Europ. J.clin.Pharmacol. iO, 425-432 (1976) © by Springer-Verlag 1976

Simple Model to Explain Effects of Plasma Binding and Tissue Binding on Calculated Volumes of Distribution, Apparent Elimination Rate Constants and Clearances*

J. G. Wagner

College of and Upjohn Center for Clinical , The University of Michigan, Ann Arbor, Michigan, USA

Received: March 26, 1976, accepted: June 10, 1976

Summary. A simple pharmacokinetic model, incorporating linear plasma protein bind- ing, linear tissue binding, and first order elimination of free (unbound) , was studied. If Clp is the plasma , Vf is the "true" volume of distribu- tion of free drug, B is the apparent elimination rate constant, ~ is the fraction of the drug which is free in plasma, f is the fraction of the drug which is free in the entire body, kf is the intrinsic elimination rate constant for free drug, and A~B is the initial amount of drug which is bound to tissues, then the model in- dicates that the following relationships hold: (I) Cl~ = Vf~ kf; (2) ~ = f kf; and Vdex t = (o/f) Vf. Only o, and not f, can be measured experimentally. Dividing Clp by provides an estimate of the intrinsic clearance of free drug, Vfkf. A plot of Vdext versus ~ has an intercept equal to Vf, and the ratio of the slope/intercept is" an estimate of A~/A~, where A~ is the initial amount of free drug (equalo to Vf times initial concentration of free drug in plasma). Thus, an estimate of A~B may be obtained. Dividing the intrinsic clearance by Vf provides an estimate of kf. Thus, theoretlcally,• est±mates • of Vf, kf, ATBO and f may be obtained. The variables are not separated when ~ is plotted versus ~, and curvature of such plots is expec- ted; no useful information is obtained from such plots.

Key words: Plasma protein binding, tissue binding, of free drug, intrinsic rate constant for free drug, amount bound to tissues.

There has been much discussion in the possible effects of plasma protein bind- literature on the effects of plasma pro- ing, but also discussed the possible ef- tein binding on the distribution, elim- fects of tissue binding. He cited data ination, and activity of (Anton showing that cardiac glycosides which are and Solomon, 1973; Benya and Wagner, highly bound to plasma are also 1975; Coffey, 1972; Garrett, 1972; Keen, highly bound to tissues. A similar sit- 1972; KrHger-Thiemer, 1966 and 1968; Levy uation has been reported for diphenhy- and Yacobi, 1974; Martin, 1965; Wagner, dramine (Wagner, 1973 and Albert et el., 1971 and 1975) and these citations are 1975) and for in the rat (Benya not intended to be complete. The review and Wagner, 1975; Yacobi and Levy, 1975). of Keen (1971) not only discussed the Several authors (Coffey, 1972; Garrett, 1972; Keen, 1971; Kr~ger-Thiemer, 1966 and 1968; Levy and Yacobi, 1974; Martin, 1965; Yacobi and Levy, 1975) have devel- oped mathematical and pharmacokinetic mPartly supported by Public Health Service Grant models and equations which incorporate 5-P-II-GM 1559 and partly by Grant IROIAAOO683- plasma protein binding. Others (DiSanto, OIAI from the National Institute on Abuse 1971 and Wagner, 1971 and 1975) have de- and Alcoholism, Rockville, Maryland. veloped models to illustrate possible ef- 426

fects of tissue binding. Bischoff and CTB- molar concentration of tissue bound Dedrick (1968) developed a flow rate- drug at time t. limited model, which incorporated both D - the dose of drug administered by plasma protein binding and binding to bolus intravenous injection. several different types of tissues; they f - the fraction of the total amount of assumed that the relationship for bind- drug in the body which is free. ing to plasma proteins could also be used (defined by equation 14). for tissue binding; equations used were kf - the intrinsic elimination rate con- those of the Langmuir type. Hermann (1975) stant for free drug (dimension of showed that the binding of ouabain and I/time). digitoxin by human erythrocytes obeyed Kp - the association constant for protein the Langmuir equation. When the dissoci- bound drug (Kp = kl/k2). ation constant is much greater than the K T - the association constant for tissue drug concentration, the Langmuir equation bound drug (KT = k3/k4) • collapses to a linear binding equation. m - the number of binding sites for drug In inter-individual comparisons of a on tissue protein. group of one species of animal or man it n - the number of binding sites on plas- would be advantageous if one could pre- ma protein. dict, for example, the plasma clearance P - molar concentration of plasma pro- of a drug by measurement of the fraction tein. of the drug which is free in plasma. The - the fraction of drug which is free question arises as to which are the ap- in plasma (defined by equation 19). propriate pharmacokinetic parameters to t - time. correlate with this fraction. The pur- t" - some specific time. pose of this report is to determine t I - a lag time needed in real life to which relationship or relationships have establish model conditions. a firm footing in pharmacokinetic theory, T - molar concentration of tissue which rather than the alternative approach of binds the drug. just making empirical correlations. The Vdare a - a volume of distribution defined physical model approach also suggests by equation 34. when one might expect a trend line to be Vdext - a volume of distribution defined linear or curved; often, scatter in data, by equation 39. as a result of just biological variation Vf - the "true" volume of distribution of and assay error, does not allow one to free drug (dimension of liters). adequately determine when a trend line V T - the volume of tissue which binds should be linear or curved. drug (liters).

THEORETICAL Assumptions Symbolism I. When the dose, D, is given by bolus intravenous injection, the drug instan- Af - amount (moles) of free drug at time t. taneously is partitioned into three por- tions, namely A~B , A~B , and A~. A~ - initial amount (moles) of free drug 2. Binding to both plasma protein and at time zero. tissue is linear and independent of drug ApB- amount (moles) of plasma protein concentration. bound drug at time t. 3. The volume occupied by plasma pro- initial amount (moles) of plasma A~B- teins is ignored and hence ApB = VfCpB protein bound drug at time zero. ATB- amount (moles) of tissue bound drug and Af = VfCf. 4. The free drug is eliminated accord- at time t. ing to first order kinetics with rate initial amount (moles) of tissue A~B- constant equal to kf. bound drug at time zero. 5. Only free drug is available for 8 - apparent elimination rate constant, elimination. such that -d in Cp/dt = 8 Clp- plasma clearance ~defined by equa- Model tion 31). Cf - molar concentration of free (unbound) The simple model includes only binding drug at time t. of drug to one class of sites on one C~ - initial molar concentration of free plasma protein, binding of drug to one (unbound) drug at time zero. class of sites on one type of tissue and Cp - molar total plasma concentration of one fluid volume. However, the approach drug (defined by equation 24) at taken with the simple model could be time t. readily extended to more complicated mod- Cpo _ initial molar total plasma concen- els as was done by DiSanto (1971) and tration of drug at time zero. Wagner (1971) with analogous nonlinear CpB- molar concentration of plasma pro- models. The conclusions reached with the tein bound drug at time t. simple model would be extrapolatable to 427 more complicated models of a similar type. The model is shown schematically in Scheme 1. dt VT Af + n Kp P + m K T T I Volume = Vf Volume = V T Eq. (11) Plasmakl.~__Plasma k3 Protein---~Protein + Freel+ Tissue~_~___Tissue Substituting from equation 5 for Af bound k 2 drug I k% bound and cancelling the Vf's on both sides drug I drug yields equation 12 from equation 11. dCf I~ kf - Cf dt VT + n Kp + m K T T ~f Scheme i. Modified "one compartment open model" incorporating linear tissue binding and linear plasma protein binding = B Cf Eq. (12) where B is the apparent elimination rate constant and is given by equation Equations 13. kf For linear binding: B = VT CpB = n Kp P Cf Eq. (I I + n Kp P + m K T T V-~ ApB = Vf CpB = n Kp P Vf Cf = nKp P Af Eq. (2 Eq. (13) CTB = m K T T Cf Eq. (3 If we define f as the fraction of the total drug in the body which is free, ATB = V T CTB = m K T T V T Cf then: VT = m KTTv-~fAf Eq. (4 Af I f: Af + ApB + ATB ApB ATB Af = Vf Cf Eq. (5) I + m-f + A-7-

Mass balance gives: Eq. (14) Af + ApB + ATB + kfl~'Af- dt = D From equation 2 we obtain: Eq. (6) ApB In equation 6, the term containing the Af - n Kp P Eq. (I 5) integral represents the amount of drug From equation 4 we obtain: eliminated between time zero and time t'. Differentiation of equation 6 with re- ATB VT spect to time yields equation 7. if - m K T T V--f Eq. (16) dAf dApB dATB d--t-- + d~ + d~ + kf Af = O Eq. (7) Substituting from equations 15 and 16 into equation 14 gives equation 17. Now, f = I dAPB dApB dAf dAf Eq. (17) = - (n Kp P)-- VT dt dAf dt dt I + n Kp P + m K T T V--~ Eq. (8) Substituting from equation 17 ~nto dATB dATB dAf V T dAf equation 13 gives equation 18. dt = dAf dt - (m K T T ~ff) dt 8 = f kf Eq. (18) Eq. (9) Thus, according to this simple model, Substituting from equations 8 and 9 the apparent elimination rate constant, into equation 7 yields: 8, is a product of the intrinsic elimi- dAf V T nation rate constant for free drug, kf, dt [I + n Kp + m K T T ~ff] + kf Af = O and the fraction of the total amount of drug in the body which is free, f. Eq. (I O) If we define a as the fraction of the drug which is free (unbound) in the vol- Rearrangement of equation 10 gives ume Vf (and plasma is assumed to be re- equation 11. presentative of this fluid), then: 428

Cf Vf Cf The initial condition at time zero U -- Cf + CpB Vf Cf + Vf CpB for the model is given by equation 27. Af I ÷A B=D Eq (27) Eq. (I 9) Af + ApB Ap B 1+- In equation 27, A~, A~B , and A~B are Af the initial amounts existing as free Thus, drug, drug bound to tissues, and drug bound to plasma proteins, respectively, ApB I and D is the dose administered. In real I + - Eq. (20) Af life it would take some finite time, even when the values of k I and k 3 are large, Substituting from equation 20 into to attain this "initial condition", but equation 14 and rearranging gives: this lag time, tl, would most probably be small relative to the measured time f = ~ Eq. (21) courses of most drugs. The model assumes I + (ATB)~ such partitioning, as indicated by equa- Af tion 27, has already occurred at time zero. Hence, in real life the only dif- Substituting for f in equation 21 ference would be that the D in equation gives equation 22. 27 would be replaced by A~, the apparent initial amount of drug in the body, such = ~ kf Eq. (22) that A~ < D, and t would be replaced by B (ATB ) (t - tl). + Af An estimate would be

Equation 22 indicates that for inter- A~ = D - Vfkf f~l Cfdt. individual comparisons of a drug in one o It may be readily seen that Cp is also species of animal or in man, a plot of the apparent elimination rate constant, given by equation 28. ~, versus the fraction of drug which is free in plasma, ~, will be gently curved ° I A~ + A~B D-- Eq. (28) over a small range of ~ values, even Cp = A~ + A~B + A~B Vf when the ratio ATB/A f is constant from individual to individual of the species. Equations 14 and 19 also apply to the With the scatter expected in such a plot initial condition, as well as at any (because of expected non-constancy of time t, hence substituting from those the ATB/A f ratios) such curvature would be difficult to detect with real data. equations and cancelling the A~'s yields equation 29. Integration of equation 12 yields equation 23. f A~ + A~B - Eq. (29) Cf = C~ e -Bt Eq. (23) A~ + A~B + A~B

The total plasma concentration, Cp, Substituting from equation 29 into is given by equation 24, after substi- equation 28 gives equation 30. tuting from equation I for CpB. Cp = Cf + CpB = (I + n Kp P) Cf The plasma clearance, Cip, is usually Eq. (24) calculated as the ratio of %he dose to Multiplying both sides of equation 23 the total area under the total plasma by (I + n Kp P) and utilizing equation concentration-time curve, as indicated 24 gives equation 25. by equation 31. D Cp = (I + n Kp P) C~ e -~t = C~ e -~t Cip - Eq. (31) f0 Cp dt Eq. (25) where Integration of equation 25 between the limits of t = 0 and t = ~, followed C~ = (I + n Kp P) C~ Eq. (26) by substitution for C~ from equation 30, substitution for 8 from equation 18 and Hence, the model predicts that the cancellation of the f's gives equation apparent elimination rate constant, ~, 32.

obtained from total plasma concentrations O is the same as that obtained from free I0 Cp dt Cp D drug concentrations. 8 Vf o kf Eq. (32) 429

Substituting for the integral in equa- ing and tissue binding of the drug. When tion 31 from equation 32 and cancellation plasma protein binding is very low (i.e. of the D's yields equation 33. ~÷ I and A~B/D+ 0), and tissue binding is large [i.e. A~B/D÷ I), then this Clp = Vf ~ kf Eq. (33) bracketed portion will be much larger than unity, and Vdare a >> Vf. If a = 0.5 Thus, according to the model, the cal- and (A~B + A~B)/D = 0.5, then the brack- culated plasma clearance, CID, is the eted portion would be equal to unity and product of the intrinsic clearance of Vdarea = Vf. Even for a drug which is free drug, Vf kf, and the fraction of both very highly plasma protein bound the drug which is free in plasma, ~. For and tissue bound, Vdare a could be equal inter-individual comparisons a plot of to Vf; for example, if ~ = 0.02 and (A~B Clp versus ~ should yield a straight + A~B)/D = 0.98, then the bracketed por- line, passing through the origin, with tion would be equal to unity. Usually, slope equal to Vf kf, providing that the of course, the bracketed portion would intrinsic clearance of free drug is con- not be equal to unity, hence either stant from one individual to the next. Vdare a > Vf or Vdarea < Vf. Thus, the Alternatively, and preferred, the plasma calculated volume of distribution is not clearance of each individual may be di- equal, usually, to the volume of distrib- vided by the fraction of drug free in ution of free drug. plasma for that individual to obatin in- The extrapolated volume of distribu- dividual estimates of the intrinsic tion, Vdext , is calculated with equation clearance, then these analyzed statisti- 39. cally to obtain a measure of dispersion. D Vdarea is usually calculated with Vdext - C~ Eq. (39) equation 34. o Substituting for C D from equat±on 30 Vdarea : D - Clp Eq. (34) into equation 39 yields equation 40. f0 Cp dt Vdext = (~) Vf Eq. (40) Substituting from equation 18 for B and from equation 33 for Clp into equa- By comparing the right hand sides of tion 34, and cancellation of the kf's, equations 35 and 40 we see that for this gives equation 35. model:

Vdare a : (f) Vf Eq. (35) Vdext = Vdare a Eq. (41)

Equation 35 indicates that in the ab- Equation 41 is the expected result sence of both plasma protein binding and for this modified "one compartment open tissue binding (i.e. ~ = f = I), Vdare a model". If there was more than one fluid = Vf. However, equation 35 also indicates compartment and/or distribution to some that Vdare a is not a simple function of tissues took an appreciable time, than a ~, as is clarified below. multicompartmental model would be re- Rearrangement of equation 27 gives quired, and, in such cases, Vdext > equation 36. Vdarea- From equations 14 and 19 it may read- : D-A B- Eq (36) ily be shown that: VT (CTB) Substituting from equation 27 and 36 - I + (~ff) Cp Eq. (42) into equation 14, and rearrangement f yields equation 37. Equation 42 indicates that ~ ~ f. Ah Also, rearrangement of equation 34 gives f = I D D Eq. (37) equation 43. Hence, the model agrees with classical concepts. Substituting for f from equation 37 into equation 35 yields equation 38. Cip = Vdare a B Eq. (43)

Keen (1971) recommended that the vol- Vdarea = A~ B A~ B Vf Eq. (38) ume of distribution be calculated from free drug concentrations, rather than I D D from total drug concentrations. If this is done, then the volume calculated, The portion of equation 38 marked off based on the simple model, is Vf/f. The by square brackets can have a value rang- same result would be obtained by divid- ing from much less than unity to much ing the volumes, Vdext and Vdarea, esti- greater than unity, depending upon the mated from total drug concentrations, by relative degrees of plasma protein bind- ~. It should be noted that such volumes 430 are still dependent upon relative degrees ATB A~B of plasma protein binding and tissue - Eq. (51) Af A~ binding, as pointed out by Keen (1971). The product of the apparent elimination Since• the ratlo . ATB/A o of ±s . known from rate constant, 8, and such a volume equations 47 and 51, and A A~ is known (estimated from free drug concentrations) from equation 50, then A~B is obtained is, by use of equation 18, equal to the with equation 52. intrinsic clearance of free drug, as in- o o dicated by equation 44. A~B - ATB/Af Vf C~ Eq. (52) Vf Intrinsic clearance = (~-) (f kf)

= Vf kf SIMULATIONS

Eq. (44) Simulation ~ ~ equation 22 - A plot of the If there are multicompartmental phar- apparent elimination rate constant, ~, macokinetics, then the volume calculated versus the fraction free in plasma, o, from free drug concentrations should be does not separate the variables as may the modified Vdarea, rather than Vdext , be seen from equation 22. Even in those unless, as is sometimes the case, the cases where the ratio ATB/A f is constant, two volumes are essentially identical. while o varies, plots of ~ versus o will Estimation of Vf and ATB/A f - Substituting be curved as may be seen in Figure I. from equation 20 into equation 14 yields The greater the value of the ratio ATB/Af, equation 45. The latter is also obtained the greater the curvature. For low val- by rearranging equation 21. ues of ATB/Af, the plots are nearly lin- ear. In real life, one would expect I f - Eq. (45) ATB/A f to vary, and not be constant, from I ATB one individual to the next. This would -- + a Af cause scatter on the plots of B versus o. Evaluation of such "scattered data" Substituting from equation 45 into by linear regression analysis, would equation 40 and simplifying gives equa- lead to an apparent positive intercept tio 46. on the ordinate. But, the intercept and slope values obtained from such plots (ATB ) Vdext : [I + o ] Vf Eq (46) would have no physical significance. Af Thus, correlations of B with o are notre- commended. This would also apply to any Equation 46 indicates that a plot of transformation, such as half-life. Vdext versus ~ will be linear with inter- Simulation ~th equation 46 - Two sets of cept equal to Vf, and slope equal to data were generated by letting ATB/A f (ATB/A f) Vf. Hence, ATB/A f is obtained equal 10 and 50, then varying o over the with equation 47. range 0.004 to 0.02. For a given value

ATB _ Slope _ (ATB/Af) Vf of o, the value of "f" was calculated Af Intercept Vf with equation 45, then the corresponding value of o/f, equivalent to Vdext/Vf, was Eq. (47) calculated using equation 40. Results are shown in Figure 2. Extrapolation of Estimation of kf and f - Since Vf is known the linear plots to the ordinate yielded (see above) and the intrinsic clearance, a value of unity, equal to Vf/Vf in this Vf kf, is known by dividing Clp by case. The slopes of the lines are the (equation 33), then an estimate of kf is same as the assigned values of ATB/Af, obtained from equation 48. namely 50 and 10. Suppose the value of Intrinsic Clearance Vf kf Vf was 0.6 i/kg. Then, using the value kf = Vf - Vf of ATB/A f = 50, the slope of the Vdext versus ~ plot would be 30, and the inter- Eq. (48) cept would be 0.6. Hence, the calculated Now that kf and B are known, then "f" value of ATB/A f would be slope/intercept may be estimated by rearranging equation = 30/0.6 = 50. If there was considerable 18 to give equation 49. scatter on a plot of Vdext versus ~, this would indicate variation in the ratio f _ B kf Eq. (49) ATB/A f from one individual to the next.

Estimation of A~B - If we write equation DISCUSSION 5 for the zero time condition, then equa- tion 50 is obtained. Also, since binding The simple model employed incorporated is linear, equation 51 holds. linear tissue binding, linear plasma pro- = vf (5o) tein binding, and first order elimination 43l

°°I /7 2.0 0.008 [ bl~ 16 5o/

I1 0.0 6 [ II 1.2 /,o 0.004 [ 0.002 I- 0.4 / I I I I I I I I I I 0 0.4 0.8 1.2 1.6 2,0 0 O# Q8 1.2 1.6 tO o- x 102 o x lO 2 Fig. i. Results of simulations with equation 22 Fig. 2. Results of simulations with equation 40 (see text). Indicates curvature obtained when (see text). Numbers on lines refer to assigned apparent elimination rate constant plotted versus values of the ratio ATB/A f fraction free in plasma. Numbers on curves refer to assigned values of the ratio ABT/A f

of free (unbound) drug, hence any con- are not separated, and the plot will have clusions are limited to real systems with a gentle curvature. If, also, ATB/A f these properties. The most important varies appreciably from one individual equations derived are 18, 22, 33, 40, to the next, then there will be consider- and 46 through 52, which require equa- able scatter on the 6 versus ~ plots. tions 14, 19, 31, 34 and 39 for interpre- These types of plots are not recommended. tation. Since the binding of drugs to plasma With respect to inter-individual com- proteins is extremely rapid (Keen, 1971) parisons, the model suggests the follow- and the binding of some drugs to tissues ing: is also extremely rapid (see Wagner, 1973 (I) Dividing the plasma clearance, Clp, for in the rat; and Benya (estimated from total plasma concentra- and Wagner, 1975 for warfarin in the rat), tions) by the fraction of drug free in one might expect the model to hold for plasma, a, will provide estimate of the low doses in some cases. It should be intrinsic clearance of free drug, Vf kf. noted that if one derives equations for The values obtained from a panel of sub- a similar model, but with several dif- jects or patients, or a group of animals, ferent types of tissues, the same general may then be treated statistically to ob- form for equation 17 is obtained, except tain a mean and some measure of disper- that there are additional terms in the sion, such as the coefficient of varia- denominator for each type of tissue (see tion. Alternatively, one could plot Clp Wagner, 1971). versus g and the slope of the least The equations derived in this article squares linear regression line would be and the conclusions reached differ ap- an estimate of Vf kf for the population. preciably from some of those of Keen (2) A plot of the extrapolated volume (1971), Coffey (1972), Krfiger-Thiemer of distribution, Vdext , versus fraction free (1966 and 1968), Levy and Yacobi (1974) in plasma, a, has an intercept equal to and Yacobi and Levy (1975). The reason Vf and a slope equal to (ATB/Af) Vf, for the differences is the inclusion of hence the ratio ATB/A f is obtained by both tissue binding and plasma protein dividing the slope by the intercept. Once binding in the model and derivations in V~ and ATB/A f are known, then kf, f, and the present article, whereas previous A~B are readily obtained, providing there authors cited were principally concerned is not much variation in ATB/A{. only with plasma protein binding. It is (3) The apparent eliminatio~ rate con- hoped that this article will aid in ob- stant, B, is equivalent to the product taining additional pharmacokinetic in- of'the intrinsic elimination rate con- formation when a drug is administered by stant for free drug, kf, and the frac- bolus intravenous injection to either a tion of the total drug in the body which panel of subjects or patients or to a is free, symbolized by f. It should be group of animals of one species. Also, noted that B is not equal to ~ kf, unless it discloses some possible pitfalls in there is absolutely no tissue binding. the analysis of such pharmacokinetic When B is plotted versus a, the variables data. 432

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