EUROPEAN JOURNAL OF DRUG METABOLISM AND PHARMACOKINETICS, 1993, Vol. 18, No. I, pp. 71-88

Pharmacokinetics of endogenous substances: Some problems and some solutions A. MARZO! and A. RESCIGN02 I Sigma Tau S.p.A., Pomezia, Rome, Italy 2School ofPharmacy, University ofParma, Parma, Italy

Keywords: Endogenous substances, phannacokinetics, homeostatic equilibria, pharmacokinetic models, non linear pharmacokinetic processes, endogenous pool, baseline concentration

SUMMARY The paper deals with the most relevant aspects related to the phllDD8COkinetics of endogenous substances. Two different views are presented in order to focus on two aspects of the problem, the physiological background of these substances and the need for empirical or tailored models to process phllDD8COkinetic data. Very often endogenous substances follow saturable enzyme biotransformation, reversible intcJ:conversion, active and diffusional transports, renal threshold, endogenous synthesis plus dietary supply with possible balanccment between these two factors. feedback processes, asymmetric distribution with specific body storage, and gender differences. These mechanisms allow the body to preserve and restore homeostatic equilibria of endogenous substances. The most relevant problem in pharmacokinetics of these substances is the presence of baseline concentration which needs to be carefully defined also for possible rhythms related to age, sex, diet, night and day periods. Theoretical considerations are presented for the management of pharmacokinetic analysis of these substances, which only rarely follow linear processes. Throughout the text various practical examples are considered.

INTRODUCTION According to a narrow definition, endogenous substances are those developed or originated within the organism. In this paper the above definition is extended to include substances not synthesized in the body but present in the organism and basically involved in physiological and biochemical processes, for instance inorganic ions, because from a pharmacokinetic point of view they behave like the narrowly defined endogenous substances. Table I lists the main endogenous substances divided into 9 classes. Most endogenous substances are administered to humans in common diet, or as dietary integrators, or

Please send reprint requests to : Prof. Antonio Marzo, Department of Pharmacokinetics and Dmg Metabolism, Sigma Tau S.p.A., Via Pontina Km 30 400,00040 Pomezia, Rome, Italy

as drugs. In certain cases they have been very thoroughly investigated from a metabolic point of view (glucose, fats, aminoacids), but in general little information is available about their metabolism and pharmacokinetics. When endogenous substances are involved, particular attention should be paid to kinetic investigation, because very often these drugs follow saturable enzyme processes, active and diffusional transports, renal threshold, reversible systemic interconversion, endogenous synthesis plus dietary supply, feedback processes, asymmetric distribution with specific body storage, and gender differences. In these cases the conventional compartmental models are poorly predictive and in certain cases misleading, and thus a non-compartmental or a tailored kinetic model should be considered as more appropriate (1). In this paper some specific problems connected to pharmacokinetics of endogenous substances are described and possible solutions are suggested.

78

Eru. J. Drug Metab. Pharmacokinet., 1993, No.1

Table I: Main endogenous substances divided into 9 classes. 1. Water and Ions

2+Zn2+, Fe 2+/F3+ 3+ cr, rrr-, HC03- , N 0H20, K+, Na +, Ca 2+, Mg, e , AI, 3, Phosphates 2. Lipids and Derivatives Glycerides, fatty acids, phospholipids, cholesterol, bile acids, prostaglandins, thromboxanes, glycolipids, gangliosides 3. Mono-, Oligo-, Poly-saccharides, Muco-polysaccharides and Derivatives Glucose, glucosamine, lactose, sucrose, mannitol, galactose, galactosamine, fructose, sorbitol, glycogen, hyaluronate, chondroitin and dcrmatan sulphate, polygalacturonic acid, pectin, heparin, sialic acid 4. Aminoaclds and Derivatives Essential and non essential aminoacids, glutathione, camitine and acylcamitines, choline and acetylcholine, spermine and spcnnidine, urea, NH3, histamine, serotonin, dopamine, epinine 5. Hormones Steroidal hormones: corticosteroids, progesterone, estradiol, testosterone Aminoacid related hormones: thyroxine, adrenalin, noradrenalin, melatonin Peptidic hormones: relaxin, pamthormone, insulin, glucagon, oxytocin, vasopressin (antidiuretic hormone), melanotropin, somatotropin, corticotropin, follicle stimulating hormone, luteinizing hormone, prolactin, gonadotropin 6. Vitamins Vitamin A, vitamin B complex (thiamin, riboflavin, nicotinate and nicotinamide, pantothenate, pyridoxal pool, biotin), vitamin C (ascorbate), vitamin E, vitamin K, vitamin D 7. Nucleotides, Coenzymes and Related Compounds CoASH, NADINADP, ATP/ADP/AMP/cAMP, GMP/cGMP, S-adenosylmethionine, DNA, RNA 8. Tri- and Di-Carboxylic Acids Citrate, succinate, fumarate, maleate, tartrate, a-ketoglutaratc 9. Purines, Pyrimidines, Nucleosides and Related Compounds Guanine, adenine, adenosine, guanosine, cytosine, thymine, uracil, inosine, hypoxanthine, xanthine, orotate, uric acid.

GENERAL CONSIDERATIONS

excretion can be considered a useful mirror of bioavailability, assuming that no net body storage

occurs.

Homeostatic equilibrium and its preservation The body possesses homeostatic mechanisms which preserve the baseline concentration of endogenous substances, and which can restore it when impaired. At steady state, the production, i.e. the sum of endogenous synthesis, dietary intake and pharmacological doses, is equal to the dismission, i.e, the sum of excretion and biotransformation.

PRODUCTION

DISMISSION

dietary intake

+

excretion

endogenous synthesis

+

+

biotransformation

pharmacological dose

This equivalence in some cases has fewer terms, thus simplifying the bioavailability investigations; this is the case for instance with ions which are not synthesized nor biotransfonned, therefore the urinary

Iron is a peculiar example as it has not an effective excretion route, apart from the loss of blood, and thus the body strictly controls its enteral absorption. In healthy men about 1 mg of iron is lost and is absorbed in a day; in women this amount may be higher due to the loss of blood with menstruation (2). For essential aminoacids the above equivalence involves only the dietary. intake and the sum of excretion and biotransformation. With the L-carnitine moiety the equivalence is between: (a) the sum of dietary intake and endogenous synthesis; and (b) renal excretion, as it is not biotransformed apart from the intereonversion within the same family producing L-camitine acyl esters (3). When the production is assured by endogenous synthesis plus dietary intake, the possible balance between these two factors must be carefully considered. In certain cases a fraction of the endogenous substance produced could be retained in the body pool, as for instance when the drug is administered tocorrect a deficiency. In this case the fraction not dismissed should be evaluated with an appropriate method.

A. Marzo & A. Rescigno, PKofendogenous substances

Whole-body content and distribution of endogenous substances Water is the biggest endogenous pool, constituting, on average, 70% of body weight For pharmacokioetic and metabolic studies of endogenous substances exogenously administered, some relevant data are the whole-body content, the presence of a pool with uni- or bi-directional interconversion, the presence of specific storage. Whole dynamic pools can participate in the preservation of homeostatic equilibrium, as in the case of the L-carnitine family (3) and the pyridoxal pool (4). Some pools are in a very quick dynamic evolution as in the case of ATPIADPI AMP or acetylcholine/ choline systems. A healthy human body synthesizes in a day as much ATP as its own body weight (5). Some highly dynamic pools are biochemically related to slower pools, such as choline which participates in the highly dynamic pool of the cholinergic system, in the phospholipids and in citicoline (6); or acetyl-S-CoA which, inter alia, is transferred from acetyl-L-carnitine to choline (7). Apart from the case of water discussed above, the body contains 50 mglkg of iron in males and 37 mglkg in females - 64% as hemoglobin, 11% as myoglobin and enzymes and 25% stored in liver, spleen and bone marrow (2). As much as 15-20 g of L-caroitine are asymmetrically distributed in the body, 98% being present in skeletal and cardiac muscles (8). Similarll, some ions are asymmetrically distributed, K+ and Mg + being more concentrated in intracellular and Na + in extracellular areas (9).

The role of renal clearance The saturable tubular reabsorption process is the key mechanism allowing the body to control the homeostatic equilibrium of most endogenous substances, at least those excreted via the urine. Typical examples are the components of the L-caroitine family. most ions, some amiooacids (3, 9, 10). These substances possess a renal threshold close to plasma concentration. The threshold of glucose is 1.8 gil against the normal baseline concentration of -0.9 gil; this allows the body to retain a fundamental energetic substrate (11). With endogenous substances which are excreted with a saturable tubular reabsorption, renal clearance increases with plasma concentration and/or the dose administered, leading to typical dose-dependent

pharmacokinetics (3).

79

The active transepytelial crossing of an endogenous substance is a common mechanism which leads on the one hand to a saturable intestinal absorption and on the other hand to a saturable tubular reabsorption. Thus an endogenous substance cleared via urine, when administered at pharmacological doses able to saturate active transports, is absorbed through a passive diffusion, usually slower than the active process, and is excreted with increased renal clearance as the plasma concentration exceeds renal threshold. This simple, effective mechanism operates with various endogenous substances preserving their optimum homeostatic equilibrium (3, 12). When administering low oral doses of these substances, usually plasma concentration is poorly affected or not at all, whereas urinary excretion and thus renal clearance are (12). In the case of K+, bioequiva1ence studies were carried out measuring only urinary excretion which is a good bioavailability mirror (13).

Problems in evaluating absolute bioavailability The basic mechanisms involved in the preservation of homeostatic equilibrium often generate serious problems in evaluating bioavailability. In fact when detenniniog the absolute bioavailability, the non-linear processes operating with endogenous substances prevent the use of the ratio AUCo./AUCLv. due to the variable clearance (1, 14). The presence of a baseline concentration calls for a baseline subtraction. For substances cleared via urine, the net cumulative urinary excretion could be a useful bioavailability mirror, as, for instance, with L-caIDitioe moiety and potassium (13). In the presence of a multicomponent interexchanging pool, the different or very different plasma concentration after i, v. and oral routes and the saturable tubular reabsorption could lead to misleading results. even when using the os/i.v. cumulative urinary excretion ratio. The os/i.v. net AUC ratio could be used to assess absolute bioavailability in a protocol leading to superimposable clearance variations after oral and I,v, acfministration. This usually occurs with a slow i, v. infusion of a given dose that generates clearance variations similar to those encountered with an oral dose, which usually should be higher than the i.v, dose.

80

Eur. J. DrugMetab. Phormacokinet., 1993, No.1 ability is evaluated, the baseline subtraction is a mandatory procedure. The following examples clarify this view. We suppose thatthe same dose is administered by oral and tv. routes, producing AUC values as follows: baseline 300, i.v. 1000 (net 7(0), oral 400 (net 1(0). Absolute bioavailability results as follows:

Single dose and repeated dose regimen If the whole production (endogenous synthesis + dietary intake + pharmacological dose) equals the amount dismissed (biotransformation + exaetion), a singe dose and a repeated dose regimen should result in a superimposable behavior. When a fraction of the whole production is retained in the body, a repeated dose regimen should result in a significant increase of body storage and baseline concentration. With a substance not biotransformed and cleared via urine, the urinary recovery after an tv. administration can verify whether the body retains a fraction of the dose administered. In all cases the repeated dose regimen in comparison with the single dose administration is a basic approach for pharmacokinetics of endogenous substances, mainly in target population where the increase of endogenous pool often is the main objective.

400- 300

1000- 300 • 100= 14.29%

1800 1600

with baseline subtraction (correct procedure)

LH • Luteinizing hormone

1400 1200 J

..<

~ 1000

,j,:

c:J

z 800 600 400 200 0

How baseline concentration can be managed in phannacokinetic investigations Baseline concentration could be stable or could vary with age, sex, diet, or could have a specific rllythm as for instance with female sexual honnones and their multicomponent dynamic equilibrium (Fig. 1) or with melatonin which shows a quick reversible daily rllythm (Fig. 2) (15, 16). The baseline concentration and its possible rllythms must be carefully ascertained before planning a phannacokinetic investigation. This may be easily done with L-camitine and acetyl-L-eamitine, but it may be difficult to define when it is low or very low. The analytical method must be carefully validated in order to assay baseline concentrations. As the body preserves baseline concentrations and, when impaired by an exogenous dose, it restores previous situations, the time needed to have a complete restoration is an essential parameter in phannacokinetic studies of these substances. When measuring absolute bioavailability or the net area under the curve or urinary excretion, the observed concentrations must be corrected for the presence of a baseline. Several problem are involved in this procedure. In the following examples we have assumed that no compensation occurs between an oral pharmacological dose and endogenous synthesis of the given endogenous substance. When the absolute bioavail-

1200 1000 800

...I

...

:E

c:J

z

FSH - Folliclestimolating hormone

600

400 200 0

600 ~O ...I

...

:E

" C1.

ESTROGENS

400 300

200 100

0 28 24

PROGESTERONE

20 ...I

~

c:J

z

16 12

8 4

0 15

10 115 20 DAY OF MENSTRUAl CYCLE

Fig. 1 : The rhythm of hypophysial and lutcinic hormones governing the menstrual cycle. [Reproduced with permission from Wiele R.L.V., DynmfurthL (1973) : Pbarmacol. Rev., 25, 189-207].

A. Marzo & A. Rescigno, PK of endogenous substances (u)

800

~600 E

.ec:

.r; 400 o

ii

v

~200

011.00

12.00

16.00 20.00

24.00 04.00 08.00

(h)

800

81

We have simulated a whole value measured and 3 different baseline subtractions (n = 10), where A is a baseline value close to SO of the whole measurement, B is a baseline value about 4-times higher than SO of the whole value, and C is a baseline value similar to B, but with data constructed in order to have net values close to each other. The coefficient of variation is 11.2% with the whole value, 14.9% with net value in column A, 32.9% with net value in column B, and 11.5% with the net value in colwnn C. With reference to an endogenous substance possessing a well dermed baseline situation with some fluctuations, not possessing peculiar rhythms, experimental protocols for pbarmacokinetic investigations must consider the following items:

Whether to give an oraldoseinfasting or non-fasting status In humans overnight fasting usually does not affect the baseline situation, thus the fasting should be selected in the protocol In rats, overnight fasting can produce serious modifications due to the fact that rats don't feed during the day. Thus in rats a non-fasting status could be the best choice. 08.00

12.00

16.00 20.00

24.00 04.00 08.00

Time (h)

Fig. 2 : Tuned serum concentration of melatonin in 8 healthy women with normal mcnstroal cycles (filled circles) and 4 women taking contraceptive pills (filled diamonds), (a) on 213 March and (b) on 16/17 March. [Reproduced with permission from Webley G.E., Melel H., Willey K.P. (1985) : J. Endocrino!., 106, 387-394].

400

without traction cedure)

300 1000 ·100=30.00%

with a zero dose and no baseline subtraction (a paradoxical result)

1000 • 100 = 40.00%

baseline (incorrect

subpro-

When investigating comparative bioavailability, the baseline subtraction can generate some problems. Several things may happen; the baseline value can be close to SO of the total value, or higher and in the last case data dispersion is a crucial point as it can produce an arithmetic reason for a bioinequivalence. The examples provided in Table IT should clarify this complex problem.

Baseline measurement in a single dosestudy In humans an appropriate series of blood samplings and a complete urine collection (0-12 h, 12-24 h) the day before the dose assure a well defined baseline (3). The same can be performed in rats, using a group of control animals.

Baseline measurement in a steady-state study When different doses or formulations are compared at steady state, a good baseline observation can be carried out in a period lasting as long as the period for steady-state observations. For instance in a doseproportionality investigation with three dose levels and one baseline observation, four periods each lasting 5-7 days with a randomized cross-over sequence without wash-out can be followed. This procedure allows complete data to be collected in a period of 20-30 days, but a pilot investigation should focus on whether a tailed effect would seriously impair this scheme and how to perform the phannaeokinetic analysis. A lasting tailed effect could be prevented by introducing a wash-out period. In this case the study will last 35-50 days, which is a very long period.

Eur. J. Drug Metab. Pharmacokinet., 1993, No.1

82

In the presence of a marked rhythm, as occurs for

instance with female sexual hormones and melatonin, the experimental protocol should be carefully tailored considering the rhythm, the doses and the variations of plasma concentrations. More complex is the case when a compensatory effect occurs between exogenous dose and endogenous synthesis. This case is examined in more detail in the next section.

where

x = [Xl

X2 ... xnl

is the row vector of the state variables xi (t), X2(t), ..., Xn(t) representing the amount of substance present in each compartment,

KI -k12 ... -kIn]

K = -k.21

K2 .•• -k.2n . . . . [ -knl -kD2 ...

Kn

THEORETICAL CONSIDERATIONS is the n x n matrix of the turnover rates, Ki, and the

transfer rates, kij, and

Appropriate pharmacokinetic analysis

R= [n 12 ... rnl

As examined in previous sections, traditional pharmacokinetic models are not appropriate with endogenous substances. The most appropriate model should be selected case by case, according to the specific behavior of the given endogenous substance. In the following sections this view is extensively discussed, and the mathematical rationales which are basically involved in the model selection are provided.

is the row vector of the production rates, n(t), n(t), ..., rn(t), of each compartment At steady state, dX


Xss°K=R,

A linear pharmacokinetic model If the absorption, distribution, metabolism, and elimin-

ation of a drug follow strictly linear processes, we can describe them by a matrix equation, (17):

:=-XOK+R

[1]

where Xs.r is the value of vector X at steady state. From this last identity, multiplying on the right by T we get

Xss=RoT, where

T=K-I

Table II : Simulation of how the baseline subtraction can modify data dispersion of net values obtained. Baseline A is close to SD of whole values, baseline B is 4-times higher, baseline C is close to B, but less disperse. Values

measured

Mean SD %CV

Baseline A

Net A

18 16 14 25

160 180 190 140 155 135 148 175 162 148

19 22 21 24 23

142 164 176 115 127 116 126 154 138 125

159 17.8 11.2%

21 4.3 20.5%

138 20.7 14.9%

28

Baseline B

Net B

88

72 90

90

103 122 79 82

Baseline

Net

C

C

94

66 70 76 56 66 56 56 76 66 66

110 114 84 89 79

105 78 93

87 18 76 53 54 70 84 55

93 13.6 14.5%

66

94

21.7 32.9%

11.5 12.2%

94

92 99

96 82

65 7.5 11.5%

A. Marzo & A. Rescigno, PKof endogenous substances is the matrix of the permanence times and residence times of the compartmental system (18).

is a diagonal matrix of constant coefficients bi, 1>2..... bu. Equation [1] now becomes:

Suppose now that, while the compartmental system is at steady state, we administer a bolus represented by vector A, where

and ai is the amount of substance administered to compartment i at time t = O. The system Equation [1] does not formally change, but should be completed by the initial condition

=-X· K+ (Xref-X)· B

[4]

The integral of Equation [1] with initial condition [2] is

f.o R(1)· exp[-(t -1)K] • dr [3]

are possible on vector R.

The rateofendogenous production is constant If R does not change with X, then Identity [3] becomes X = (Xu + A) • exp(-tK) + R •

f.o exp[-(t - 1)K] • dr

[5]

Xss • (K + B) = Xref • B Equation 4 can be written in the fonn: dX (it=-(X-XIS) · (K+B)

[2]

X(O) = Xss + A

Several hypotheses

:

and the steady-state value Xss of the state variables is given by:

A = [al 1I2 ... an]

X = (Xss + A)· exp(-tK) +

83

[6]

and its integral is: X - Xu = A • exp [-t (K + B)]

[7]

Again subtraction of the base values from the measured values of the state variables leads to an exponential function. but in this case matrix K has been substituted by matrix K + B. i.e. to the turnover rate Ki of each compartment has been added the coefficient bi measuring the rate at which that compartment is controlled. In other wonts. by subtracting the base values. the compartments whose concentration is regulated by endogenous production will appear to have turnover rates larger than their real value.

Permanence timeandresidence time

thence, X = (Xu + A) • exp(-tK) + R· K-1 [I-exp(-tK)]

By integrating all terms of Equation [6] from 0 to t we

get

and finally, X(t)-X(O)=-

X - Xss = A • exp(-tK) This identity shows that by simply subttaeting from the measured values of the state variables the corresponding base values. one gets the same result as though there were no endogenous production.

The rate ofendogenous production is a linear function ofthe concentration Suppose now that R is proportional to the deviation of X from a fixed value Xref, provided that all elements of R are always positive definite; we can write R = (Xref- X) ·B where

0 0]

bl B= 01>2 [

0

. . . .

o

O •••bn

f.o (X-Xss}dt·

(K + B)

when t increases. X(t) approaches its steady-state value. therefore: -A=- [

o

(X-Xss)dt· (K+ B)

thence [

o

{X-Xss}dt=A. (K+Br l

This identity shows that the element of row i and column j of the inverse of matrix K + B is equal to the area under the curve Xj(t) - Xj(oo) when the drug is administered as a unit bolus at time 0 in compartment I, Matrix (K + Brl is analogous to matrix T = K- 1 fonned by the permanence and residence times of all compartments (19); we may call its elements• pseudo-permanence times and pseudo-residence times. Observe that if we had conducted the experiment

Eur. J. Drug Metab. Pharmacokinet., 1993, No.1

84

with a labeled carrier-free substance. i.e, in trace amount. there would have been no baseline to subtract and B in the above equation would have disappeared.

A simple example Suppose now that a certain endogenous substance has turnover rates 1.0 in the plasma compartment and 0.2 in the peripheral compartment. transfer rates 0.5 from the plasma to the peripheral compartment and 0.2 from the peripheral compartment to the plasma. and that it is synthesized in the plasma at a fixed rate equal to 20. All these rates have the dimension l, (timer with an arbitrary unit With these hypotheses we have the differential equations:

this system is indistinguishable from the previous one. because the steady-state values are the same and the rate of production at steady state is the same. If now we administer the same exogenous dose as before. the rate of endogenous production decreases because xi has increased, but slowly the steady state is reached again. In this case the differential equations are. dxl dt = -Xl + O.2xz + (60 - xi) dxz dt = +O.5XI - O.2xz and from sampling the plasma we get xi =

40 + 9.7 exp (-2.05t) + 0.26exp (-O.15t)

i.e. the same baseline but quite different slopes; for the AUCweget

dxl

(it = -Xl + 0.2xz + 20

(

dxz dt = +O.5xI - O.2xz

(XI(T)-XI(OO»)dT =

i;5

+ ~:i~ = 6.47

which divided by the dose gives the pseudopermanence time 0.65. more than 3-times smaller than

At steady state we can observe the values of Xl and Xl that make:

the real permanence time. The rate ofendogenous production is an almost

dxl = dxz =0 dt dt

linear function ofthe concentration

i.e. the steady-state values.

xi = 40, xz = 100 If now we administer a dose 10 of that same substance as a bolus and we measure it in the plasma. we can fit the resulting values to the bi-exponential function: xr (t) = 40 + 8.92 exp (- l.llt) + 1.08 exp (- 0.09t) after subtraction of its base value and integration from 00 we get.

oto

(

(XI(T)-XI(oo))dT =

~:~ + ~:~: = 20.04

This value divided by the dose is 2.0. i.e. exactly the permanence time in the plasma compartment. as given by the well-known fonnuIa:

1 TI

=

KI 1- klZ e QI KI K2

Now consider a two-compartment system as before. but with the only difference that the endogenous substance in the plasma is synthesized at a rate equal to the difference 60 - xr. At steady state

Suppose again that R is proportional to the deviation of X from a fixed value ~f. but that there is a possibility that at times some elements of vector ~f­ X become negative; in these cases it is realistic to suppose that the corresponding elements of vector R become zero. In this case Equation [1] becomes:

dX • (it=-Xe K + R where vector R· is not a linear function of X any more. This is the case for instance if in the previous example we use a dose that makes xi larger than the reference value 60. Suppose that, everything else being equal. we administer a dose D = 30. The differential equations of this system are now: dxl dt = -Xl

+ 0.2X2

dxl dt = -Xl

+ 0.2X2 + (60 - Xl) for Xl < 60

for Xl ~ 60

dxz dt = +O.5XI - O.2xz with initial conditions xi =

70. xz = 100

A. Marzo & A. Rescigno, PK of endogenous substances With these values of xi and X2, the derivative of Xl is negative while the derivative of X2 is positive. The values of xr and X2 go through three different phases. In the first one, Xl decreases and X2 increases, until dxz/dt = 0; at this point begins the second phase, when both xr and X2 decrease; at some point xr reaches the reference value 60 and the third phase begins, with xi and X2 still decreasing, but with a concomitant endogenous production of the drug; the third phase ends when both dxl/dt = 0 and dx2ldt = 0, i.e. when the steady state is reached. If we were sampling xi during the first two phases, we could get the true phannacokinetic parameters of this system without subtracting any base value; from the values of xi sampled during the third phase we should subtract the base value and find the pseudo-parameters as shown in the previous example. But of course it is extremely difficult to discover where the third phase begins.

The determination of bioavailability

85

where V is the initial volume of distribution. Integration of Equation [9] with condition [10] yields:

Jf

o

(v..

1 (Div -<:(t)+Css) (c(r)-clS)dr= K

By definition

lim c(t) = CIS t~

therefore

Div [ o (c(r)-ea)dr = KV Now suppose that a dose Dos is given per os; Equation [9] becomes

de =-K. (c(t)-css)+ r(t) dt ~ V where r(t) is the rate at which the exogenous substance reaches the plasma. and the initial condition is c(O) = Css. By integration we get c(t)-Css=-K·

The endogenous rate of production is constant

f.o

It is a well known fact (14) that (

o

f.

0

r(r)dr

(c(r) - ea)dr =

K~ (

r(r)dr

[12]

c(r ) dr = F· D

a

where c(t) is the concentration measured after a bolus administration of a dose D, F is the fraction absorbed, and a is the clearance, provided that this last quantity is constant, i.e. it does not depend upon c and t, If K is the turnover rate of the dmg under investigation and r is the constant rate of production, we can write:

de - = -K • c(t) + r dt

[8]

at steady state the derivative is zero, therefore the steady-state value Css of c(t) is r css=K

Now the integral on the right-hand side is the amount of exogenous substance that reaches the plasma, namely F • Dos; therefore:

{AUCo~oa

D: ·lAUCo~oa 1e=F D'

[13]

IV

The AVe from zero to a finite value or t We return now to Equation [11] and write it as

_1. {AU~t I. =_1 (1 _V • c(t) - Css) Div KV Div IV

=_1_ • c(O)- c(t) KV

[14]

c(O)-css

similarly we can write Equation [12] as:

and Equation 8 can be written :

(c(r)-css)dr+ Vi

thence

The AVe from zero to infinity

[

[11]

=-K· (c(t)-css)

If a dose Div is given i. v. as a bolus at time t the initial condition is

Div V

c(O)=css+-

[9] =

0,

r.

1 1 r(r)dr - . {AUCo~t 1 = • ......;,0_ _ Dos os KV Dos The fraction absorbed F is not equal to the ratio . D IV

[10]

Dos •

{AUCo-MIos {AUCo-+t [IV

Eur. J. DrugMetab. Pharmacokinet., 1993, No.1

86

as in Identity [13]. That ratio must be corrected for the tenn c(O)- c(t) c(O)-css in Equation 14, which is easy to compute, and for the tenn

o

o

Area under the curveofendogenous products Two fonnulations are considered bioequivalent, if it can be shown, with a predetermined level of contidence, that

[ r(r)dr [

kinetic parameters that are usually considered as indicators of the rate and extent of absorption, are AUC, Cmu and T max• We shall examine them in sequence.

r(r)dr

< AUCI < 0.80 - AUC2 - 1.25

which is much more difficult to evaluate, unless the absorption is very fast The endogenous rate ofproduction is controlled

We have seen before that when the rate of endogenous production is a linear function of the concentration of the substance in some pools, the system behaves as though its matrix K were substituted by the matrix K + B. We can describe this system with the differential equation

where AUCI and AUC2 are the areas under the curves CI(t) and C2(t), the concentrations observed after admjnistration of each of the two fonnulations in similar conditions. The rationale for this rule is that those areas are supposed to be proportional, inter alia, to the fraction of drug absorbed. The problem here is whether the baseline must be subtracted from CI(t) and C2(t). If we subtract the baseline, remembering Equation 12, we have:

~~ =-K· C + b· (Cref-C(t») where Cref is a fixed value and b is a factor of proportionality, measuring the rate of endogenous production. The steady-state value eu of the concentration is given by K • CBs = b(Cref- eu) therefore the above equation can be written

(

(C2(r)-C2(oo»)dr=

=

thence, with the reasonable hypothesis that dose, volume of distribution and turnover rate are the same for the two fonnulations,

-de =-(K + b)· (c(t) - css) dt The results of the previous section we simply substitute K + b for K.

are still valid if

Because both integrals {AUCo~t}jv

and {AUCo-+t}os

as computed in the previous section, are proportional to K, in the present case they will be proportional to K + b, therefore their ratio remains exactly the same.

The determination of bioequivalence A similar problem arises in the detennination of bioequivalence. Two different fonnulations of the same drug product are considered bioequivalent if their rate and extent of absorption do not show a signfficant difference when administered at the same dose under similar conditions (20). The pharmaco-

Now baseline? from 0 to following

r. r.

what happens if we don't subtract the Of course we cannot carry the integration 00. but if we chose a time t large enough. the approximation should be valid:

FIDI cI(r)dr =--+CI(OO). t o KIVI

o

C2(r)dr =

F2D2 + C2(00).t

K2V2

thence, with the same hypothesis as before.

r. r.

o

o

cI(r)dr C2(r)dr

FI+A F2+A

A. Marzo & A. Rescigno, PKofendogenous substances where A = KIVlcl(co). t 01

KzV2C2(co). t Dz

The larger A is, the closer to one is the ratio of the two AVC's. For FI and F2 we expect some values between 0.1 and 1, and close to each other; for 1, the duration of the observations, we expect a value several times larger than the turnover time, therefore KIt and K2t must be larger than 2 or 3; the dose must be a few times larger than the amount of endogenous substance present; therefore we expect A to be a number of the order of magnitude one. Figure 3 shows some values of the ratio (FI+A)I (F2+A)for different values of A. A large baseline, if not subtracted, makes two fcnnulations appear to be more equivalent than they actually are. The variance of the observed concentrations does not change by subtracting the baseline, but the coefficient of variation does, therefore a large baseline, if not subtracted. simulates a larger level of confidence.

Other parameters usedin bioequivalence studies Other parameters used for the determination of bioequivalence are the value of maximum concentration and the time of maximum concentration. When computing the ratio of the maximum concentration of two different formulations, we have the same problem as in the case of AVC, namely the larger the baseline, the closer to one is this ratio. The time of maximum concentration of course is not influenced by the baseline, except when the baseline fluctuates; but in general this parameter is a 1.4,.------------------,

1.3

~1.2~-

~1.1, 1

. • • . • . • • • • • -- • • • • • . • • • • - ••••

~

-

.

• - ...............•.••.••••.••••.•

87

very poor indicator of the rate of absorption, Another pharmacokinetic parameter has been proposed (21) for evaluating the bioequivalence of two formulations, namely the distance Sz between two plasma concentration curves; this parameter is defined by:

sJr. «1(1)-">(1»): d1]!h l(

(cI(r)+C2(r») dr

The closer Sz is to zero, the more equivalent are the two formulations. The numerator on the right-hand side of course is not influenced by the baseline. The denominator apparently is, but on closer scmtiny it appears that the effect of the baseline can easily be offset In fact Sz is never considered by itself, but compared with the distance between CI(t), respectively C2(t), and a fraction of its standard deviation. Now if the corresponding values of CI(t) and C2(t) are not too different (if they are, bioequivalence is excluded a prion), the denominators in these three distances are very close, and only their numerators affect the result

CONCLUSIONS As extensively discussed, pharmacokinetic and bioavailability investigations with endogenous substances must be carefully planned. Specific problems, such as non-linear processes, the presence of an endogenous pool, gender differences, homeostatic equilibrium and the rule of baseline concentration must be taken into account Consequently, both the experimental protocol and an adequate model should be appropriately selected. In most cases available software cannot manage pharmacokinetic data of endogenous substances. In this review, the authors have presented two different views to focus on two aspects of the problem, the physiological background of these substances and the need for an empirical or a tailored model for pharmacokinetic data analysis. Several cases are discussed in this review, but the whole problem remains a very complex one.

0. 9 ' - - - - - - - -2- - - - -3- - - - - '4 o A

Fig. 3 : Ratio of two AUC's for different values of A. (Fl .. 0.95, F2 .. 0.79).

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