stable, but computationally more complex least-squares methods discussed in the following section. 5.3.2. Least-Squares M e t h o d s Jbr Numerical Deconrolution A judgement as to the success of any procedure of estimating an input rate R(t) can be based on noting how well the response predicted by the estimated input rate compares with the observed response from which it was calculated. If /~(t) denotes a particular estimate of the input rate R(t). we can calculate by means of eqn(80) the corresponding response (~(t) which would be observed if the input rate were in fact /~(t): at

CJ(t) = [

R ( T ) G 6 ( t - T) dT.

192)

d0

If /~(t)is close to R(t) we expect (~(t) to be close to G(t). The degree of "closeness" can be expressed by the weighted sum of squares of the deviations of (~,(t) from G(t), S =

~

wi(Gi - Gi) 2.

(93)

i-I

In this equation G~ is the observed response at time t~ and (~ is the response predicted by eqn(92) at time t~. The weighting function w~ determines the relative weight to be placed on the deviation at time q. By the least-squares method the best estimate /~(t) of the true input rate Rlt) is chosen to be that which minimizes S. or predicts the response G(t) which is closest to the observed G(t) as judged by the available data. To proceed further with this approach it is necessary to specify a class of functions from which /~ is to be selected. Suppose a particular function f is chosen which is a function of time and a number of parameters 0~. 02 . . . . . (tk: i,e. R(t) = f (t,O>O2 . . . . . Ok).

(94)

Introducing this expression for R in eqn(92) gives an expression for G as a function of the parameters 0j. Substituting this expression for G~ in eqn(93), after evaluating at time t~, gives an expression for S which is of the form S = U(O>02 . . . .

00.

(95)

Assessment of rate and exlent of drug absorption

151

That is, S is a function of the parameters 0j. These parameters can be chosen to minimize S. Introducting the minimising values of the parameters into eqn(94) gives the required estimate of the input rate. In some cases, depending on the form of the function f and the means chosen to represent G~, U in eqn(95) may be an explicit function of the parameters. In other cases, this might not be possible (if the integral in eqn(92) cannot be evaluated analytically). Nevertheless, for particular values of the parameters a value for S can be calculated (starting with a numerical evaluation of the integral of eqn(92)), which is sufficient for S to be regarded as a function of the parameters. In terms of computer implementation it is of little consequence whether S is obtained by evaluating an explicit expression or by a more lengthy numerical procedure. As an example suppose that it is assumed that absorption follows first-order kinetics, with

l~(t) = k,FDe -kJ

(96)

where k, and F are estimates of the first-order absorption rate constant and fraction of the administered dose D which is absorbed. With/~(t) given by eqn(96), eqn(92) becomes

d(t) = kaFD f~ e-k"TGa(t -- T) dT. Evaluating this expression at times eqn(93) gives

t i and introducing the resulting expression into

S = ~',., wi koFD

e-k"rG~(ti- T ) d T -

Gi

(97)

i=1

which is a function of the parameters ka and F. The values of ka and F which minimise F are introduced into eqn(96) to give the estimated R(tl. Numerical examples based on simulated data, with the integral in eqn[97) evaluated numerically, are given by Cutler ( 1978b~. If G~ is expressed as a poly-exponential function N

G~=

~ Bje -b~'

08)

j=l

the integral in eqn(97) can be evaluated analytically to give an explicit expression for S: Bj

S=,=1~

w"[k~FD{j~=I~--BJ b j e - b " ' - ( j ~ = l k , - b j ) e - k ° " } - G ' ]

2

(99)

The use of eqn(99) involves prior determination of the constants Bj and b~ by fitting eqn(98) to the experimental data on G6. With these parameters fixed, eqn(99) expresses S as a function of the two absorption parameters ka and F. The use of a least-squares deconvolution method requires prior assumptions concerning the mathematical form of the input function. When little or no information is available on the nature of the input function a polynomial function of time might be a suitable approximation, since polynomials are sufficiently flexible to represent a wide range of functions with reasonable accuracy. Adequate approximations to a first-order input and to an input arising from the cube-root dissolution law were demonstrated, with simulated data, by Cutler (1978c). It was also shown that a polynomial approximation may be useful in indicating the general form of the input function even in cases in which a polynomial gives a poor representation. A rectangular input function was examined, of the form / ~ ( t ) = R 0 f o r 0 ~ < t ~ t*.

152

DAVID CUTLI!R

The input rate is constant, equal to Ro, up to time t* when it ceases abruptly. Due to the discontinuity at t* this function cannot be well represented by a polynomial. Attempts to estimate this input function using a polynomial approximation were unsuccessful but revealed the source of the difficulty (the discontinuity at t*) with higher degree polynomials being roughly constant before decreasing sharply in the vicinity of t*. This is a fairly extreme test of a polynomial approximation which suggests that this method is sufficiently robust to be useful in a wide range of situations. In principle, a polynomial approximation can be obtained by introducing the desired polynomial expression in eqn(92), with subsequent minimisation of S given by eqn(93). This procedure may lead to instability (Gamel et al., 1973) due to ill-conditioning of the resulting equations. This problem can be overcome by a method which develops the required polynomial by a procedure using orthogonal functions (Cutler, 1978c).

5.4. ASSESSMENT OF ABSORPTION RATE BY CURVE FITTING METHODS

Curve fitting methods involve fitting the plasma concentration-time data following the test dose to an analytical function which includes parameters which account for the absorption process and the disposition process. The absorption rate is constructed by identifying, on the basis of certain assumptions, the parameters in the fitted function which refer to the absorption process. The method will be illustrated for the case of a one compartment model with firstorder absorption. In this case the plasma concentration is given by C(t) --

kaFD (e -k+ (ko - k) V

e-k~t).

(100)

Fitting experimental data on the assumption that it follows eqn(100) gives an expression of the form C(t} = B l e -b'' + B2e -~'-'

(101)

where B~, B2, hL, and b2 are constants. We take bl > b2. If it is assumed that absorption is rapid, so that k a > k, comparison of eqn(100) with eqn(101) gives k, = ht, thus providing a value for the absorption rate cortstant (and the absorption rate profile exp(-],:a t) if required). Conversely, the assumption that absorption is slow, with k~ < k gives the value ka = b 2.

An obvious pitfall with this approach is in the identification of ka with either bl or b2. If the only evidence available is the data to which eqn(101) is fitted, this identification is always uncertain. A further problem is that of "vanishing" exponentials, which has been discussed by Ronfield and Benet (1977). If disposition follows the two-compartment model a first-order absorption process with rate constant equal to the sum of the exit rate constants from the peripheral compartment, an equation of the form of eqn(101) arises. In this case, neither bx nor b2 can be identified as the absorption rate constant k,. It is impossible to distinguish this case from the one compartment case described by eqn(100) on the basis of extravascular data alone. These difficulties also arise when disposition follows multi-compartment kinetics. The possibility of mis-identification of k, with one of the exponential coefficients increases as the number of compartments increases. As the number of exponential coefficients needed to describe the extravascular data increases a further problem arises, due to the nonuniqueness of exponential coefficients fitted to experimental data (Westlake, 1971). Saunders and Natunen (1973) investigated this problem and proposed a curve fitting method which provided stable parameter estimates when two-compartment disposition is followed. Some of the problems associated with curve fitting methods are avoided when intravenous data are also available. However, when this is so, the mass balance or deconvolution methods appear to be preferable.

Assessment of rate and extent of drug absorption

153

5.5. USE OF STATISTICAL MOMENTS TO CHARACTERIZE ABSORPTION RATES

This approach follows when absorption is regarded as a stochastic process. The information provided by the statistical moments of the absorption process (we will be concerned with the mean and variance) is the same as the information provided on a probability distribution by its statistical moments. This information is less than a complete description of the time course of the absorption process, but has the advantage of summarising briefly salient features of the absorption rate in just the same way as a mean and variance give a useful, but incomplete, description of a probability distribution. In characterizing an absorption rate the statistical moments compare favorably with the parameters tmax and Cmax discussed in Section 5.1. The statistical moments which characterize the absorption process can be calculated from the time profile of the absorption rate if this is known---e.g, following the procedures described in the preceding sections. Here we develop procedures for estimating the statistical moments directly from experimental data without first determining the absorption rate. The absorption process may be recognised as a stochastic process by considering the fate of a molecule introduced at the absorption site (say, the gut) at time zero. It is not possible to predict at what time an individual molecule will be absorbed, nor is it possible to determine from observation when an individual molecule is absorbed. Nevertheless when a large number of molecules are presented at the absorption site (as a dose of drug) they behave collectively in a predictable manner (to the extent that an absorption process is reproducible). The predictability of the population, with the unpredictability of the individual behaviour, is the essential feature of a stochastic process. To express this situation mathematically we write p(t) to denote the probability density function for absorption; i.e. p(t)dt denotes the probability that a molecule introduced at time zero is absorbed (enters the general circulation) in the time interval (t,t + dt). If n is the total number of molecules absorbed, by the frequency interpretation of probability np(t)dt is the number of molecules absorbed in the time interval (t,t + dt). In the notation used previously, n = FD where F is the fraction of the administered dose D which is absorbed. In terms of the absorption rate R~bs(t), the amount of drug absorbed in the time interval (t,t + dt) is Rabdt)dt; consequently,

p(t) = Rabdt)/FD.

(102)

Thus, the absorption rate, corrected for the absorbed dose FD, can be interpreted as the probability density function for absorption of a single molecule. We may now define MRT~b~ -

1

/

FD d o

tR~bs(t) dt

(103)

as the Mean Residence Time of a molecule at the absorption site. The Variance of the Residence Time of a molecule at the absorption site may be defined as VRTab~ - FD1 f o (t - MRTabs)2R~bs(t) dt.

(lO4)

Higher moments can also be defined but due to calculation errors it is unlikely that these can be of practical use. We now consider how MRTabs and VRTabs can be calculated from experimental plasma concentration-time data (Cutler, 1978d; Yamaoka, et al., 1978). The following relationships are derived in the Appendix for the case in which disposition is a linear process (the restriction of linearity does not apply to the absorption process itself). C~ is the plasma concentration following a unit impulse input into the general circulation, and

154

DAvu~ CUTLIiR

C is the plasma concentration following the test dose.

MRTc = MRTc. + MRT~b~

(105)

VRTc = VRTc~ + VRT.b~

(106)

....--::,.,.,/:o-.,.,

,,07,

where

and VRTc =

:o"

(t - M R T c ) 2 C ( t ) dt

I::

C(t) dt

(108)

with corresponding expressions for MRTc, and VRTc, obtained by replacing C with C~ in eqns(107) and (108). A physical interpretation of the moments MRTo MRTc: VRTc and VRT G is not required for our purposes; these may be regarded simply as numerical quantities which can be derived from experimental data according to the definitions of eqns(107), (108) and the corresponding definitions for MRTc~ and VRT G. If an interpretation is required: MRTc, is the mean residence time in the body for a molecule introduced into the general circulation and MRTc is the mean residence time of a drug molecule in the body (including the absorption site) following the test dose. VRTco and VRTc represent the corresponding variances of the residence time of a drug molecule in the body. Equations (105) and (106) demonstrate the additive property which makes the statistical moments useful in characterizing the absorption process. If plasma concentrations of the drug are measured following an i.r. dose and the test dose, evaluation of MRTc and MRTc,~ gives MRT~b~ = MRTc - MRT G.

(109)

from eqn(105). Evaluation of VRTc and VRTc,, allows calculation of VRT.b~ = VRTc - VRT G.

(110)

from eqn(106). MRT~b~ and VRT,,b~ give an immediate general indication of the absorption rate; rapid absorption is associated with a small value of MRT~b~, a small value of VRT.,~ indicates that most of the absorbed drug is absorbed over a short time period. These parameters may also give information on the parameters of an assumed absorption mechanism. For example, if absorption is assumed to follow first-order kinetics, with absorption rate constant ka. we have MRT~b~ = 1/ka and VRT~B~ = 1/k~ on evaluating eqn(103) and eqn(104) with R(t) = k , F D e x p ( - k d ) . Thus, k~ can be calculated: the relationship VRT~b~ = MRT~b~ may give some indication whether first-order absorption kinetics are followed. A significant depature from this relationship is incompatible with the absorption process being first-order. Statistical moments may also be used to directly compare a test dose with a reference dose without the need of an intravenous study. Using the subscripts x and s to denote the test formulation and a standard formulation, eqn(109) gives MRT,b~,.x = MRTc.x - MRTc~ and MRT,b.,.~ = MRTc.~ - MRT G assuming that MRTc~ is the same in both studies (i.e., that there is no change in the

Assessment of rate and extent of drug absorption

155

disposition of the drug between studies). Combining the last two equations, MRT~b~.~ - MRT~h~.~ = MRTc.~ - MRTc..~.

(111)

This equation shows that differences in the mean residence times at the absorption site between the two formulations can be calculated from their respective plasma concentration-time curves. Similarly, from eqn(110), VRT~b~.~ - VRT~b~.~ = VRTc.x - VRTc..~.

(i 12)

Equations (1 ll) and (112) offer significant advantages over the use of tm~,as a means of comparing absorption rates in the absence of intravenous data. The mean and variance are well-defined parameters which are independent of the disposition of the drug. tm~, depends on the disposition of the drug as well as on the absorption process and except for the simplest of cases the relative contribution of absorption and disposition to the observed value cannot be determined. If absorption following both the test and standard formulations is first-order, with absorption rate constants k~.~ and k,., respectively, MRT~b~.x - MRT~b~.~ = 1/k,,,.~ - l/k~.~. If t~.x = ln2/k~.x and t~.~ = ln2/k~.~ denote the half-times for absorption for the test and standard formulations, (MRT~b~.~ - MRT~b~.~) = (t._,.x - t~.,0/In2

(113)

allows calculation of the difference in the absorption half-times. Similarly, for the variances, (VRT~b~.x - VRT~b~.~) = [(t,,.x) 2 - (t,,..~12]/(In2)2.

(114)

In theory, the last two equations may be combined to give estimates of the individual absorption half-times. After some manipulation, we have

t~,~, =

2 [ A M + AM

and ln2[AV =

] -

A M

where AM = MRTab~.x - MRT,b~,, and AV = VRTabs.x - VRTab~.s. Errors in estimating A M , and in particular AE are likely to restrict the application of these equations. The use of statistical moments requires estimation from experimental data of integrals such as those appearing in eqn(107) and eqn(108). As in estimation of AUC (Section 4) this involves two steps, the first being estimation of the integral over the time range spanned by the data, while the second estimates the integral after the last data point by an extrapolation procedure. If the plasma concentration declines exponentially after the last data point C(t,~),t~, with C(t) = C(tm) - ~ t - t m ) ,

for

MRTc in eqn(107) is given by (Yamaoka et al., 1978). MRTc = S1/AUC

J.P.T. 14 2

("

t t> tin.

(115)

156

DAVID CtrTLJiR

with AUC =

C(t) dt

Sx =

tC(t) dt

and

=

tC(t)dt

+ ~

tm+

{1161

.

AZ

The integral in the last equation can be evaluated by the methods discussed in Section 4.1 ; with concentrations C(ti) at times ti the product t~C{t~) is formed before applying the chosen integration method. As with estimation of the extrapolation term of AUC, eqn(116) requires that the single exponential decline assumed in eqnl115) is obeyed for a sufficient period before the last data point to enable evaluation of 2z from the terminal slope of lnC vs t. With eqn(1151, to represent the plasma concentration after the last data point VRTc in eqn(108) can be written VRTc

=

S2/AUC

-

(S1/AUC) 2

with $2 =

fo

=

t 2 C ( t ) dt

t2C(t) dt + ~ - - 0

t~, + ~ -

XZ

+

.

(117)

AZ

The integral in this equation can be evaluated by forming the products t {C~, from the data C(t~), t;, before applying the integration methods discussed in Section 4.1. If the entire plasma concentration-time curve is fitted to a poly-exponential expression, N

C(t) =

~

B i e -h''

j=l

where B~ and b; are constants, N

AUC =

~

Bj/bj

j=l N

s, = E j=l

and N

$2 =

Z

j=l

2BilbO.

It was noted in Section 4.2 that in estimating AUC the extrapolation beyond the last data point may be the source of large and unpredictable errors. In estimating moments, this difficulty becomes more severe as the functions to be integrated, tC(tl and tZC(tl, decrease more slowly than C(t). Yamaoke et al. (1978) have estimated "cutoff errors" (the contribution of the extrapolated portion of the integral); when C is measured to 57/o of its maximum value, the cutoff errors in AUC, M R T and VRT were about 5, 10 and 40% respectively, which fall to about 1, 2 and 10% respectively when C is measured to 1% of its maximum value. The error in extrapolation can be expected to be smaller than these

Assessment of rate and extent of drug absorption

157

figures in practice, provided the extrapolation term is estimated with an error less than 1009~,. However, additional errors arise in estimating the integral over the time range spanned by the data. Estimation errors become progressively larger with higher statistical moments which will probably preclude their practical use. 5.6. ESTIMATION OF ABSORPTION LAG-TIME

In most cases following extravascular administration, particularly following oral administration, there is a delay, called the lag-time, between the administration of the drug and its first appearance in the general circulation. The magnitude of the lag-time may be of intrinsic interest, as a measure of a physical factor (such as a delay in stomach emptying) influencing absorption. In other cases, calculation of the lag-time is a prerequisite of absorption rate calculations, since neglect of the lag-time (or assuming it to be negligibly small) may result in an error in the calculated absorption rate. This is also the case in estimation of the extent of absorption; an error in an estimate of the area under the plasma concentration-time curve during the first time interval will result if the lag-time is inaccurately estimated. The main difficulty in estimating the absorption lag-time is that in most cases there is insufficient data at early times. If an accurate estimate is required there is no substitute for frequent sampling over the early stages following absorption. A sampling interval of the same order of magnitude as the required accuracy in the lag-time estimate is needed. The simplest method for estimating the absorption lag-time is by graphical extrapolation of the plasma concentration-time data to intersect the time axis. This procedure may be automated by assuming the early time data to follow a quadratic function (Saunders and Natunen, 1973) C'=ctt 2 +fit+ where co, fl and ~ are constants, tla~ is given by solving the equation C' = 0. Another approach is to include the lag-tlme as a parameter in an expression fitted to the data by a least-squares procedure. For a poly-exponential fit, m

C(t) = ~

B~ e - b z ' - ' ' ' o

for t ~> t~,g

j=l

= 0

for t < tlag

where the parameters Bj, bj and tl~g are to be estimated. When t~g has been estimated the time scale can be translated to the new time scale t' = t-ttas with subsequent calculations on the new time scale with zero lag-time. APPENDIX 1. INPUT-RESPONSERELATIONSHIPFOR A LINEAR SYSTEM Consider a unit dose of drug presented at a point L in the body, at time zero, as an impulse or bolus (that is, "instantaneously", or in practice very rapidly compared with the rate of the disposition processes). Let Gj(t) be the response at time t which results from this input. If an amount M is introduced as a bolus at time zero, and if the system is linear, the resulting response is M G~(t); if the amount M is introduced at time T(0 <~ T ~< t), and if the system is time-invariant (i.e. G~(t) does not depend on the time at which the unit impulse is introduced) the resulting response is G6(t - T). Note that t - T is the time elapsed after the amount M is introduced. Now consider a continuous input at the rate R(T). Over the short time interval (T, T + dT) the input R(T) results in an amount R(T)dTbeing introduced, which, for small dT, approximates an impulse input. This leads to a contribution R(T)G~(t - T)dT to the response at time t. By virtue of the linear property the total response at time t is made up of the sum of all such contributions for 0 ~< T _< t; i.e. G(t) ~

~

R(T)G6(t - T ) d T

O<~T<~t

where G(t) is the observed response at time t. This approximation becomes exact as we pass to the limit d T---* O. The summation is replaced by an integral to give Glt) =

I2

R(T)G~lt - T) dT.

(AI)

15g

DAVID CUTLER

It is often convenient to express cqn(Al) in terms of the Laplace transforms of the functions involved. By a standard result of Laplace transform theory (see, e.g. Spiegel, 1965) eqn(Al) gives

,q{s) = r(s).qa(s)

{A2)

where ~l(s),rls) and gas) are the Laplace transforms of G(t~. R(t) and G,,lt). The Laplace transform x(sl of a function X{t) is defined as

f

x(s) =

~ e "X(t)dt.

IA3)

Therefore, lira ,' x l s•h' =

f X(tldt.

(A4)

s~O

From cqnlA2). lim '

" f I lira q' l" I hm,g~,

s~O

s 40

s ~0

and using the result stated in eqn(A4),

L

L

G(tldt =

L

R(tldt

GAt~dt.

IA5)

2. RELATIONSHIPS FOR STATISTICAL MOMENTS The Laplace transform of a function Xlt} is given by eqn(A3). Differentiating with respect to s, dx/ds = - fo ~ t e ~'X(t)dt

(A6}

and therefore, x

lim I d x d s ~, = -

L

tX(t}dt,

tA7)

Differentiating cqn(A61 with respect to s.

d2~: ds z = f f

tee -'tX(t)dt

and therefore, lim i d 2 x d s : ] = s~O

L

t2Xltldt.

(A8}

We now use these results to relate the m o m e n t s of G. R and G, b.~ means of eqnlA21. Differentiating eqnlA2) with respect to s.

dg.'ds = rdg~/ds + g~dr ds.

(A91

Using eqn(A2k 1 dg

1 dg~

g ds

ga ds

1 dr r ds

giving r , lim ~dg, ds, ~

lim [dg~/dsl

I r lira ,d .ds~- I

s~O

s~O

s~O

lim ,9, ' ' s~O

-

lim 'Lga,

+

s~O

lim ,r,'/

(AIOI

s~O

Applying eqn IA4) and eqn (A7}. writing G, G~ and R for X, eqn (A10) becomes

+

or

M R T a = M R T ~ + MRT,b~

(All)

in the notation used in Section 5.5 of the text. Equation(105) of the text follows when the response is interpreted as the plasma concentration of the drug. Differentiating eqn(Agl with respect to s gjves

d29/ds 2 = r d2 g,ffds 2 + 2(dggds)(dr/ds) + g~ d2r/ds 2.

Assessment of rate and extent of drug absorption

159

Using eqn(A2), 1 d2.q ,q ds 2

dr]

1 d2,% 2fl d.%]fl •q~ d~

+

1 d2r

L,q,~ ~JL;: ~

+ r ~ls~

and therefore lim Id29/dsZ I

lim, 2

['lim',d.%/ds',']~lim~,dr/dsl -]

,2

fli

Id2r/ds2', 1

s~0

lim ',9~I [ lim ',,qa') [ lim Irl ,-o L ' '" ' Jg x-o J Using eqns (A4). (A7) and (A8k writing G. G, and R for X. cqnlAI2) becomes

[ L

lim ',0'~

s~O

J'0 t2G(t) dt J'0"t2G dt) dt =

.

fo~ G(Odt

.

.

.

.

.

j tGo(t)dt fo" tR(t)dt 4-

.

2

f j G61t)dt

.

.

.

.

.

.

.

.

lim :rl ,-o

] _l

(AI2)

fo t2R(t)dt 4-

fo" G,(t)dt f f R(t)dt

(A13)

--

fo" R(t)dt

The variance of the response G is defined as

fo~ (t - MRT6)2G(t)dt VRT~ =

fo:" t2G(t)dt =

fo G(t)dt . . . . -~. . . . .

.

MRT~. fj

G(t)dt -

(AI4)

_

Similarly. fo ~ (t - MRTo,)ZG~(t)dt - - - - - ~.........

VRT~=

Jo

G6(tJdt

fo ~ t:Ga(t) dt ........

M RT~.,.

(AIS)

MRT~2b'"

(A16)

fo ~ G,~(t) dt

and

VRTa~'"

=

fo ~ (t - MRT~t,OZR(t)dt

fo ~' tZR(t)dt

Jof R(t)dt

a(f) R(t)dt

Introducing eqns(Al4). (AI5) and (A16) into eqnlAl3) gives VRT~ + MRT~ = VRTo., + MRT~, + VRT,b, + MRT~Zt,, + 2MRTooMRT,t,,.

(A17)

From eqn(All) we have MRT~ = M R T ~ + MRT~t,, + 2MRT~,MRT,t,,. Using this result, eqn(A171 gives VRTa = VRT~., + VRT~b ~.

(A18)

Equation(106J follows when the response is interpreted as the plasma concentration of the drug.

REFERENCES AMIDON, G.. PAUL M. and WELLING. P. (1975) Model-independent methods in pharmacokinetics: theoretical considerations. Math. Biosci. 25: 259-272. BENET, L. and CHIANO. C-W. (1972) The use and application of deconvolution methods in pharmacokinetics. In: Abstracts of the 13th National Meetin 9 of the A.Ph.A., Academy of Pharmaceutical Sciences. Chicago. 1972. BOXENBAUM, H. and KAPLAN. S. (1975) Potential source of error in absorption rate calculations. J. Pharmacokin. Biopharm. 3: 257-264. CHIOU, W. (1978) Critical evaluation .of potential error in pharmacokinetic studies of using the linear trapezoidal rule method for the calculation of the area under the plasma level time curve. J. Pharmacokin. Biopharm. 6: 539-546. CHIOU, W. (1980) New compartment and model-independent method for rapid calculation of drug absorption rates. J. pharm. Sci. 69: 57-62. CURLER. D. (1978a) Linear systems analysis in pharmacokinetics. J. Pharmacokin. Biopharm. 6:265 282. CUTLER. D. (1978bj Numerical deconvolution by least squares: use of prescribed functions. J. Pharma('ol,in Biopharm. 6: 227-241. CUTLER. D. (1978c) Numerical deconvolution by least squares: use of polynomials to represent the input function. J. Pharmacokin. Biopharm. 6: 243-263. CL'TLER, D. (1978dl Theory of the mean absorption time. an adjunct to conventional bioa~ailability studies. J. Pharm. Pharmac. 30: 476~,78. GAMEL. J.. ROUSSEAU. W., KAT8OLL C. and MESEL E. (1973j Pitfalls in digital computation of the impulse response of vascular beds from indicator-dilution curves. Circ. Res. 3 2 : 5 1 6 523.

160

DAVID CUTLER

HWANG, S. and KWAN, K. (1980) Further considerations on model-independent bioavailability estimation. J. pharm. Sci. 69: 77-80. JUSKO, W., KOUP, J, and ALVAN, G. (1976) Non-linear assessment of phenytoin availability. J. Pharmacokin. Biopharm, 4: 327-336, KWAN, K. and TILL. A. (1973) Novel method for bioavailability assessment. J. pharm. Sci. 62:1494 1497, KWAN, K., BONDI, J. and YEH, K. 11975) Bioavailability assessment under quasi- and nonsteady state conditions I: theoretical considerations. J. pharm. Sci. 64: 1639-1642. LALKA, D. and FELDMAN. H. (1974) Absolute drug bioavailability: approximation without comparison to parenteral dose for compounds exhibiting perturbable renal clearance. J. pharm. Sci. 63: 1812. L(x), J. and RIEGELMAN, S. (19681 New method for calculating the intrinsic absorption rate of drugs. J, pharm. Sci. 57: 918-928. MARTIS, L. and LEVY, R. (19731 Bioavailability calculations for drugs showing simultaneous first-order and capacity-limited elimination kinetics. J. Pharmacokin. Biopharm. I: 283-294. OIL S. and JUNG, D. t1979) Bioavailability under variable renal clearance conditions. J. pharm. Sci. 68: 128-129. RIEGELMAN, S., LOo, J. and ROWLAND, M. ~19681 Shortcomings in pharmacokinetic analysis by conceiving the body to exhibit properties of a single compartment. J. pharm. Sci. 57:117 123. RESCIGNO, A. and SEGRE, G. (1966) Drug and Tracer Kinetics, Blaisdell, Mass. RONFIELD, R. and BENET, L. (19771 Interpretation of plasma concentration time curves after oral dosing. J. pharm. Sci. 66: 178"-180. SAtJNDERS, L. and NATUNEN, T. (1973) A stable method for calculating oral drug absorption rate constants with two compartment disposition. J. Pharm. Pharmuc. 25: Suppl., 44P-51P. SMOLEN. V. (1976a) Theoretical and computational basis for drug bioavailability determinations using pharmacological data I. General considerations and procedures. J. Pharmacokin. Biopharm. 4:337 353. SMOLEN, V. (1976b) Theoretical and computational basis for drug bioavailability determinations using pharmacological data II. Drug input-response relationships. J. Pharmacokin. Biopharm. 4: 355-375. SPIEGEL, M. (1965) Theory and Problems of Laplace Tran.~rms. Schaum's Outline Series, McGraw-Hill, VAUGHAN, D. and DENNIS, M. (1978) Mathematical basis of point-area deconvolution method for determining in cico input functions. J. pharm. Sci. 67:663 665. WAGNER, J. (1974) Application of the Wagner-Nelson absorption method to the two compartment open model. J. Pharmacokin. Biopharm. 2: 469~1.86. WAGNER, J. (1975) Application of the Loo-Riegelman absorption method. J. Pharmucokin. Biopharm. 3: 51-57. WAGNER. J. and AVERS, J. (1977) Bioavailability assessment: methods to estimate total area (AUC O--zcl and total amount excreted {A~I and importance of blood and urine sampling scheme with application to digoxi n. J. Pharmacokin. Biopharm. 5: 533-557. WAGNER. J. and NELSON, E. (19631 Per cent absorbed time plots derived from blood level a n d o r urinary excretion data. d. pharm. Sci. 52: 610-611, WESTLAKE, W. (1971) Problems associated with analysis of pharmacokinetic models. J. pharm. Sci. 60:882 885. YAMAOKA, K.. NAKAGAWA, T. and UND. T. 11978~ Statistical moments in pharmacokinetics. J. Pharmacokin. Biopharm. 6:547 558. YEH, K. and KWAN, K. (19761 Bioavailability assessment under quasi- and nonsteady-state conditions lI: study designs. J. pharm. Sci. 65:512 517. YEH, K. and KWAN, K. ~19781 A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation. J. Pharmacokin. Biopharm. 6:79 98,