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Simple Model to Explain Effects of Plasma Protein Binding and Tissue

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Simple Model to Explain Effects of Plasma Protein Binding and Tissue Europ. J.clin.Pharmacol. iO, 425-432 (1976) © by Springer-Verlag 1976 Simple Model to Explain Effects of Plasma Protein Binding and Tissue Binding on Calculated Volumes of Distribution, Apparent Elimination Rate Constants and Clearances* J. G. Wagner College of Pharmacy and Upjohn Center for Clinical Pharmacology, 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) drug, was studied. If Clp is the plasma clearance, 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, volume of distribution 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 drugs (Anton showing that cardiac glycosides which are and Solomon, 1973; Benya and Wagner, highly bound to plasma proteins 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 warfarin 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 Alcohol 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.
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