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Determination of Volume of Distribution at Steady State with Complete Consideration of the Kinetics of Protein and Tissue Binding in Linear Pharmacokinetics
Determination of Volume of Distribution at Steady State with Complete Consideration of the Kinetics of Protein and Tissue Binding in Linear Pharmacokinetics LEONID M. BEREZHKOVSKIY Roche Palo Alto LLC, 3431 Hillview Avenue, Palo Alto, California 94304
Received 8 April 2003; revised 6 August 2003; accepted 9 August 2003
ABSTRACT: The assumption of an instant equilibrium between bound and unbound drug fractions is commonly applied in pharmacokinetic calculations. The equation for the calculation of the steady-state volume of distribution Vss from the time curve of drug concentration in plasma after intravenous bolus dose administration, which does not assume an immediate equilibrium and thus incorporates dissociation and association rates of protein and tissue binding, is presented. The equation obtained Vss ¼ (Dose/ AUC)*MRTu looks like the traditional equation, but instead of mean residence time MRT calculated using the total drug concentration in plasma, it contains mean residence time MRTu calculated using the plasma concentration of the unbound drug. The equation connecting MRTu and MRT is derived. If an immediate equilibrium between bound and unbound drug fractions occurs, MRTu and MRT are the same, but in general, MRTu is always smaller than MRT. For drugs with high protein affinity and slow dissociation rate MRTu may be of an order of several hours smaller than MRT, so that Vss can be considerably overestimated in the traditional calculation. ß 2004 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 93:364–374, 2004
Keywords: steady-state volume of distribution; protein and tissue binding kinetics; dissociation rate; mean residence time
INTRODUCTION Traditional equation for calculation of volume of distribution at steady state Vss from the time course of total drug concentration in plasma after intravenous bolus dose administration Vss ¼ (Dose/AUC)*MRT includes an assumption of an immediate equilibrium between bound and unbound drug fractions in plasma. The purpose of this work is to obtain an equation for the steadystate volume of distribution for a linear pharmacokinetic system with complete consideration of the kinetics of protein and tissue binding (no
Correspondence to: Leonid M. Berezhkovskiy (Telephone: 650-855-6398; Fax: 650-855-5020; E-mail: [email protected])
assumption of an instant binding equilibrium) along with the kinetics of a whole system. The values of Vss calculated from the derived equation are compared with that calculated from the traditional one. For drugs with high protein affinity and slow dissociation rate the traditional calculation may yield substantially overestimated values of the volume of distribution at steady state. Volume of distribution at steady state is a commonly calculated parameter in pharmacokinetic studies. It is defined as a ratio of a total quantity of drug in the body Ass to the total concentration of drug in plasma Cp,ss at steady state condition, that is, when the system is subjected to the constant rate drug infusion into plasma, so that all concentrations, which describe the drug distribution in the body become unchanged (reach steady-state values). Thus,
Journal of Pharmaceutical Sciences, Vol. 93, 364–374 (2004) ß 2004 Wiley-Liss, Inc. and the American Pharmacists Association
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JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
Vss ¼ Ab;ss =Cp;ss :
ð1Þ
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
In other words Vss corresponds to the equivalent volume in which a drug is distributed into the body. The quantity of drug in the body at steady state can be calculated as the sum of quantities in each organ (compartment) Ab;ss ¼ S Ct;ss Vt þ Cp;ss Vp , where Ct,ss and Vt are total drug concentration in the organ tissue at steady state and organ volume, respectively, and Vp is the plasma volume. Substituting Ass in eq. 1 presents Vss as a sum of plasma and tissue volume X Vss ¼ Vp þ Ptp Vt ; ð2Þ where Ptp ¼ Ct;ss =Cp;ss is the tissue–plasma partition coefficient of the organ. Equation 2 is used as a starting point for the for the analysis of the influence of protein binding on the distribution volume,1,2 as well as for calculation of Vss from the physicochemical properties of drug and tissues.3 It is a substantial achievement of pharmacokinetics to figure out that such a complicated parameter as Vss (eqs. 1 and 2) can be calculated straightforwardly from the time course of the plasma drug concentration following a single intravenous bolus dose. The history of a problem is well documented in the review article of Wagner.1 One recent issue is the problem of obtaining the steadystate volume of distribution for the nonlinear protein binding;4 Vss is concentration dependent in this case. Wagner8 showed that for a linear pharmacokinetic system Vss can be determined directly from the plasma concentration–time curve following intravenous bolus injection X X 2 Vss ¼ D Ci =l2i = ð3Þ Ci =li ; where Ci are the coefficients and li are the exponents in the polyexponential equation describing plasma concentration Cp(t) following bolus intravenous injection of dose D X C p ðt Þ ¼ Ci expðli tÞ: ð4Þ The number of parameters Ci and li in eqs. 3 and 4 depend on the number of compartments of the n-compartment open mammillary model for which these equations were applied.1 This model considers the pharmacokinetic system as a central compartment (plasma) from which the drug can be eliminated or reversibly transferred to peripheral compartments, and each rate is the first order. Oppenheimer et al.5 and Benet and Galeazzi6 described noncompartamental method of determi-
365
nation of the steady-state volume of distribution directly from Cp(t), which utilizes exclusively areaunder-the-curve calculations 0 1 12 Z1 Z Vss ¼ D tCp ðtÞdt=@ Cp ðtÞdtA 0
0
¼ D*AUMC=ðAUCÞ2 ¼ ðD=AUCÞ*MRT:
ð5Þ
The third and forth equalities in eq. 5 are standard pharmacokinetic values: AUC is the area under plasma concentration–time curve, AUMC is the area under the first moment of the concentration-time curve, and MRT is the mean residence time. They are all defined here through the total drug plasma concentration profile Cp(t), which is obtained from a regular pharmacokinetic study after a single bolus intravenous dose of drug AUC ¼
Z1
Cp ðtÞdt;
ð6Þ
0
AUMC ¼
Z1
tCp ðtÞdt;
ð7Þ
0
MRT ¼
Z1 0
tCp ðtÞdt=
Z1
Cp ðtÞdt ¼ AUMC=AUC:
0
ð8Þ Equation 5 is commonly used in pharmacokinetic calculations. It is interesting to mention that in eq. 3 for Vss the term S Ci/li ¼ AUC and the term S Ci/l2i ¼ AUMC for the concentration–time curve Cp(t) given by the series, eq. 4. Then Wagner’s eq. 3 for Vss is just identical to the model independent eq. 5 X X 2 Vss ¼ D Ci =l2i = Ci =li ¼ D*AUMC=ðAUCÞ2 ¼ ðD=AUCÞ*MRT:
ð9Þ
As emphasized by Wagner,7 there really are no model-independentmeanresidence timesorsteadystate volumes of distribution. Benet and Galeazzi6 pointed that the underlying two most important assumptions for eqs. 5 and 9 are linear kinetics and the exit of drug directly from plasma or central compartment, where drug was administered. Lets consider a simple example shown in Figure 1 and calculate Vss. The drug A just stays in plasma after bolus intravenous administration JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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Figure 1. One compartment model with a simple protein binding reaction.
(one-compartment model). There is one simple protein binding reaction in the system A þ P ¼ Ab with the rate constants k1 ¼ konP and k2 ¼ koff, where kon and koff are the association and dissociation rates of the reaction, and P is the protein concentration, which is assumed to be constant because of an excess of protein. The drug elimination rate kelA is considered proportional to unbound quantity A, where kel is the elimination rate constant. This model is described by two linear differential equations for the quantities of unbound A and bound Ab drug dA=dt ¼ ðk1 þ kel ÞA þ k2 Ab dAb =dt ¼ k1 A k2 Ab ;
ð10Þ
with initial condition A(t ¼ 0) ¼ D, Ab(t ¼ 0) ¼ 0. This linear system of equations can be solved using standard technique, and for the concentrations we get CA ðtÞ ¼ AðtÞ=Vp ¼ ðD=Vp Þ½ðk2 b2 Þ expðb2 tÞ ðk2 b1 Þ expðb1 tÞ=ðb1 b2 Þ
ð11Þ
Cp ðtÞ ¼ ðA þ Ab Þ=Vp ¼ ðD=Vp Þ½ðk1 þ k2 b2 Þ expðb2 tÞ ðk1 þ k2 b1 Þ expðb1 tÞ=ðb1 b2 Þ;
ð12Þ
where b1 and b2 are the roots of the quadratic equation b2 þ (k1 þ k2 þ kel) b þ k2 kel ¼ 0. Calculating AUC and AUMC according to eqs. 6 and 7 with Cp(t) given by eq. 12 and substituting in eq. 5, we obtain h i Vss ¼ Vp 1 þ k1 kel =ðk1 þ k2 Þ2 : ð13Þ Obviously, for this model (just one plasma compartment of volume Vp, and the only first term in eq. 2) we would expect to get plasma volume Vp as the volume of distribution at steady-state Vss. It is reasonable to ask why does eq. 13 have a different value of Vss. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
There is another important assumption in eq. 9 for Vss, which was not mentioned by Benet and Galeazzi,6 but was used straightforward in Wagner’s derivation:8 the elimination rate from central compartment (plasma) was considered proportional to the total drug concentration in plasma. It is commonly assumed in linear pharmacokinetics9 that there is an immediate equilibrium between bound and unbound fractions of drug in plasma, so that the time course of unbound drug concentration Cu(t) follows the total plasma concentration Cp(t) and is calculated simply as Cu ¼ Cp*fu, where the unbound fraction fu is considered to be constant determined by plasma protein and lipid binding. The elimination rate is then calculated as Cl*Cp ¼ Clu *Cu, where Cl is the total clearance of drug from plasma and Clu ¼ Cl/ fu is the clearance of unbound drug. In this situation the elimination rate is considered proportional to both the total and the unbound plasma concentrations. In the model shown in Figure 1 we considered the kinetics of protein binding reaction, not assuming it to be in an instant equilibrium; that is why we did not obtain plasma volume Vp as the steady-state volume of distribution Vss in eq. 13. To make a limiting transition to the case of an immediate equilibrium, we just need to assume that k1, k2 ! 1 with k2/k1 ¼ Kd/P, where Kd ¼ koff/ kon is the equilibrium dissociation constant, then eq. 13 clearly yields the expected value Vss ¼ Vp. To consider kinetics, rather then an instant equilibrium of protein and tissue binding, along with the kinetics of a whole system (drug exchange between organs and drug elimination) appears to be a more general approach to the problem of calculating of the volume of distribution at steady state. In this approach an instant equilibrium condition Cu(t)/Cp(t) ¼ fu ¼ const is not applicable. It seems reasonable to assume that drug elimination rate is proportional to the concentration of unbound fraction,4,10,11 because the body does not eliminate its own proteins, or does it much slower than elimination of a free drug fraction. Thus, in this article, which is correct in most cases, the bound compound is considered as protected from elimination. This assumption is completely in agreement with Wilkinson and Shand physiological approach to hepatic (eliminating organ) drug clearance,12 although it might seem that there is a contradiction here, especially for high extraction ratio drugs, when both bound and free fractions are eliminated. For a drug that does not bind to protein it was suggested12 that elimination rate of drug can be calculated as [ClintQ/(Clint þ
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
Q)]*C ¼ Clu*C, where Clint is the intrinsic clearance of unbound drug, Q is blood flow to the clearing organ, and total drug concentration C ¼ Cu in this case. According to Wilkinson and Shand theory, only unbound (free) drug is being eliminated, but ‘‘the removal of free drug leads to a dissociation of bound drug during passage through the liver and some fraction of this released drug is extracted.’’ Therefore, initially bound drug is involved in the elimination process along with an unbound one. To incorporate this process into the calculation of the elimination rate it was suggested to use the following equation for clearance Cl ¼ [Clint fuQ/(Clint fu þ Q)], so that drug elimination rate equals to Cl*C, where C is the total drug concentration. For high extraction ratio drugs (Clint fu >> Q) clearance approaches the value of eliminating organ blood flow Q, and elimination rate becomes approximately equal to its maximum value Q*C ¼ QCu/fu. For low extraction ratio drugs (Clint fu Q) elimination rate becomes Clint fuC ¼ ClintCu. In both cases elimination rate can be considered either proportional to C or Cu, because an assumption of an immediate equilibrium Cu ¼ fuC was applied. It is important that Wilkinson and Shand did not assume that elimination of bound drug happens straightforward, but it indeed occurs through preliminary instant dissociation with consequent elimination of then free drug. In this work we do not consider that there is an immediate equilibrium between bound and free drug. Free drug is being eliminated from the body and the rate of this process is given by Clu*Cu(t). In terms of the Wilkinson and Shand theory, Clu would be calculated as ClintQ/(Clint þ Q), but for the purpose of determination of Vss we do not need the particular expression for Clu. In this article we determine the steadystate volume of distribution for a linear pharmacokinetic system with drug input and exit directly from plasma or central compartment. Protein and tissue binding were considered kinetically, but were not assumed to be in an instant equilibrium. For this model the following equation for calculation of Vss from plasma concentration– time data following a single intravenous bolus dose was obtained Vss ¼ D*AUMCu =ðAUC*AUCu Þ ¼ ðD=AUCÞ*MRTu ;
ð14Þ
where AUCu, AUMCu, and MRTu are the values calculated using unbound drug concentration Cu(t) in plasma
AUCu ¼
Z1
Cu ðtÞdt
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ð15Þ
0
AUMCu ¼
Z1
t Cu ðtÞdt
ð16Þ
0
MRTu ¼
Z1 0
tCu ðtÞdt=
Z1
Cu ðtÞdt ¼ AUMCu =AUCu
0
ð17Þ Application of eq. 14 gives the anticipated value Vss ¼ Vp for the example shown in Figure 1. Equation 14 differs from eqs. 5 and 9 by using the term MRTu instead of MRT, which would be the same if an immediate equilibrium for plasma protein and tissue binding is assumed. In the next two sections eq. 14 is derived, as well as the equation for MRTu. The obtained results and examples as well as modification of eq. 14 for Vss, when possible direct (not after dissociation) elimination of bound drug occurs along with free drug elimination, are considered in the Discussion section.
EQUATION FOR THE STEADY-STATE VOLUME OF DISTRIBUTION Let us consider a linear pharmacokinetic system with drug input and elimination from the central compartment (plasma) only. Such a system is characterized by a set of concentrations Ci ¼ Ai/Vi, which correspond to different units of volumes Vi , where the drug is distributed, and containing the quantities of drug Ai. Vi is not necessarily the volume of the whole compartment (or body organ), but a volume of the phase with concentration Ci inside the compartment. For instance, it can be a volume of fat in plasma. We do not need to specify possible binding and exchange reactions in this section. Once linear kinetics is considered, the following system of equations for drug quantities, which correspond to a single intravenous bolus administration is applied X dAi =dt ¼ Bij Aj ð18Þ where the matrix B corresponds to concrete mechanism of drug distribution and elimination. We consider only the case of the central drug input. It is also assumed that the drug is irreversibly JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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removed from the system (by a first-order rate process) and the temperature of the system (body) is constant, so that all kinetic rates incorporated in matrix B of eq. 18 are constant. Let the first variable A1 in eq. 18 be the quantity of unbound drug in plasma A1 ¼ Au. The initial conditions for eq. 18 are Au(t ¼ 0) ¼ D, Ai ðt ¼ 0Þ ¼ 0ði 6¼ 1Þ. When the case of continuos drug input into plasma (P.O.), given by the rate R(t), is considered, the first equation of the system 18 becomes X dApo B1j Apo ð19Þ u =dt ¼ j þ RðtÞ; and the initial conditions for the system are Aipo (t ¼ 0). To calculate Vss ¼ Ass/Cp,ss, we need to obtain the steady-state quantity and total plasma concentration for the condition of constant drug input rate R(t) ¼ Ro. We consider, as discussed in the introduction, that elimination occurs only from central compartment and its rate Clu*Cu is proportional to the concentration of unbound drug in plasma. Thus the total quantity of drug in the body at instant t will be Ab ðtÞ ¼ Ro t Clu
Zt
Cpo u ðtÞdt:
ð20Þ
0
The general solution of the system of eq. 18 for the bolus drug input is Ai(t) ¼ Saikexp(lkt),13,14 where lk are the eigenvalues of the matrix B. In this section we are interested only in the concentrations of total Cp(t) and unbound Cu(t) drug fractions in plasma, which thus can be written as X Cp ðtÞ ¼ Ci expðli tÞ; ð21Þ Cu ðtÞ ¼
X
di expðli tÞ:
ð22Þ
For an arbitrary drug input rate R(t) into a linear pharmacokinetic system system, as in eq. 19, a general expression that connects the concentration Cpo(t) generated by the continuous input with the impulse input (intravenous bolus injection) concentration C(t) can be applied po
C ðtÞ ¼ ð1=DÞ
Zt
RðtÞCðt tÞdt:
ð23Þ
0
The equation above is known as Duhamel’s or convolution integral.14 It can be conveniently used in pharmacokinetics to obtain drug absorbtion rates and to derive different equations.15–17 Equation 23 just expresses the fact that each portion of drug, which enters the system, conJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
tributes independently to the total drug concentration. Thus, if a portion R(t)dt entered the body at the time interval from t to t þ dt, then an input to the total concentration at time t, that comes from this portion, which was in the body during the time interval t t, is R(t)dt*C(t t)/D. Here, C(t)/D is the concentration profile for a unit dose of drug. Integration with respect to the whole time interval from 0 to t leads to eq. 23. Using eq. 23 for the case of constant drug input rate R(t) ¼ Ro, from eqs. 21 and 22 we get X Cpo ðCi =li Þ½1 expðli tÞ; ð24Þ p ðtÞ ¼ ðRo =DÞ Cpo u ðtÞ ¼ ðRo =DÞ
X ðdi =li Þ½1 expðli tÞ:
ð25Þ
The value of Clu in eq. 20 for Ab(t) can be expressed from the mass conservation law D ¼ Clu
Z1
Cu ðtÞdt ¼ Clu
X
ðdi =li Þ:
ð26Þ
0
Substituting eq. 25 for Cpo p ðtÞ into eq. 20 and combining with eq. 26, yields h i X X Ab ðtÞ ¼ Ro = ðdi =li Þ * di =l2i ½1 expðli tÞ: ð27Þ At steady state, from eqs. 24 Pand 27, we obtain Cp;ss ¼ Cpo ð t ! 1 Þ ¼ ð R =D Þ =li Þ; Ab;ss ¼ Ab o P ðCi p P ðt ! 1Þ ¼ ½Ro = ðdi =li Þ* di =l2i . Then the steady state volume of distribution is h X i ðCi =li Þ * Vss ¼ Ab;ss =Cp;ss ¼ D= hX i X di =l2i = ðdi =li Þ : Noticing that inPequation above,Paccording to eqs. 21 P and ðCi =li Þ ¼ AUC; ðdi =li Þ ¼ 22,2 P AUCu and di =li = ðdi =li Þ ¼ AUMCu =AUCu , we finally obtain the equation for calculation of the volume of distribution at steady state from the time course of plasma drug concentration following a single intravenous bolus dose D Vss ¼ ðD=AUCÞ AUMCu =AUCu ¼ ðD=AUCÞ*MRTu :
ð28Þ
Equation 28 can also be derived, as shown in the Appendix, without applying polyexponential expressions 21 and 22, but just using general convolution integral, eq. 23, to obtain the steadystate plasma concentration and the total quantity of drug in the body.
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
ON THE CONNECTION BETWEEN MEAN RESIDENCE TIMES CALCULATED USING THE TOTAL AND UNBOUND DRUG CONCENTRATIONS
where Cu(t) ¼ Au(t)/Vpw. At the steady-state condition, that is, dai,ss/ dt ¼ 0, which is also an equilibrium condition for the reactions 29, eq. 30 yields
To obtain the relation between mean residence times calculated using the total and unbound drug concentrations, we need to consider the drug kinetics in plasma. This includes protein binding and partitioning into lipid phase. There are several kinds of proteins in plasma, and albumin is the most abundant among them. Each reactions is characterized by the forward and backward microscopic rate constants kiþ, ki, and may be written as A Ð ai ;
369
ai;ss ¼ Au;ss kþ i =ki :
The unbound drug quantity ratio qu, and the unbound drug concentration ratio fu, defined at steady state condition are X X ai;ss ¼ 1= 1 þ =k qu ¼ Au;ss = Au;ss þ kþ i i ð34Þ fu ¼ Cu;ss =Cp;ss ¼ Au;ss =Vpw = h i X ai;ss =Vp ¼ qu =vw: Au;ss þ
ð29Þ
ð35Þ
where ai is the notation of the compound in bound state. There is also possible drug exchange between plasma and organs, due to unbound drug diffusion.1 For the quantity of bound drug in plasma ai(t) we have the following kinetic equation
Combining eq. 33 with eqs. 34 and 35 yields the connection between areas under the curve for total and unbound drug concentrations
dai =dt ¼ kþ i A u ki a i ;
It is interesting to mention that AUC and AUCu are related through the equilibrium parameter fu, that is, eq. 36 is valid no matter if an instant equilibrium between bound and unbound drug fractions occurs. In other words, while Cu(t) 6¼ fuCp(t) during the time course of a drug, the connection between integrated values (eq. 36) still holds. The same is therefore valid for the values of total and unbound plasma clearance Cl ¼ D/AUC, Clu ¼ D/AUCu, so that Clu ¼ Cl/fu, while elimination rate CluCu(t) 6¼ Cl*Cp(t). To obtain the area under the first moment of the total plasma concentration–time curve, we use eq. 32 for total plasma concentration
ð30Þ
where Au is the quantity of unbound drug in plasma. The initial conditions, which correspond to intravenous bolus dose D are Au(t¼ 0) ¼ D and ai(t ¼ 0) ¼ 0. Integration of eq. 30 with respect to time t from zero to infinity gives kþ i
Z1
Au ðtÞdt ¼
0
k i
Z1
ai ðtÞdt:
ð31Þ
0
The concentration of unbound compound in plasma Cu ¼ Au/Vpw, where Vpw ¼ Vpvw is the volume of plasma water, which contains unbound compound, and vw ¼ Vpw /Vp is the water fraction of total plasma volume. The value of vw is around 0.95.3 The total quantity of drug in plasma is Au þ Sai and therefore the total plasma concentration of drug can be written as X X ai =Vp ¼ vw Au þ ai =Vpw : Cp ðtÞ ¼ Au þ ð32Þ Then for the area under total plasma concentration curve, with an account of eqs. 31 and 32, we have Z1 Z1 X AUC ¼ Cp ðtÞdt ¼ vw Cu ðtÞdt 1þ kþ i =ki dt; 0
0
ð33Þ
AUCu ¼ fu AUC:
AUMC ¼ 1=Vp
ð36Þ
Z1 X t Au þ ai dt:
ð37Þ
0
Multiplying eq. 30 by time t and integrating from zero to infinity with the account of initial conditions and eq. 31, we can express the second term in the equation above through Au(t) Z1 0
tai ðtÞdt ¼
Z1 0
Z1 h 2 i tAu dt þ kþ = k Au dt i i 0
ð38Þ Substitution of eq. 38 in eq. 37 transforms it to Xh 2 i AUMC ¼ AUMCu =fu þ vw AUCu kþ : i = ki JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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BEREZHKOVSKIY
Finally, the equation above combined with eqs. 34, 35, and 36 yields the equation that connects mean residence times calculated using the unbound and total drug concentrations, MRTu ¼ AUMCu/AUCu and MRT ¼ AUMC/AUC Xh X 2 i kþ MRTu ¼ MRT = 1þ kþ i = ki i =ki : ð39Þ The mean residence time MRT in eq. 39 can be calculated from total plasma concentration–time curve as commonly done, while the second term requires the knowledge of possible reactions and their rate constants. In general, MRTu is always smaller than MRT, but if the assumption of an instant equilibrium is made, that is, kiþ, ki ! 1, and kiþ/ki ¼ const, then according to eq. 39, MRTu becomes equal to MRT.
DISCUSSION The obtained eq. 28 for the steady-state volume of distribution contains the mean residence time MRTu, which is calculated using the concentration of unbound drug in plasma. The value of MRTu really equals to the mean time, which the drug spends in the system after intravenous bolus input (assuming that the drug is irreversibly removed from the body), when elimination rate is proportional to the concentration of unbound compound. Indeed, if elimination rate E(t) is EðtÞ ¼ Clu Cu ðtÞ;
ð40Þ
then the quantity of drug that exits the body in the time interval from t to t þ dt equals to E(t)dt. Therefore, the probability to stay in the body during the time interval from t to t þ dt is Z1 EðtÞdt= EðtÞdt; 0
and the average time T that a drug remains in the body can be written as T¼
Z1
tEðtÞdt=
0
Z1
EðtÞdt:
ð41Þ
0
Substitution for E(t) from eq. 40 in eq. 41 yields T ¼ MRTu ¼
Z1
tCu ðtÞdt=
0
¼ AUMCu =AUCu :
Z1
Cu ðtÞdt
It is important to mention that the equation for the steady-state volume of distribution presented in the form ð43Þ Vss ¼ ðD=AUCÞ*T; where T is the true mean residence time given by eq. 41, is correct in the case when both unbound and bound compounds are exposed to the first order elimination, so that EðtÞ ¼ Clu*Cu ðtÞ þ Clb*Cb ðtÞ; where Cb(t) is the plasma concentration of bound compound. The validity of eq. 43 can be proved in the same manner as eq. 28 was derived, but using the equation above for elimination rate and eq. 41 as a general definition of the mean residence time T. Thus, eq. 43 appears to be general for a linear pharmacokinetic system with the central input and elimination. In this equation the mean residence time T should be considered according to the elimination mechanism, that is, calculated from eq. 41 with an appropriate expression for the elimination rate E(t). If elimination rate is just proportional to the unbound drug concentration, then T ¼ MRTu, and MRT calculated using the total plasma concentration does not have clear interpretation in this case, it is related to T through eq. 39. Mean residence time T becomes equal to MRT only if elimination rate can be assumed proportional to the total drug concentration in plasma. This is a valid assumption, as discussed in detail below, for the compounds with fast dissociation rates from proteins. Equations 28 and 39 for Vss do not contain explicitly the rate constants characterizing peripheral kinetics. Peripheral kinetics goes into the equation for Vss through the integrated values of AUC, AUMC, AUCu, and AUMCu calculated using the time course of drug in plasma. Drug kinetics in plasma appears in the equation for Vss directly through the rate constants kiþ, ki, as well as indirectly through areas under the curve. Let us consider a possible difference between MRT and MRTu, which would result in an overestimated values of Vss. To simplify the problem, lets assume that there is just one reaction of protein binding in plasma A þ P ¼ Ab. Then, according to eq. 39 þ MRT MRTu ¼ 1=k 1 = 1 þ k1 =k1 ¼ ð1=koff Þ=ð1 þ Kd =PÞ;
0
ð42Þ
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
ð44Þ
where (as explained in the introduction) koff ¼ k 1, kþ ¼ k P and K ¼ k /k . 1 on d off on
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
The factor 1/(1 þ Kd/P) in the above equation is the value of bound drug fraction fb ¼ 1 fu. Obviously, the difference MRT MRTu will be greater for highly bound compounds (fu ! 0, Kd/ P 1). Then MRT MRTu & 1/koff, and is basically determined by the dissociation rate koff of the bound drug from the protein. The value of dissociation constant can vary significantly from approximately 10 to 105 s1, corresponding to dissociation half-life, ln 2/koff, from less than 1 s to several hours. Usually highaffinity binding, which is characterized by a small value of koff, requires specific orientation of the ligand to interact with the protein. This results in slower on-rates, eventually producing Kd ’ 1– 100 nM. The value of the on-rate commonly varies from 10 to 106 M1s1 (maximum possible value, which corresponds to the Smoluchowski limit, is about 109 M1s1). The value of 1/koff in eq. 44 should be of an order of MRT to have significant influence on the difference between MRT and MRTu; thus, the effect is more likely to be observed for the drugs with shorter MRT values. Let us consider an example of a drug with MRT ¼ 0.5 h. For koff ¼ 103 s1 (1/koff & 17 min), kon ¼ 1.4*102 M1s1 and P ¼ 679 mM (average albumin concentration in human plasma), we get 1/(1 þ Kd/ P) ’ 0.99, and eq. 44 yields MRTu ¼ 13 min. Thus, the correct value of the steady-state volume of distribution, calculated with MRTu instead of MRT, will be 2.3 times smaller than Vss calculated using traditional eq. 5. The rate constants for protein binding are not commonly measured and require complex instrumentation to be obtained.18 The value of koff & 0.04 min1 ¼ 6.7*104 s1 (1/koff ¼ 25 min) was observed by Yuan et al.19 in the study of the kinetics of compound dissociation from rat plasma proteins after adding an excessive amount of dextrane-coated charcoal to equilibrated solution. The time curve obtained for the amount of compound remaining in solution was byexponetial: the fast exponent corresponded to a quick charcoal adsorption of initial free fraction, while the slow one corresponded to the rate of the protein– compound complex dissociation. For this compound the difference between MRT and MRTu should be about 0.5 h. Usually the binding of small molecules, which may have several binding sites on a protein, is rapidly reversible. In this case, MRT and MRTu are virtually the same. The large molecules, which could form several hydrogen bonds binding to the protein, can exhibit the difference of several hours
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between MRT and MRTu. The average time that the drug spends in the body after bolus dose input may be relatively long just because of a slow dissociation rate from protein in plasma, but not necessarily due to peripheral partitioning or low clearance Clu. But the true value of mean residence time, MRTu, could be considerably less than MRT calculated over the total compound concentration in plasma. For example, if MRT ¼ 3 h and MRTu ¼ 1 h, the steady-state volume of distribution calculated using MRT, eq. 5, will be three times greater than the true one calculated with MRTu according to eq. 14. In this situation the comparison of Vss obtained from physicochemical properties,3 which is supposed to generate the value equal to that from eq. 14, with Vss calculated from traditional eq. 5 may not be valid. For the drug that just stays in plasma and participate in one protein binding reaction, as considered in the introduction (Fig. 1), the obtained eq. 13 provides a convenient straightforward estimation of the koff and kel, which could lead to a substantially greater value of Vss than the true one given by eqs. 1 or 14. Equation 13 can be written as h i Vss ¼ Vp 1 þ k1 kel =ðk1 þ k2 Þ2 h i ¼ Vp 1 þ ðkel =koff Þ*ðkd =PÞ=ð1 þ Kd =PÞ2 : ð45Þ For example, for the rat intravenous bolus injection study, if drug total clearance is about 80% of the liver blood flow (medium level clearance), that is Cl ¼ 44 mL/(kg min) and Vp ¼ 0.045 L/kg, we get kel ¼ Cl/(Vpfu) ¼ 0.97/fumin1. If the drug protein binding is 90% ( fu ¼ 0.1), that is, Kd/P ’ 0.1 and koff & 0.04 min1, as observed by Yuan et.al.,19 the calculated value of Vss, according to eq. 45, will be Vss 23Vp. Thus, the steady-state volume of distribution calculated using traditional eq. 5 will overestimate the true value of Vss by an order of magnitude (23 times), suggesting a noticeable tissue distribution of drug, which actually does not happen. It is not possible to predict whether traditional calculation would substantially overestimate the value of Vss based just on the drug extraction ratio. As follows from eqs. 39, 44, or 45, the difference in Vss strongly depends on the dissociation rate koff of the drug from protein, while drug extraction ratio is determined only by relative values of hepatic (or clearing organ) blood flow Q and fuClint ( fu provides equilibrium characteristics Kd ¼ koff/kon, but not koff separately). On the other hand, as mentioned above, the shorter MRT values, or bigger kel values, according JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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BEREZHKOVSKIY
to an example given by eq. 45, would lead to an overestimated values of Vss obtained from traditional calculation. This suggests that a drug, which eliminates faster from the body is more likely to exhibit the difference in Vss calculation. Thus, the drugs with higher intrinsic clearance and that are therefore more likely to be high extraction ratio drugs, could be the ones for which traditional calculation of Vss would yield substantially overestimated values. In general, it is clear that both fast elimination and small value of dissociation rate constant could result in the absence of immediate equilibrium between bound and free drug, so that binding kinetics become important in calculation of the steady-state volume of distribution. But small enough value of dissociation rate constant koff (which also leads to high level of protein binding) is the crucial factor that determines possible significant overestimation of Vss calculated by using traditional eq. 5. The possibility of having dissociation rate constant koff 104 s1 (1/koff 3 h) can be estimated using transition state theory.20 The dissociation rate constant of the compound–protein complex is calculated as koff ¼ ðo=2pÞ expðEo =kTo Þ;
ð46Þ
where o is the oscillation frequency corresponding to the potential well created by compound– protein interaction, Eo is the energy of dissociation, k is the Boltzmann’s constant, and To is the absolute temperature. Lets approximate the interaction potential all the way to the top of the barrier as quadratic, that is, E(x) ¼ mo2x2/2, where x is the distance from the bottom of potential well (position of minimal energy) and m is the mass of the compound molecule. Assum˚ corresponds to the ing that deviation of xo ’ 1 A energy of the dissociation barrier Eo (so that dissociation occurs), we get Eo ¼ mo2x2o /2, or o ¼ (2Eo/m)1/2/xo. Then eq. 46 becomes koff ¼ ð2pxo Þ1 ð2Eo =mÞ1=2 expðEo =kTo Þ:
ð47Þ
For the compound with molecular weight of 350 g/mol and the dissociation energy Eo ¼ 22 kcal/ mol eq. 47 yields koff ¼ 6.8 105 s1. The value of dissociation energy of 22 kcal/mol is about the energy of three strong hydrogen bonds. For example, the formation of three or more hydrogen bonds commonly occurs for peptide binding to the major histocompatibility complex proteins, long dissociation half-life t1/2 ¼ ln2/koff 1 to 100 h is not unusual for this system.21 Thus, for the comJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
pound that could possibly form three strong hydrogen bonds binding to plasma proteins, it is likely to have the value of dissociation rate constant small to the extent that would lead to a substantial overestimation of the steady-state volume of distribution and the mean residence time calculated using traditional eqs. 5 and 8.
CONCLUSION The equation for calculation of volume of distribution at steady-state, Vss, with complete consideration of the kinetics of protein and tissue binding is obtained. Traditional equations for Vss and mean residence time, MRT, are based on the assumption of an instant equilibrium between the bound and free drug in plasma. It is shown that for drugs with high protein affinity and slow dissociation rate constant (koff 103 s1) the suggested equations predict the steady-state volume of distribution and mean residence time considerably less than that obtained from the traditional calculation. Drug binding to the protein with the formation of three or more strong hydrogen bond would likely result in such a low value of the dissociation rate constant. Drugs with high intrinsic clearance are more likely to yield overestimated values of Vss and MRT calculated by using the traditional equations. They require not as small values of koff as low extraction ratio drugs for a considerable deviation from an instant binding equilibrium to occur.
ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. Bill Fitch (Roche Palo Alto) for assistance in the preparation of this manuscript.
APPENDIX According to a general eq. 23, the concentration of drug Cpo(t) in plasma, when the drug is administered into plasma at constant input rate R(t) ¼ Ro is written as po
C ðtÞ ¼ ðRo =DÞ
Zt
Cðt tÞdt;
ð48Þ
0
where C(t) is the drug concentration in plasma after bolus input of the drug dose D.
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
The steady-state volume of distribution is calculated according to eq. 1, Vss ¼ Ab,ss/Cp,ss. We consider the case when drug elimination occurs only from central compartment and its elimination rate Clu*Cu is proportional to the concentration Cu of unbound drug in plasma. Then according to the mass conservation law D ¼ Clu
Z1
Let us obtain the quantity of drug in the system at steady state. At instant t it can be written as Ab ðtÞ ¼ Ro t Clu
Zt
0 0 Cpo u ðt Þdt :
ð50Þ
According to eq. 48, the integral in the equation above can be written as 2 3 Zt Zt Zt 0 0 4 Cu ðt0 tÞdtÞ5dt0 : ð51Þ Cpo u ðt Þdt ¼ ðRo =DÞ 0
0
Cu ðt tÞdtÞdt ¼
0
Zt0
Cu ð xÞdx
0
Z1
Cu ð xÞdx
Z1
Cu ð xÞdx
t0
0
¼ D=Clu
Cp ðt tÞdt
0
Z1
Cp ð xÞdx ¼ Ro AUC=D
ð55Þ
0
Finally, from eqs. 54 and 55 the desired equation for the steady state volume of distribution is obtained ¼ ðD=AUCÞ MRTu : Equation 54 written as MRTu ¼ Ab,ss/Ro can be used to calculate the mean residence time from the steady-state condition.
REFERENCES
0
Using the new variable x ¼ t0 t and the mass balance condition, eq. 49, the integral in the brackets in eq. 51 can be transformed to
¼
Z1
Vss ¼ Ab;ss =Cp;ss ¼ D*AUMCu =ðAUC*AUCu Þ
0
Zt0
Cp;ss ¼ ðRo =DÞ
ð49Þ
0
0
The total concentration of plasma at steady state can be expressed according to eq. 48 as
¼ ðRo =DÞ Cu ðtÞdt ¼ Clu AUCu :
373
Z1
Cu ð xÞdx:
ð52Þ
t0
By combining eqs. 51 and 52 and substituting into eq. 50, this equation becomes 2 3 Z t Z1 Ab ðtÞ ¼ ðRo =AUCu Þ 4 Cu ð xÞdx5dt0 : 0
ð53Þ
t0
Integration in eq. 53 can be done by substituting Cu(x) with Cu(x)y(x t0 ), where y(xt0 ) is the step function, y(x t0 ) ¼ 1 for x t0 and y(x t0 ) ¼ 0 for x < t0 . Then integration with respect to x can be carried out from 0 to infinity. Integrating first with respect to t0 from 0 to 8, and then with respect to x, finally yields Ab;ss ¼ Ab ðt ! 1Þ ¼ Ro AUMCu =AUCu :
ð54Þ
1. Wagner JG. 1981. History of pharmacokinetics. Pharmacol Ther 12(3):537–562. 2. Jusco JW, Chiang ST. 1982. Distribution volume related to body weight and protein binding. J Pharm Sci 71(4):469–470. 3. Poulin P, Theil F-P. 2002. Prediction of pharmacokinetics prior to in vivo studies. 1. Mechanismbased prediction of volume of distribution. J Pharm Sci 91(1):129–156. 4. Cheng H, Gillespie R. 1996. Volumes of distribution and mean residence time of drugs with linear tissue distribution and binding and nonlinear protein binding. J Pharmacokinet Biopharm 24(4):389– 402. 5. Oppenheimer JH, Schwartz JL, Surks MI. 1975. Determination of common parameters of iodothyronine metabolism and distribution in man by noncompartmental analysis. J Clin Edocrinol Metab 41:319–324. 6. Benet LZ, Galeazzi RL. 1979. Noncompartmental determination of the steady-state volume of distribution. J Pharm Sci 68(8):1071–1074. 7. Wagner JG. 1988. Types of mean residence times. Biopharm Drug D ispos 9(1):41–57. 8. Wagner JG. 1976. Linear pharmacokinetic equations allowing direct calculation of many needed pharmacokinetic parameters from the coefficients and exponents of polyexponential equations which have been fitted to the data. J Pharmacokinet Biopharm 4(5):443–467. 9. Rowland M, Tozer TN. 1995. Clinical pharmacokinetics, concepts and applications, 3rd ed. Baltimore: Williams & Wilkins. pp 179–180. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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10. Wagner JG. 1985. Effect of protein binding on steady-state equations. J Pharmacokinet Biopharm 13(5):559–560. 11. McNamara PJ, Gibaldi M, Stoeckel K. 1983. Volume of distribution for a drug (ceftrioxone) exhibiting concentration dependent protein binding. Eur J Clin Pharmacol 25:399–405. 12. Wilkinson GR, Shand DJ. 1975. A physiological approach to hepatic drug clearance. Clin Pharmacol Ther 18(4):377–390. 13. Veng-Pedersen P. 1980. Model-independent method of analyzing input in linear pharmacokinetic systems having polyexponential impulse response I: Theoretical analysis. J Pharm Sci 69(3):298– 305. 14. Korn GA, Korn TM. 1961. Mathematical handbook for scientists and engeneers. New York: McGrawHill, Inc. 15. Vaughan DP, Tucker GT. 1976. General derivation of the ideal intravenous drug input required to achieve and maintain a constant plasma concen-
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
16. 17.
18.
19. 20. 21.
tration. Theoretical application to lignocaine thereapy. Eur J Clin Pharmacol 10:433–440. Cutler DJ. 1978. Linear system analysis in pharmacokinetics. J Pharmacokinet Biopharm 6:265–282. Veng-Pedersen P. 1980. Novel deconvolution method for linear pharmacokinetic systems with polyexponential impulse response. J Pharm Sci 69(3):312–318. Markgren P-O, Schaal W, Hamalainen M, Karlen A, Hallberg A, Samuelsson B, Danielson UH. 2002. Relationship between structure and interaction kinetics for HIV-1 protease inhibitors. J Med Chem 45:5430–5439. Yuan J, Yang DC, Birkmeier J, Stolzenbach J. 1995. J Pharmacokinet Biopharm 23(1):41–55. Levine I. 1978. Physical chemistry. New York: McGraw-Hill, Inc. Berezhkovskiy LM. 1999. The analysis of peptide affinity and its binding kinetics to DR1DW1 major histocompatibility complex protein. J Biophys Chem 77:183–194.
Received 8 April 2003; revised 6 August 2003; accepted 9 August 2003
ABSTRACT: The assumption of an instant equilibrium between bound and unbound drug fractions is commonly applied in pharmacokinetic calculations. The equation for the calculation of the steady-state volume of distribution Vss from the time curve of drug concentration in plasma after intravenous bolus dose administration, which does not assume an immediate equilibrium and thus incorporates dissociation and association rates of protein and tissue binding, is presented. The equation obtained Vss ¼ (Dose/ AUC)*MRTu looks like the traditional equation, but instead of mean residence time MRT calculated using the total drug concentration in plasma, it contains mean residence time MRTu calculated using the plasma concentration of the unbound drug. The equation connecting MRTu and MRT is derived. If an immediate equilibrium between bound and unbound drug fractions occurs, MRTu and MRT are the same, but in general, MRTu is always smaller than MRT. For drugs with high protein affinity and slow dissociation rate MRTu may be of an order of several hours smaller than MRT, so that Vss can be considerably overestimated in the traditional calculation. ß 2004 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 93:364–374, 2004
Keywords: steady-state volume of distribution; protein and tissue binding kinetics; dissociation rate; mean residence time
INTRODUCTION Traditional equation for calculation of volume of distribution at steady state Vss from the time course of total drug concentration in plasma after intravenous bolus dose administration Vss ¼ (Dose/AUC)*MRT includes an assumption of an immediate equilibrium between bound and unbound drug fractions in plasma. The purpose of this work is to obtain an equation for the steadystate volume of distribution for a linear pharmacokinetic system with complete consideration of the kinetics of protein and tissue binding (no
Correspondence to: Leonid M. Berezhkovskiy (Telephone: 650-855-6398; Fax: 650-855-5020; E-mail: [email protected])
assumption of an instant binding equilibrium) along with the kinetics of a whole system. The values of Vss calculated from the derived equation are compared with that calculated from the traditional one. For drugs with high protein affinity and slow dissociation rate the traditional calculation may yield substantially overestimated values of the volume of distribution at steady state. Volume of distribution at steady state is a commonly calculated parameter in pharmacokinetic studies. It is defined as a ratio of a total quantity of drug in the body Ass to the total concentration of drug in plasma Cp,ss at steady state condition, that is, when the system is subjected to the constant rate drug infusion into plasma, so that all concentrations, which describe the drug distribution in the body become unchanged (reach steady-state values). Thus,
Journal of Pharmaceutical Sciences, Vol. 93, 364–374 (2004) ß 2004 Wiley-Liss, Inc. and the American Pharmacists Association
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JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
Vss ¼ Ab;ss =Cp;ss :
ð1Þ
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
In other words Vss corresponds to the equivalent volume in which a drug is distributed into the body. The quantity of drug in the body at steady state can be calculated as the sum of quantities in each organ (compartment) Ab;ss ¼ S Ct;ss Vt þ Cp;ss Vp , where Ct,ss and Vt are total drug concentration in the organ tissue at steady state and organ volume, respectively, and Vp is the plasma volume. Substituting Ass in eq. 1 presents Vss as a sum of plasma and tissue volume X Vss ¼ Vp þ Ptp Vt ; ð2Þ where Ptp ¼ Ct;ss =Cp;ss is the tissue–plasma partition coefficient of the organ. Equation 2 is used as a starting point for the for the analysis of the influence of protein binding on the distribution volume,1,2 as well as for calculation of Vss from the physicochemical properties of drug and tissues.3 It is a substantial achievement of pharmacokinetics to figure out that such a complicated parameter as Vss (eqs. 1 and 2) can be calculated straightforwardly from the time course of the plasma drug concentration following a single intravenous bolus dose. The history of a problem is well documented in the review article of Wagner.1 One recent issue is the problem of obtaining the steadystate volume of distribution for the nonlinear protein binding;4 Vss is concentration dependent in this case. Wagner8 showed that for a linear pharmacokinetic system Vss can be determined directly from the plasma concentration–time curve following intravenous bolus injection X X 2 Vss ¼ D Ci =l2i = ð3Þ Ci =li ; where Ci are the coefficients and li are the exponents in the polyexponential equation describing plasma concentration Cp(t) following bolus intravenous injection of dose D X C p ðt Þ ¼ Ci expðli tÞ: ð4Þ The number of parameters Ci and li in eqs. 3 and 4 depend on the number of compartments of the n-compartment open mammillary model for which these equations were applied.1 This model considers the pharmacokinetic system as a central compartment (plasma) from which the drug can be eliminated or reversibly transferred to peripheral compartments, and each rate is the first order. Oppenheimer et al.5 and Benet and Galeazzi6 described noncompartamental method of determi-
365
nation of the steady-state volume of distribution directly from Cp(t), which utilizes exclusively areaunder-the-curve calculations 0 1 12 Z1 Z Vss ¼ D tCp ðtÞdt=@ Cp ðtÞdtA 0
0
¼ D*AUMC=ðAUCÞ2 ¼ ðD=AUCÞ*MRT:
ð5Þ
The third and forth equalities in eq. 5 are standard pharmacokinetic values: AUC is the area under plasma concentration–time curve, AUMC is the area under the first moment of the concentration-time curve, and MRT is the mean residence time. They are all defined here through the total drug plasma concentration profile Cp(t), which is obtained from a regular pharmacokinetic study after a single bolus intravenous dose of drug AUC ¼
Z1
Cp ðtÞdt;
ð6Þ
0
AUMC ¼
Z1
tCp ðtÞdt;
ð7Þ
0
MRT ¼
Z1 0
tCp ðtÞdt=
Z1
Cp ðtÞdt ¼ AUMC=AUC:
0
ð8Þ Equation 5 is commonly used in pharmacokinetic calculations. It is interesting to mention that in eq. 3 for Vss the term S Ci/li ¼ AUC and the term S Ci/l2i ¼ AUMC for the concentration–time curve Cp(t) given by the series, eq. 4. Then Wagner’s eq. 3 for Vss is just identical to the model independent eq. 5 X X 2 Vss ¼ D Ci =l2i = Ci =li ¼ D*AUMC=ðAUCÞ2 ¼ ðD=AUCÞ*MRT:
ð9Þ
As emphasized by Wagner,7 there really are no model-independentmeanresidence timesorsteadystate volumes of distribution. Benet and Galeazzi6 pointed that the underlying two most important assumptions for eqs. 5 and 9 are linear kinetics and the exit of drug directly from plasma or central compartment, where drug was administered. Lets consider a simple example shown in Figure 1 and calculate Vss. The drug A just stays in plasma after bolus intravenous administration JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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Figure 1. One compartment model with a simple protein binding reaction.
(one-compartment model). There is one simple protein binding reaction in the system A þ P ¼ Ab with the rate constants k1 ¼ konP and k2 ¼ koff, where kon and koff are the association and dissociation rates of the reaction, and P is the protein concentration, which is assumed to be constant because of an excess of protein. The drug elimination rate kelA is considered proportional to unbound quantity A, where kel is the elimination rate constant. This model is described by two linear differential equations for the quantities of unbound A and bound Ab drug dA=dt ¼ ðk1 þ kel ÞA þ k2 Ab dAb =dt ¼ k1 A k2 Ab ;
ð10Þ
with initial condition A(t ¼ 0) ¼ D, Ab(t ¼ 0) ¼ 0. This linear system of equations can be solved using standard technique, and for the concentrations we get CA ðtÞ ¼ AðtÞ=Vp ¼ ðD=Vp Þ½ðk2 b2 Þ expðb2 tÞ ðk2 b1 Þ expðb1 tÞ=ðb1 b2 Þ
ð11Þ
Cp ðtÞ ¼ ðA þ Ab Þ=Vp ¼ ðD=Vp Þ½ðk1 þ k2 b2 Þ expðb2 tÞ ðk1 þ k2 b1 Þ expðb1 tÞ=ðb1 b2 Þ;
ð12Þ
where b1 and b2 are the roots of the quadratic equation b2 þ (k1 þ k2 þ kel) b þ k2 kel ¼ 0. Calculating AUC and AUMC according to eqs. 6 and 7 with Cp(t) given by eq. 12 and substituting in eq. 5, we obtain h i Vss ¼ Vp 1 þ k1 kel =ðk1 þ k2 Þ2 : ð13Þ Obviously, for this model (just one plasma compartment of volume Vp, and the only first term in eq. 2) we would expect to get plasma volume Vp as the volume of distribution at steady-state Vss. It is reasonable to ask why does eq. 13 have a different value of Vss. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
There is another important assumption in eq. 9 for Vss, which was not mentioned by Benet and Galeazzi,6 but was used straightforward in Wagner’s derivation:8 the elimination rate from central compartment (plasma) was considered proportional to the total drug concentration in plasma. It is commonly assumed in linear pharmacokinetics9 that there is an immediate equilibrium between bound and unbound fractions of drug in plasma, so that the time course of unbound drug concentration Cu(t) follows the total plasma concentration Cp(t) and is calculated simply as Cu ¼ Cp*fu, where the unbound fraction fu is considered to be constant determined by plasma protein and lipid binding. The elimination rate is then calculated as Cl*Cp ¼ Clu *Cu, where Cl is the total clearance of drug from plasma and Clu ¼ Cl/ fu is the clearance of unbound drug. In this situation the elimination rate is considered proportional to both the total and the unbound plasma concentrations. In the model shown in Figure 1 we considered the kinetics of protein binding reaction, not assuming it to be in an instant equilibrium; that is why we did not obtain plasma volume Vp as the steady-state volume of distribution Vss in eq. 13. To make a limiting transition to the case of an immediate equilibrium, we just need to assume that k1, k2 ! 1 with k2/k1 ¼ Kd/P, where Kd ¼ koff/ kon is the equilibrium dissociation constant, then eq. 13 clearly yields the expected value Vss ¼ Vp. To consider kinetics, rather then an instant equilibrium of protein and tissue binding, along with the kinetics of a whole system (drug exchange between organs and drug elimination) appears to be a more general approach to the problem of calculating of the volume of distribution at steady state. In this approach an instant equilibrium condition Cu(t)/Cp(t) ¼ fu ¼ const is not applicable. It seems reasonable to assume that drug elimination rate is proportional to the concentration of unbound fraction,4,10,11 because the body does not eliminate its own proteins, or does it much slower than elimination of a free drug fraction. Thus, in this article, which is correct in most cases, the bound compound is considered as protected from elimination. This assumption is completely in agreement with Wilkinson and Shand physiological approach to hepatic (eliminating organ) drug clearance,12 although it might seem that there is a contradiction here, especially for high extraction ratio drugs, when both bound and free fractions are eliminated. For a drug that does not bind to protein it was suggested12 that elimination rate of drug can be calculated as [ClintQ/(Clint þ
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
Q)]*C ¼ Clu*C, where Clint is the intrinsic clearance of unbound drug, Q is blood flow to the clearing organ, and total drug concentration C ¼ Cu in this case. According to Wilkinson and Shand theory, only unbound (free) drug is being eliminated, but ‘‘the removal of free drug leads to a dissociation of bound drug during passage through the liver and some fraction of this released drug is extracted.’’ Therefore, initially bound drug is involved in the elimination process along with an unbound one. To incorporate this process into the calculation of the elimination rate it was suggested to use the following equation for clearance Cl ¼ [Clint fuQ/(Clint fu þ Q)], so that drug elimination rate equals to Cl*C, where C is the total drug concentration. For high extraction ratio drugs (Clint fu >> Q) clearance approaches the value of eliminating organ blood flow Q, and elimination rate becomes approximately equal to its maximum value Q*C ¼ QCu/fu. For low extraction ratio drugs (Clint fu Q) elimination rate becomes Clint fuC ¼ ClintCu. In both cases elimination rate can be considered either proportional to C or Cu, because an assumption of an immediate equilibrium Cu ¼ fuC was applied. It is important that Wilkinson and Shand did not assume that elimination of bound drug happens straightforward, but it indeed occurs through preliminary instant dissociation with consequent elimination of then free drug. In this work we do not consider that there is an immediate equilibrium between bound and free drug. Free drug is being eliminated from the body and the rate of this process is given by Clu*Cu(t). In terms of the Wilkinson and Shand theory, Clu would be calculated as ClintQ/(Clint þ Q), but for the purpose of determination of Vss we do not need the particular expression for Clu. In this article we determine the steadystate volume of distribution for a linear pharmacokinetic system with drug input and exit directly from plasma or central compartment. Protein and tissue binding were considered kinetically, but were not assumed to be in an instant equilibrium. For this model the following equation for calculation of Vss from plasma concentration– time data following a single intravenous bolus dose was obtained Vss ¼ D*AUMCu =ðAUC*AUCu Þ ¼ ðD=AUCÞ*MRTu ;
ð14Þ
where AUCu, AUMCu, and MRTu are the values calculated using unbound drug concentration Cu(t) in plasma
AUCu ¼
Z1
Cu ðtÞdt
367
ð15Þ
0
AUMCu ¼
Z1
t Cu ðtÞdt
ð16Þ
0
MRTu ¼
Z1 0
tCu ðtÞdt=
Z1
Cu ðtÞdt ¼ AUMCu =AUCu
0
ð17Þ Application of eq. 14 gives the anticipated value Vss ¼ Vp for the example shown in Figure 1. Equation 14 differs from eqs. 5 and 9 by using the term MRTu instead of MRT, which would be the same if an immediate equilibrium for plasma protein and tissue binding is assumed. In the next two sections eq. 14 is derived, as well as the equation for MRTu. The obtained results and examples as well as modification of eq. 14 for Vss, when possible direct (not after dissociation) elimination of bound drug occurs along with free drug elimination, are considered in the Discussion section.
EQUATION FOR THE STEADY-STATE VOLUME OF DISTRIBUTION Let us consider a linear pharmacokinetic system with drug input and elimination from the central compartment (plasma) only. Such a system is characterized by a set of concentrations Ci ¼ Ai/Vi, which correspond to different units of volumes Vi , where the drug is distributed, and containing the quantities of drug Ai. Vi is not necessarily the volume of the whole compartment (or body organ), but a volume of the phase with concentration Ci inside the compartment. For instance, it can be a volume of fat in plasma. We do not need to specify possible binding and exchange reactions in this section. Once linear kinetics is considered, the following system of equations for drug quantities, which correspond to a single intravenous bolus administration is applied X dAi =dt ¼ Bij Aj ð18Þ where the matrix B corresponds to concrete mechanism of drug distribution and elimination. We consider only the case of the central drug input. It is also assumed that the drug is irreversibly JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
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removed from the system (by a first-order rate process) and the temperature of the system (body) is constant, so that all kinetic rates incorporated in matrix B of eq. 18 are constant. Let the first variable A1 in eq. 18 be the quantity of unbound drug in plasma A1 ¼ Au. The initial conditions for eq. 18 are Au(t ¼ 0) ¼ D, Ai ðt ¼ 0Þ ¼ 0ði 6¼ 1Þ. When the case of continuos drug input into plasma (P.O.), given by the rate R(t), is considered, the first equation of the system 18 becomes X dApo B1j Apo ð19Þ u =dt ¼ j þ RðtÞ; and the initial conditions for the system are Aipo (t ¼ 0). To calculate Vss ¼ Ass/Cp,ss, we need to obtain the steady-state quantity and total plasma concentration for the condition of constant drug input rate R(t) ¼ Ro. We consider, as discussed in the introduction, that elimination occurs only from central compartment and its rate Clu*Cu is proportional to the concentration of unbound drug in plasma. Thus the total quantity of drug in the body at instant t will be Ab ðtÞ ¼ Ro t Clu
Zt
Cpo u ðtÞdt:
ð20Þ
0
The general solution of the system of eq. 18 for the bolus drug input is Ai(t) ¼ Saikexp(lkt),13,14 where lk are the eigenvalues of the matrix B. In this section we are interested only in the concentrations of total Cp(t) and unbound Cu(t) drug fractions in plasma, which thus can be written as X Cp ðtÞ ¼ Ci expðli tÞ; ð21Þ Cu ðtÞ ¼
X
di expðli tÞ:
ð22Þ
For an arbitrary drug input rate R(t) into a linear pharmacokinetic system system, as in eq. 19, a general expression that connects the concentration Cpo(t) generated by the continuous input with the impulse input (intravenous bolus injection) concentration C(t) can be applied po
C ðtÞ ¼ ð1=DÞ
Zt
RðtÞCðt tÞdt:
ð23Þ
0
The equation above is known as Duhamel’s or convolution integral.14 It can be conveniently used in pharmacokinetics to obtain drug absorbtion rates and to derive different equations.15–17 Equation 23 just expresses the fact that each portion of drug, which enters the system, conJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
tributes independently to the total drug concentration. Thus, if a portion R(t)dt entered the body at the time interval from t to t þ dt, then an input to the total concentration at time t, that comes from this portion, which was in the body during the time interval t t, is R(t)dt*C(t t)/D. Here, C(t)/D is the concentration profile for a unit dose of drug. Integration with respect to the whole time interval from 0 to t leads to eq. 23. Using eq. 23 for the case of constant drug input rate R(t) ¼ Ro, from eqs. 21 and 22 we get X Cpo ðCi =li Þ½1 expðli tÞ; ð24Þ p ðtÞ ¼ ðRo =DÞ Cpo u ðtÞ ¼ ðRo =DÞ
X ðdi =li Þ½1 expðli tÞ:
ð25Þ
The value of Clu in eq. 20 for Ab(t) can be expressed from the mass conservation law D ¼ Clu
Z1
Cu ðtÞdt ¼ Clu
X
ðdi =li Þ:
ð26Þ
0
Substituting eq. 25 for Cpo p ðtÞ into eq. 20 and combining with eq. 26, yields h i X X Ab ðtÞ ¼ Ro = ðdi =li Þ * di =l2i ½1 expðli tÞ: ð27Þ At steady state, from eqs. 24 Pand 27, we obtain Cp;ss ¼ Cpo ð t ! 1 Þ ¼ ð R =D Þ =li Þ; Ab;ss ¼ Ab o P ðCi p P ðt ! 1Þ ¼ ½Ro = ðdi =li Þ* di =l2i . Then the steady state volume of distribution is h X i ðCi =li Þ * Vss ¼ Ab;ss =Cp;ss ¼ D= hX i X di =l2i = ðdi =li Þ : Noticing that inPequation above,Paccording to eqs. 21 P and ðCi =li Þ ¼ AUC; ðdi =li Þ ¼ 22,2 P AUCu and di =li = ðdi =li Þ ¼ AUMCu =AUCu , we finally obtain the equation for calculation of the volume of distribution at steady state from the time course of plasma drug concentration following a single intravenous bolus dose D Vss ¼ ðD=AUCÞ AUMCu =AUCu ¼ ðD=AUCÞ*MRTu :
ð28Þ
Equation 28 can also be derived, as shown in the Appendix, without applying polyexponential expressions 21 and 22, but just using general convolution integral, eq. 23, to obtain the steadystate plasma concentration and the total quantity of drug in the body.
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
ON THE CONNECTION BETWEEN MEAN RESIDENCE TIMES CALCULATED USING THE TOTAL AND UNBOUND DRUG CONCENTRATIONS
where Cu(t) ¼ Au(t)/Vpw. At the steady-state condition, that is, dai,ss/ dt ¼ 0, which is also an equilibrium condition for the reactions 29, eq. 30 yields
To obtain the relation between mean residence times calculated using the total and unbound drug concentrations, we need to consider the drug kinetics in plasma. This includes protein binding and partitioning into lipid phase. There are several kinds of proteins in plasma, and albumin is the most abundant among them. Each reactions is characterized by the forward and backward microscopic rate constants kiþ, ki, and may be written as A Ð ai ;
369
ai;ss ¼ Au;ss kþ i =ki :
The unbound drug quantity ratio qu, and the unbound drug concentration ratio fu, defined at steady state condition are X X ai;ss ¼ 1= 1 þ =k qu ¼ Au;ss = Au;ss þ kþ i i ð34Þ fu ¼ Cu;ss =Cp;ss ¼ Au;ss =Vpw = h i X ai;ss =Vp ¼ qu =vw: Au;ss þ
ð29Þ
ð35Þ
where ai is the notation of the compound in bound state. There is also possible drug exchange between plasma and organs, due to unbound drug diffusion.1 For the quantity of bound drug in plasma ai(t) we have the following kinetic equation
Combining eq. 33 with eqs. 34 and 35 yields the connection between areas under the curve for total and unbound drug concentrations
dai =dt ¼ kþ i A u ki a i ;
It is interesting to mention that AUC and AUCu are related through the equilibrium parameter fu, that is, eq. 36 is valid no matter if an instant equilibrium between bound and unbound drug fractions occurs. In other words, while Cu(t) 6¼ fuCp(t) during the time course of a drug, the connection between integrated values (eq. 36) still holds. The same is therefore valid for the values of total and unbound plasma clearance Cl ¼ D/AUC, Clu ¼ D/AUCu, so that Clu ¼ Cl/fu, while elimination rate CluCu(t) 6¼ Cl*Cp(t). To obtain the area under the first moment of the total plasma concentration–time curve, we use eq. 32 for total plasma concentration
ð30Þ
where Au is the quantity of unbound drug in plasma. The initial conditions, which correspond to intravenous bolus dose D are Au(t¼ 0) ¼ D and ai(t ¼ 0) ¼ 0. Integration of eq. 30 with respect to time t from zero to infinity gives kþ i
Z1
Au ðtÞdt ¼
0
k i
Z1
ai ðtÞdt:
ð31Þ
0
The concentration of unbound compound in plasma Cu ¼ Au/Vpw, where Vpw ¼ Vpvw is the volume of plasma water, which contains unbound compound, and vw ¼ Vpw /Vp is the water fraction of total plasma volume. The value of vw is around 0.95.3 The total quantity of drug in plasma is Au þ Sai and therefore the total plasma concentration of drug can be written as X X ai =Vp ¼ vw Au þ ai =Vpw : Cp ðtÞ ¼ Au þ ð32Þ Then for the area under total plasma concentration curve, with an account of eqs. 31 and 32, we have Z1 Z1 X AUC ¼ Cp ðtÞdt ¼ vw Cu ðtÞdt 1þ kþ i =ki dt; 0
0
ð33Þ
AUCu ¼ fu AUC:
AUMC ¼ 1=Vp
ð36Þ
Z1 X t Au þ ai dt:
ð37Þ
0
Multiplying eq. 30 by time t and integrating from zero to infinity with the account of initial conditions and eq. 31, we can express the second term in the equation above through Au(t) Z1 0
tai ðtÞdt ¼
Z1 0
Z1 h 2 i tAu dt þ kþ = k Au dt i i 0
ð38Þ Substitution of eq. 38 in eq. 37 transforms it to Xh 2 i AUMC ¼ AUMCu =fu þ vw AUCu kþ : i = ki JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
370
BEREZHKOVSKIY
Finally, the equation above combined with eqs. 34, 35, and 36 yields the equation that connects mean residence times calculated using the unbound and total drug concentrations, MRTu ¼ AUMCu/AUCu and MRT ¼ AUMC/AUC Xh X 2 i kþ MRTu ¼ MRT = 1þ kþ i = ki i =ki : ð39Þ The mean residence time MRT in eq. 39 can be calculated from total plasma concentration–time curve as commonly done, while the second term requires the knowledge of possible reactions and their rate constants. In general, MRTu is always smaller than MRT, but if the assumption of an instant equilibrium is made, that is, kiþ, ki ! 1, and kiþ/ki ¼ const, then according to eq. 39, MRTu becomes equal to MRT.
DISCUSSION The obtained eq. 28 for the steady-state volume of distribution contains the mean residence time MRTu, which is calculated using the concentration of unbound drug in plasma. The value of MRTu really equals to the mean time, which the drug spends in the system after intravenous bolus input (assuming that the drug is irreversibly removed from the body), when elimination rate is proportional to the concentration of unbound compound. Indeed, if elimination rate E(t) is EðtÞ ¼ Clu Cu ðtÞ;
ð40Þ
then the quantity of drug that exits the body in the time interval from t to t þ dt equals to E(t)dt. Therefore, the probability to stay in the body during the time interval from t to t þ dt is Z1 EðtÞdt= EðtÞdt; 0
and the average time T that a drug remains in the body can be written as T¼
Z1
tEðtÞdt=
0
Z1
EðtÞdt:
ð41Þ
0
Substitution for E(t) from eq. 40 in eq. 41 yields T ¼ MRTu ¼
Z1
tCu ðtÞdt=
0
¼ AUMCu =AUCu :
Z1
Cu ðtÞdt
It is important to mention that the equation for the steady-state volume of distribution presented in the form ð43Þ Vss ¼ ðD=AUCÞ*T; where T is the true mean residence time given by eq. 41, is correct in the case when both unbound and bound compounds are exposed to the first order elimination, so that EðtÞ ¼ Clu*Cu ðtÞ þ Clb*Cb ðtÞ; where Cb(t) is the plasma concentration of bound compound. The validity of eq. 43 can be proved in the same manner as eq. 28 was derived, but using the equation above for elimination rate and eq. 41 as a general definition of the mean residence time T. Thus, eq. 43 appears to be general for a linear pharmacokinetic system with the central input and elimination. In this equation the mean residence time T should be considered according to the elimination mechanism, that is, calculated from eq. 41 with an appropriate expression for the elimination rate E(t). If elimination rate is just proportional to the unbound drug concentration, then T ¼ MRTu, and MRT calculated using the total plasma concentration does not have clear interpretation in this case, it is related to T through eq. 39. Mean residence time T becomes equal to MRT only if elimination rate can be assumed proportional to the total drug concentration in plasma. This is a valid assumption, as discussed in detail below, for the compounds with fast dissociation rates from proteins. Equations 28 and 39 for Vss do not contain explicitly the rate constants characterizing peripheral kinetics. Peripheral kinetics goes into the equation for Vss through the integrated values of AUC, AUMC, AUCu, and AUMCu calculated using the time course of drug in plasma. Drug kinetics in plasma appears in the equation for Vss directly through the rate constants kiþ, ki, as well as indirectly through areas under the curve. Let us consider a possible difference between MRT and MRTu, which would result in an overestimated values of Vss. To simplify the problem, lets assume that there is just one reaction of protein binding in plasma A þ P ¼ Ab. Then, according to eq. 39 þ MRT MRTu ¼ 1=k 1 = 1 þ k1 =k1 ¼ ð1=koff Þ=ð1 þ Kd =PÞ;
0
ð42Þ
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
ð44Þ
where (as explained in the introduction) koff ¼ k 1, kþ ¼ k P and K ¼ k /k . 1 on d off on
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
The factor 1/(1 þ Kd/P) in the above equation is the value of bound drug fraction fb ¼ 1 fu. Obviously, the difference MRT MRTu will be greater for highly bound compounds (fu ! 0, Kd/ P 1). Then MRT MRTu & 1/koff, and is basically determined by the dissociation rate koff of the bound drug from the protein. The value of dissociation constant can vary significantly from approximately 10 to 105 s1, corresponding to dissociation half-life, ln 2/koff, from less than 1 s to several hours. Usually highaffinity binding, which is characterized by a small value of koff, requires specific orientation of the ligand to interact with the protein. This results in slower on-rates, eventually producing Kd ’ 1– 100 nM. The value of the on-rate commonly varies from 10 to 106 M1s1 (maximum possible value, which corresponds to the Smoluchowski limit, is about 109 M1s1). The value of 1/koff in eq. 44 should be of an order of MRT to have significant influence on the difference between MRT and MRTu; thus, the effect is more likely to be observed for the drugs with shorter MRT values. Let us consider an example of a drug with MRT ¼ 0.5 h. For koff ¼ 103 s1 (1/koff & 17 min), kon ¼ 1.4*102 M1s1 and P ¼ 679 mM (average albumin concentration in human plasma), we get 1/(1 þ Kd/ P) ’ 0.99, and eq. 44 yields MRTu ¼ 13 min. Thus, the correct value of the steady-state volume of distribution, calculated with MRTu instead of MRT, will be 2.3 times smaller than Vss calculated using traditional eq. 5. The rate constants for protein binding are not commonly measured and require complex instrumentation to be obtained.18 The value of koff & 0.04 min1 ¼ 6.7*104 s1 (1/koff ¼ 25 min) was observed by Yuan et al.19 in the study of the kinetics of compound dissociation from rat plasma proteins after adding an excessive amount of dextrane-coated charcoal to equilibrated solution. The time curve obtained for the amount of compound remaining in solution was byexponetial: the fast exponent corresponded to a quick charcoal adsorption of initial free fraction, while the slow one corresponded to the rate of the protein– compound complex dissociation. For this compound the difference between MRT and MRTu should be about 0.5 h. Usually the binding of small molecules, which may have several binding sites on a protein, is rapidly reversible. In this case, MRT and MRTu are virtually the same. The large molecules, which could form several hydrogen bonds binding to the protein, can exhibit the difference of several hours
371
between MRT and MRTu. The average time that the drug spends in the body after bolus dose input may be relatively long just because of a slow dissociation rate from protein in plasma, but not necessarily due to peripheral partitioning or low clearance Clu. But the true value of mean residence time, MRTu, could be considerably less than MRT calculated over the total compound concentration in plasma. For example, if MRT ¼ 3 h and MRTu ¼ 1 h, the steady-state volume of distribution calculated using MRT, eq. 5, will be three times greater than the true one calculated with MRTu according to eq. 14. In this situation the comparison of Vss obtained from physicochemical properties,3 which is supposed to generate the value equal to that from eq. 14, with Vss calculated from traditional eq. 5 may not be valid. For the drug that just stays in plasma and participate in one protein binding reaction, as considered in the introduction (Fig. 1), the obtained eq. 13 provides a convenient straightforward estimation of the koff and kel, which could lead to a substantially greater value of Vss than the true one given by eqs. 1 or 14. Equation 13 can be written as h i Vss ¼ Vp 1 þ k1 kel =ðk1 þ k2 Þ2 h i ¼ Vp 1 þ ðkel =koff Þ*ðkd =PÞ=ð1 þ Kd =PÞ2 : ð45Þ For example, for the rat intravenous bolus injection study, if drug total clearance is about 80% of the liver blood flow (medium level clearance), that is Cl ¼ 44 mL/(kg min) and Vp ¼ 0.045 L/kg, we get kel ¼ Cl/(Vpfu) ¼ 0.97/fumin1. If the drug protein binding is 90% ( fu ¼ 0.1), that is, Kd/P ’ 0.1 and koff & 0.04 min1, as observed by Yuan et.al.,19 the calculated value of Vss, according to eq. 45, will be Vss 23Vp. Thus, the steady-state volume of distribution calculated using traditional eq. 5 will overestimate the true value of Vss by an order of magnitude (23 times), suggesting a noticeable tissue distribution of drug, which actually does not happen. It is not possible to predict whether traditional calculation would substantially overestimate the value of Vss based just on the drug extraction ratio. As follows from eqs. 39, 44, or 45, the difference in Vss strongly depends on the dissociation rate koff of the drug from protein, while drug extraction ratio is determined only by relative values of hepatic (or clearing organ) blood flow Q and fuClint ( fu provides equilibrium characteristics Kd ¼ koff/kon, but not koff separately). On the other hand, as mentioned above, the shorter MRT values, or bigger kel values, according JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
372
BEREZHKOVSKIY
to an example given by eq. 45, would lead to an overestimated values of Vss obtained from traditional calculation. This suggests that a drug, which eliminates faster from the body is more likely to exhibit the difference in Vss calculation. Thus, the drugs with higher intrinsic clearance and that are therefore more likely to be high extraction ratio drugs, could be the ones for which traditional calculation of Vss would yield substantially overestimated values. In general, it is clear that both fast elimination and small value of dissociation rate constant could result in the absence of immediate equilibrium between bound and free drug, so that binding kinetics become important in calculation of the steady-state volume of distribution. But small enough value of dissociation rate constant koff (which also leads to high level of protein binding) is the crucial factor that determines possible significant overestimation of Vss calculated by using traditional eq. 5. The possibility of having dissociation rate constant koff 104 s1 (1/koff 3 h) can be estimated using transition state theory.20 The dissociation rate constant of the compound–protein complex is calculated as koff ¼ ðo=2pÞ expðEo =kTo Þ;
ð46Þ
where o is the oscillation frequency corresponding to the potential well created by compound– protein interaction, Eo is the energy of dissociation, k is the Boltzmann’s constant, and To is the absolute temperature. Lets approximate the interaction potential all the way to the top of the barrier as quadratic, that is, E(x) ¼ mo2x2/2, where x is the distance from the bottom of potential well (position of minimal energy) and m is the mass of the compound molecule. Assum˚ corresponds to the ing that deviation of xo ’ 1 A energy of the dissociation barrier Eo (so that dissociation occurs), we get Eo ¼ mo2x2o /2, or o ¼ (2Eo/m)1/2/xo. Then eq. 46 becomes koff ¼ ð2pxo Þ1 ð2Eo =mÞ1=2 expðEo =kTo Þ:
ð47Þ
For the compound with molecular weight of 350 g/mol and the dissociation energy Eo ¼ 22 kcal/ mol eq. 47 yields koff ¼ 6.8 105 s1. The value of dissociation energy of 22 kcal/mol is about the energy of three strong hydrogen bonds. For example, the formation of three or more hydrogen bonds commonly occurs for peptide binding to the major histocompatibility complex proteins, long dissociation half-life t1/2 ¼ ln2/koff 1 to 100 h is not unusual for this system.21 Thus, for the comJOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 93, NO. 2, FEBRUARY 2004
pound that could possibly form three strong hydrogen bonds binding to plasma proteins, it is likely to have the value of dissociation rate constant small to the extent that would lead to a substantial overestimation of the steady-state volume of distribution and the mean residence time calculated using traditional eqs. 5 and 8.
CONCLUSION The equation for calculation of volume of distribution at steady-state, Vss, with complete consideration of the kinetics of protein and tissue binding is obtained. Traditional equations for Vss and mean residence time, MRT, are based on the assumption of an instant equilibrium between the bound and free drug in plasma. It is shown that for drugs with high protein affinity and slow dissociation rate constant (koff 103 s1) the suggested equations predict the steady-state volume of distribution and mean residence time considerably less than that obtained from the traditional calculation. Drug binding to the protein with the formation of three or more strong hydrogen bond would likely result in such a low value of the dissociation rate constant. Drugs with high intrinsic clearance are more likely to yield overestimated values of Vss and MRT calculated by using the traditional equations. They require not as small values of koff as low extraction ratio drugs for a considerable deviation from an instant binding equilibrium to occur.
ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. Bill Fitch (Roche Palo Alto) for assistance in the preparation of this manuscript.
APPENDIX According to a general eq. 23, the concentration of drug Cpo(t) in plasma, when the drug is administered into plasma at constant input rate R(t) ¼ Ro is written as po
C ðtÞ ¼ ðRo =DÞ
Zt
Cðt tÞdt;
ð48Þ
0
where C(t) is the drug concentration in plasma after bolus input of the drug dose D.
DETERMINATION OF VOLUME OF DISTRIBUTION AT STEADY STATE
The steady-state volume of distribution is calculated according to eq. 1, Vss ¼ Ab,ss/Cp,ss. We consider the case when drug elimination occurs only from central compartment and its elimination rate Clu*Cu is proportional to the concentration Cu of unbound drug in plasma. Then according to the mass conservation law D ¼ Clu
Z1
Let us obtain the quantity of drug in the system at steady state. At instant t it can be written as Ab ðtÞ ¼ Ro t Clu
Zt
0 0 Cpo u ðt Þdt :
ð50Þ
According to eq. 48, the integral in the equation above can be written as 2 3 Zt Zt Zt 0 0 4 Cu ðt0 tÞdtÞ5dt0 : ð51Þ Cpo u ðt Þdt ¼ ðRo =DÞ 0
0
Cu ðt tÞdtÞdt ¼
0
Zt0
Cu ð xÞdx
0
Z1
Cu ð xÞdx
Z1
Cu ð xÞdx
t0
0
¼ D=Clu
Cp ðt tÞdt
0
Z1
Cp ð xÞdx ¼ Ro AUC=D
ð55Þ
0
Finally, from eqs. 54 and 55 the desired equation for the steady state volume of distribution is obtained ¼ ðD=AUCÞ MRTu : Equation 54 written as MRTu ¼ Ab,ss/Ro can be used to calculate the mean residence time from the steady-state condition.
REFERENCES
0
Using the new variable x ¼ t0 t and the mass balance condition, eq. 49, the integral in the brackets in eq. 51 can be transformed to
¼
Z1
Vss ¼ Ab;ss =Cp;ss ¼ D*AUMCu =ðAUC*AUCu Þ
0
Zt0
Cp;ss ¼ ðRo =DÞ
ð49Þ
0
0
The total concentration of plasma at steady state can be expressed according to eq. 48 as
¼ ðRo =DÞ Cu ðtÞdt ¼ Clu AUCu :
373
Z1
Cu ð xÞdx:
ð52Þ
t0
By combining eqs. 51 and 52 and substituting into eq. 50, this equation becomes 2 3 Z t Z1 Ab ðtÞ ¼ ðRo =AUCu Þ 4 Cu ð xÞdx5dt0 : 0
ð53Þ
t0
Integration in eq. 53 can be done by substituting Cu(x) with Cu(x)y(x t0 ), where y(xt0 ) is the step function, y(x t0 ) ¼ 1 for x t0 and y(x t0 ) ¼ 0 for x < t0 . Then integration with respect to x can be carried out from 0 to infinity. Integrating first with respect to t0 from 0 to 8, and then with respect to x, finally yields Ab;ss ¼ Ab ðt ! 1Þ ¼ Ro AUMCu =AUCu :
ð54Þ
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