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Quasi-Equilibrium Pharmacokinetic Model for Drugs Exhibiting Target-Mediated Drug Disposition

Purpose

The aim of this study is to derive and evaluate an equilibrium model of a previously developed general pharmacokinetic model for drugs exhibiting target-mediated drug disposition (TMDD).

Methods

A quasi-equilibrium solution to the system of ordinary differential equations that describe the kinetics of TMDD was obtained. Computer simulations of the equilibrium model were carried out to generate plasma concentration-time profiles resulting from a large range of intravenous bolus doses. Additionally, the final model was fitted to previously published pharmacokinetic profiles of leukemia inhibitory factor (LIF), a cytokine that seems to exhibit TMDD, following intravenous administration of 12.5, 25, 100, 250, 500, or 750 μg/kg in sheep.

Results

Simulations show that pharmacokinetic profiles display steeper distribution phases for lower doses and similar terminal disposition phases, but with slight underestimation at early time points as theoretically expected. The final model well-described LIF pharmacokinetics, and the final parameters, which were estimated with relatively good precision, were in good agreement with literature values.

Conclusions

An equilibrium model of TMDD is developed that recapitulates the essential features of the full general model and eliminates the need for estimating drug-binding microconstants that are often difficult or impossible to identify from typical in vivo pharmacokinetic data.

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References

  1. D. E. Mager E. Wyska W. J. Jusko (2003) ArticleTitleDiversity of mechanism-based pharmacodynamic models Drug Metab. Dispos. 31 510–518 Occurrence Handle10.1124/dmd.31.5.510 Occurrence Handle12695336

    Article  PubMed  Google Scholar 

  2. H. Derendorf B. Meibohm (1999) ArticleTitleModeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: concepts and perspectives Pharm. Res. 16 176–185 Occurrence Handle10.1023/A:1011907920641 Occurrence Handle10100300

    Article  PubMed  Google Scholar 

  3. G. Levy (1994) ArticleTitlePharmacologic target-mediated drug disposition Clin. Pharmacol. Ther. 56 248–252 Occurrence Handle7924119

    PubMed  Google Scholar 

  4. E. D. Lobo R. J. Hansen J. P. Balthasar (2004) ArticleTitleAntibody pharmacokinetics and pharmacodynamics J. Pharm. Sci. 93 2645–2668 Occurrence Handle10.1002/jps.20178 Occurrence Handle15389672

    Article  PubMed  Google Scholar 

  5. D. E. Mager W. J. Jusko (2001) ArticleTitleGeneral pharmacokinetic model for drugs exhibiting target-mediated drug disposition J. Pharmacokinet. Pharmacodyn. 28 507–532 Occurrence Handle10.1023/A:1014414520282 Occurrence Handle11999290

    Article  PubMed  Google Scholar 

  6. D. E. Mager W. J. Jusko (2002) ArticleTitleReceptor-mediated pharmacokinetic/pharmacodynamic model of interferon-beta 1a in humans Pharm. Res. 19 1537–1543 Occurrence Handle10.1023/A:1020468902694 Occurrence Handle12425473

    Article  PubMed  Google Scholar 

  7. D. E. Mager B. Neuteboom C. Efthymiopoulos A. Munafo W. J. Jusko (2003) ArticleTitleReceptor-mediated pharmacokinetics and pharmacodynamics of interferon-beta1a in monkeys J. Pharmacol. Exp. Ther. 306 262–270 Occurrence Handle10.1124/jpet.103.049502 Occurrence Handle12660309

    Article  PubMed  Google Scholar 

  8. S. M. Eppler D. L. Combs T. D. Henry J. J. Lopez S. G. Ellis J.H. Yi B. H. Annex E. R. McCluskey T. F. Zioncheck (2002) ArticleTitleA target-mediated model to describe the pharmacokinetics and hemodynamic effects of recombinant human vascular endothelial growth factor in humans Clin. Pharmacol. Ther. 72 20–32 Occurrence Handle10.1067/mcp.2002.126179 Occurrence Handle12152001

    Article  PubMed  Google Scholar 

  9. G. Levy D. E. Mager W. K. Cheung W. J. Jusko (2003) ArticleTitleComparative pharmacokinetics of coumarin anticoagulants L: physiologic modeling of S-warfarin in rats and pharmacologic target-mediated warfarin disposition in man J. Pharm. Sci. 92 985–994 Occurrence Handle10.1002/jps.10345 Occurrence Handle12712418

    Article  PubMed  Google Scholar 

  10. F. Jin W. Krzyzanski (2004) ArticleTitlePharmacokinetic model of target-mediated disposition of thrombopoietin AAPS Pharm Sci 6 E9 Occurrence Handle10.1208/ps060109

    Article  Google Scholar 

  11. A. M. Segrave D. E. Mager S. A. Charman G. A. Edwards C. J. Porter (2004) ArticleTitlePharmacokinetics of recombinant human leukemia inhibitory factor in sheep J. Pharmacol. Exp. Ther. 309 1085–1092 Occurrence Handle10.1124/jpet.103.063289 Occurrence Handle14872093

    Article  PubMed  Google Scholar 

  12. D. E. Mager M. A. Mascelli N. S. Kleiman D. J. Fitzgerald D. R. Abernethy (2004) ArticleTitleNonlinear binding models suggest that abciximab pharmacokinetics and pharmacodynamics are target-mediated Clin. Pharmacol. Ther. 75 88 Occurrence Handle10.1016/j.clpt.2003.11.338

    Article  Google Scholar 

  13. D. E. Mager M. A. Mascelli N. S. Kleiman D. J. Fitzgerald D. R. Abernethy (2003) ArticleTitleSimultaneous modeling of abciximab plasma concentrations and ex vivo pharmacodynamics in patients undergoing coronary angioplasty J. Pharmacol. Exp. Ther. 307 969–976 Occurrence Handle10.1124/jpet.103.057299 Occurrence Handle14534354

    Article  PubMed  Google Scholar 

  14. J. G. Wagner ((1971)) A new generalized nonlinear pharmacokinetic model and its implications J. G. Wagner (Eds) Biopharmaceutics and Relevant Pharmacokinetics Drug Intelligence Publications Hamilton, IL 302–317

    Google Scholar 

  15. I. H. Segel (1975) Enzyme Kinetics. Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems John Wiley & Sons New York

    Google Scholar 

  16. J. D. Murray (2002) Mathematical Biology Springer New York

    Google Scholar 

  17. C. C. Linand L. A. Segel (1974) Mathematics Applied to Deterministic Problems in the Natural Sciences Macmillan Publishing Co., Inc. New York

    Google Scholar 

  18. D. Z. D'Argenio A. Schumitzky (1997) ADAPT II User's Guide Biomedical Simulations Resource Los Angeles

    Google Scholar 

  19. M. Gibaldi D. Perrier (1982) Pharmacokinetics Marcel Dekker, Inc. New York

    Google Scholar 

  20. M. J. Kemme R. C. Schoemaker J. Burggraaf M. Linden Particlevan der M. Noordzij M. Moerland C. Kluft A. F. Cohen (2003) ArticleTitleEndothelial binding of recombinant tissue plasminogen activator: quantification in vivo using a recirculatory model J. Pharmacokinet. Pharmacodyn. 30 3–22 Occurrence Handle10.1023/A:1023293325245 Occurrence Handle12800805

    Article  PubMed  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Dr. S. A. Charman, Dr. C. J. H. Porter, and Dr. A. M. Segrave for providing the LIF PK data. This study was funded in part by Grant 57980 from the National Institute of General Medical Sciences, National Institutes of Health (for W. K.) and start-up funds (to D. E. M.) from the University at Buffalo, State University of New York.

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Correspondence to Wojciech Krzyzanski.

Appendices

Appendix A

Quasi-equilibrium equations for the TMDD model

Equations(7a) and (7b) define the variables Ctot and Rtot as the concentration sums of free and bound complexes. To derive the differential equations describing them, similar operations should be performed on the equations for the TMDD model. Adding Eq. (1) to Eq. (4) will result in:

$${{\text{d}}C_{{{\text{tot}}}} } \mathord{\left/ {\vphantom {{{\text{d}}C_{{{\text{tot}}}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {\text{In}}{\left( t \right)} - k_{{\operatorname{int} }} {\text{RC}} - {\left( {k_{{{\text{el}}}} + k_{{{\text{pt}}}} } \right)}C + {k_{{{\text{tp}}}} \cdot A_{{\text{T}}} } \mathord{\left/ {\vphantom {{k_{{{\text{tp}}}} \cdot A_{{\text{T}}} } {V_{{\text{c}}} }}} \right. \kern-\nulldelimiterspace} {V_{{\text{c}}} }$$
(A1)

The RC variable can be calculated from Eq. (7a), such that:

$${\text{RC}} = C_{{{\text{tot}}}} - C$$
(A2)

After substituting Eq. (A2) for RC in Eq. (A1), it can be rearranged to Eq. (8). Similarly, to obtain the differential equation for Rtot, one needs to add Eqs. (3) and (4) and eliminate the variables RC and R using Eq. (A2):

$$R{\text{ = }}R_{{{\text{tot}}}} - {\text{RC = }}R_{{{\text{tot}}}} - {\text{(}}C_{{{\text{tot}}}} - {\text{C)}}$$
(A3)

This procedure results in differential equations for Ctot and Rtot that contain the variable C which in turn can be calculated from the quasi-equilibrium Eq. (6) after substituting Eqs. (A2) and (A3) for RC and R:

$$\frac{{C{\left[ {R_{{{\text{tot}}}} - {\left( {C_{{{\text{tot}}}} - C} \right)}} \right]}}} {{C_{{{\text{tot}}}} - C}} = K_{{\text{D}}} $$
(A4)

Equation (A4) can be multiplied side by side by the denominator of the left-hand side and rearranged to the quadratic equation for C:

$$C^{{\text{2}}} - {\text{(}}C_{{{\text{tot}}}} - R_{{{\text{tot}}}} - K_{{\text{D}}} {\text{)}}C - K_{{\text{D}}} C_{{{\text{tot}}}} {\text{ = 0}}$$
(A5)

Because KD > 0 and Ctot ≥ 0, this equation has exactly one nonnegative solution:

$$C = 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2{\left[ {{\left( {C_{{{\text{tot}}}} - R_{{{\text{tot}}}} - K_{{\text{D}}} } \right)} + {\sqrt {{\left( {C_{{{\text{tot}}}} - R_{{{\text{tot}}}} - K_{{\text{D}}} } \right)}^{2} + 4K_{{\text{D}}} C_{{{\text{tot}}}} } }} \right]}$$
(A6)

which is Eq. (11). Equation (A6) is algebraically equivalent to Eq. (A4) under assumption C ≥ 0. The differential equation for AT in the equilibrium model is identical to Eq. (2).

Appendix B

Steady-state equations for the quasi-equilibrium TMDD model

To derive the steady-state equations for the quasi-equilibrium model, we will assume that the input rate is constant [In(t) ≡ In] and the nonspecific tissue-binding site (AT) is present (ktp, kpt > 0). The derivations will hold for receptor turnover with the synthesis rate ksyn > 0 and degradation or internalization (kint + kdeg > 0). Also, we consider the model with at least one clearance mechanism (kel + kint > 0) and the binding process always present (KD > 0). At steady state, there is no change in the model variables, and therefore, the time derivatives in Eqs. (8)–(10) can be set to 0 resulting in a system of four algebraic equations with four unknowns (Css, ATss, Ctotss, and Rtotss):

$$0 = {\text{In}} - k_{{\operatorname{int} }} C_{{{\text{totss}}}} - {\left( {k_{{{\text{el}}}} + k_{{{\text{pt}}}} - k_{{\operatorname{int} }} } \right)}C_{{{\text{ss}}}} + {k_{{{\text{tp}}}} \cdot A_{{{\text{Tss}}}} } \mathord{\left/ {\vphantom {{k_{{{\text{tp}}}} \cdot A_{{{\text{Tss}}}} } {V_{{\text{c}}} }}} \right. \kern-\nulldelimiterspace} {V_{{\text{c}}} }$$
(B1)
$$0 = k_{{{\text{pt}}}} C_{{{\text{ss}}}} V_{{\text{c}}} - k_{{{\text{tp}}}} A_{{{\text{Tss}}}} $$
(B2)
$$0 = k_{{{\text{syn}}}} - {\left( {k_{{\operatorname{int} }} - k_{{\deg }} } \right)}{\left( {C_{{{\text{totss}}}} - C_{{{\text{ss}}}} } \right)} - k_{{\deg }} R_{{{\text{totss}}}} $$
(B3)
$$C_{{{\text{ss}}}} = 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2{\left[ {{\left( {C_{{{\text{totss}}}} - R_{{{\text{totss}}}} - K_{{\text{D}}} } \right)} + {\sqrt {{\left( {C_{{{\text{totss}}}} - R_{{{\text{totss}}}} - K_{{\text{D}}} } \right)}^{2} + 4K_{{\text{D}}} C_{{{\text{totss}}}} } }} \right]}$$
(B4)

Equation (B4) is equivalent to the equilibrium Eq. (6) that can be rewritten as follows:

$$C_{{{\text{totss}}}} = C_{{{\text{ss}}}} + {R_{{{\text{totss}}}} \cdot C_{{{\text{ss}}}} } \mathord{\left/ {\vphantom {{R_{{{\text{totss}}}} \cdot C_{{{\text{ss}}}} } {{\left( {K_{{\text{D}}} + C_{{{\text{ss}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {K_{{\text{D}}} + C_{{{\text{ss}}}} } \right)}}$$
(B5)

where the relationships defined by Eqs. (7a) and (7b) have been exploited. Equation (B2) allows one to express ATss as a function of Css:

$$A_{{{\text{Tss}}}} = k_{{{\text{pt}}}} V_{{{\text{ss}}}} \cdot {C_{{{\text{ss}}}} } \mathord{\left/ {\vphantom {{C_{{{\text{ss}}}} } {k_{{{\text{tp}}}} }}} \right. \kern-\nulldelimiterspace} {k_{{{\text{tp}}}} }$$
(B6)

Dividing Eq. (B2) by Vc and adding it to Eq. (B1), we cancel off the AT term and obtain:

$$k_{{\operatorname{int} }} {\left( {C_{{{\text{totss}}}} - C_{{{\text{ss}}}} } \right)} + k_{{{\text{el}}}} C_{{{\text{ss}}}} = {\text{In}}$$
(B7)

If In = 0, and because CtotssCss and ksyn > 0, then Eqs. (B3), (B5), and (B7) imply that kdeg > 0, and:

$$C_{{{\text{ss}}}} = C_{{{\text{totss}}}} = 0\,{\text{and}}\,R_{{{\text{totss}}}} = {k_{{{\text{syn}}}} } \mathord{\left/ {\vphantom {{k_{{{\text{syn}}}} } {k_{{\deg }} }}} \right. \kern-\nulldelimiterspace} {k_{{\deg }} }$$
(B8)

Later, we will assume that In > 0. Let us first consider the case where receptors are eliminated through the turnover process (kdeg > 0). Equation (B5) can be used to calculate CtotssCss and substitute for this term in Eq. (B3) yielding:

$$R_{{{\text{totss}}}} = \frac{{k_{{{\text{syn}}}} {\left( {K_{{\text{D}}} + C_{{{\text{ss}}}} } \right)}}} {{k_{{\operatorname{int} }} C_{{{\text{ss}}}} + k_{{\deg }} K_{{\text{D}}} }}$$
(B9)

The only unknown now is Css. If kint = 0, then Css can be easily calculated from Eq. (B7):

$$C_{{{\text{ss}}}} = {{\text{In}}} \mathord{\left/ {\vphantom {{{\text{In}}} {k_{{{\text{el}}}} }}} \right. \kern-\nulldelimiterspace} {k_{{{\text{el}}}} }$$
(B10)

If kint > 0, then calculation of the steady states becomes more troublesome. Css can be determined from Eq. (B3) if Eq. (B5) is used to replace Rtotss and Eq. (B7) to substitute for CtotssCss resulting in the following equation for Css:

$$0 = k_{{{\text{syn}}}} - {\left( {k_{{\operatorname{int} }} - k_{{\deg }} } \right)}{\left( {{\text{In}} - k_{{{\text{el}}}} C_{{{\text{ss}}}} } \right)}/k_{{\operatorname{int} }} - k_{{\deg }} {\left( {{\text{In}} - k_{{{\text{el}}}} C_{{{\text{ss}}}} } \right)}{\left( {K_{{\text{D}}} + C_{{{\text{ss}}}} } \right)}/{\left( {k_{{\operatorname{int} }} C_{{{\text{ss}}}} } \right)}$$
(B11)

After multiplying both sides of Eq. (B11) by kintCss, it can be rearranged to the equation for Css:

$$k_{{{\text{el}}}} k_{{\operatorname{int} }} C^{2}_{{{\text{ss}}}} + {\left[ {k_{{\operatorname{int} }} {\left( {k_{{{\text{syn}}}} - {\text{In}}} \right)} + k_{{\deg }} k_{{{\text{el}}}} K_{{\text{D}}} } \right]}C_{{{\text{ss}}}} - k_{{\deg }} K_{{\text{D}}} {\text{In}} = 0$$
(B12)

If kel = 0, then the positive solution exists only if ksyn > In:

$$C_{{{\text{ss}}}} = \frac{{k_{{\deg }} K_{{\text{D}}} {\text{In}}}} {{{\left( {k_{{{\text{syn}}}} - {\text{In}}} \right)}k_{{\operatorname{int} }} }}$$
(B13)

If kel > 0, then the positive solution is:

$$C_{{{\text{ss}}}} = \frac{{{\text{In}}k_{{\operatorname{int} }} - k_{{\deg }} k_{{{\text{el}}}} K_{{\text{D}}} - k_{{{\text{syn}}}} k_{{\operatorname{int} }} + {\sqrt {{\left( {{\text{In}}k_{{\operatorname{int} }} - k_{{\deg }} k_{{{\text{el}}}} K_{{\text{D}}} - k_{{{\text{syn}}}} k_{{\operatorname{int} }} } \right)}^{2} + 4k_{{{\text{el}}}} k_{{\operatorname{int} }} k_{{\deg }} {\text{In}}K_{{\text{D}}} } }}} {{2k_{{{\text{el}}}} k_{{\operatorname{int} }} }}$$
(B14)

Let us now consider the case kdeg = 0. Then the only clearance mechanism for bound receptors is the internalization process. If kint > 0, then Eq. (B3) implies:

$$C_{{{\text{totss}}}} - C_{{{\text{ss}}}} = {k_{{{\text{syn}}}} } \mathord{\left/ {\vphantom {{k_{{{\text{syn}}}} } {k_{{\operatorname{int} }} }}} \right. \kern-\nulldelimiterspace} {k_{{\operatorname{int} }} }$$
(B15)

and from Eq. (B7), it follows that for kel > 0 and In > ksyn:

$$C_{{{\text{ss}}}} = {{\left( {{\text{In}} - k_{{{\text{syn}}}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\text{In}} - k_{{{\text{syn}}}} } \right)}} {k_{{{\text{el}}}} }}} \right. \kern-\nulldelimiterspace} {k_{{{\text{el}}}} }$$
(B16)

Rtotss can be determined from Eq. (B8). If kel = 0, then after adding Eqs. (B3) and (B7), one can conclude that ksyn = In, thus implying:

$$C_{{{\text{tot}}}} + {A_{{\text{T}}} } \mathord{\left/ {\vphantom {{A_{{\text{T}}} } {V_{{\text{c}}} }}} \right. \kern-\nulldelimiterspace} {V_{{\text{c}}} } - R_{{{\text{tot}}}} = {\text{constant}}$$
(B17)

Hence,

$$R_{{{\text{totss}}}} {\text{ = }}C_{{{\text{totss}}}} {\text{ + }}k_{{{\text{pt}}}} \cdot {C_{{{\text{ss}}}} } \mathord{\left/ {\vphantom {{C_{{{\text{ss}}}} } {k_{{{\text{pt}}}} }}} \right. \kern-\nulldelimiterspace} {k_{{{\text{pt}}}} }{\text{ + constant}}$$
(B18)

Inserting Rtotss from Eq. (B18) and Ctotss described by Eq.(B15) into Eq. (B5) yields an equation for Css:

$${k_{{{\text{syn}}}} } \mathord{\left/ {\vphantom {{k_{{{\text{syn}}}} } {k_{{\operatorname{int} }} }}} \right. \kern-\nulldelimiterspace} {k_{{\operatorname{int} }} }{\text{ = (}}C_{{{\text{ss}}}} {\text{ + }}k_{{{\text{syn}}}} {\text{/}}k_{{\operatorname{int} }} {\text{ + }}k_{{{\text{pt}}}} \cdot {C_{{{\text{ss}}}} } \mathord{\left/ {\vphantom {{C_{{{\text{ss}}}} } {k_{{{\text{pt}}}} }}} \right. \kern-\nulldelimiterspace} {k_{{{\text{pt}}}} }{\text{ + constant)}} \cdot {C_{{{\text{ss}}}} } \mathord{\left/ {\vphantom {{C_{{{\text{ss}}}} } {{\left( {K_{{\text{D}}} {\text{ + }}C_{{{\text{ss}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {K_{{\text{D}}} {\text{ + }}C_{{{\text{ss}}}} } \right)}}$$
(B19)

which can be transformed to the following quadratic equation:

$${\left( {{\text{1 + }}{k_{{{\text{pt}}}} } \mathord{\left/ {\vphantom {{k_{{{\text{pt}}}} } {k_{{{\text{tp}}}} }}} \right. \kern-\nulldelimiterspace} {k_{{{\text{tp}}}} }} \right)}C^{2}_{{{\text{ss}}}} {\text{ + constant}} \cdot C_{{{\text{ss}}}} {\text{ - }}K_{{\text{D}}} \cdot {k_{{{\text{syn}}}} } \mathord{\left/ {\vphantom {{k_{{{\text{syn}}}} } {k_{{\operatorname{int} }} }}} \right. \kern-\nulldelimiterspace} {k_{{\operatorname{int} }} }{\text{ = 0}}$$
(B20)

Equation (B21) has only one nonnegative solution:

$$C_{{{\text{ss}}}} = \frac{1} {{2{\left( {1 + \frac{{k_{{{\text{pt}}}} }} {{k_{{{\text{tp}}}} }}} \right)}}}{\left( {{\text{constant}} + {\sqrt {{\text{constant}}^{2} + 4\frac{{k_{{{\text{syn}}}} K_{{\text{D}}} }} {{k_{{\operatorname{int} }} }}{\left( {1 + \frac{{k_{{{\text{pt}}}} }} {{k_{{{\text{tp}}}} }}} \right)}} }} \right)}$$
(B21)

This completes derivations of the steady-state solutions forthe quasi-equilibrium model, which are summarized in Table II.

Table II Steady-State Solutions for the Quasi-EquilibriumTMDD Modela

Appendix C

Derivation of Wagner nonlinear tissue binding [Eq. (16)]

For derivation of Eq. (16), the best form of the equilibrium Eq. (6) is:

$$C_{{{\text{tot}}}} = C + R_{{{\text{tot}}}} C/(K_{{\text{D}}} + C)$$
(C1)

where Rtot is constant. One can differentiate both sides of Eq.(C1) and obtain:

$$\frac{{{\text{d}}C_{{{\text{tot}}}} }} {{{\text{d}}t}} = \frac{{{\text{d}}C}} {{{\text{d}}t}} + \frac{{R_{{{\text{tot}}}} K_{{\text{D}}} }} {{(K_{{\text{D}}} + C)^{2} }}\frac{{{\text{d}}C}} {{{\text{d}}t}}$$
(C2)

Equation (8), with kint = 0, can be substituted in the left-hand side of Eq. (C2) resulting in:

$${\text{In}}{\left( t \right)} - {\left( {k_{{{\text{el}}}} + k_{{{\text{pt}}}} } \right)}C + {k_{{{\text{tp}}}} \cdot A_{{\text{T}}} } \mathord{\left/ {\vphantom {{k_{{{\text{tp}}}} \cdot A_{{\text{T}}} } {V_{{\text{c}}} }}} \right. \kern-\nulldelimiterspace} {V_{{\text{c}}} } = {\left( {1 + \frac{{R_{{{\text{tot}}}} K_{{\text{D}}} }} {{(K_{{\text{D}}} + C)^{2} }}} \right)}\frac{{{\text{d}}C}} {{{\text{d}}t}}$$
(C3)

Finally, one can solve Eq. (C3) for dC/dt and arrive at Eq. (16).

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Mager, D.E., Krzyzanski, W. Quasi-Equilibrium Pharmacokinetic Model for Drugs Exhibiting Target-Mediated Drug Disposition. Pharm Res 22, 1589–1596 (2005). https://doi.org/10.1007/s11095-005-6650-0

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Key Words

  • leukemia inhibitory factor
  • mathematical model
  • nonlinear pharmacokinetics
  • target-mediated drug disposition