Drug dosage determination and potential drug interference when multiple medical compounds must be administered simultaneous is an important long-standing problem both in practical pharmacokinetics and in theoretical drug design modeling. Very simple, and mostly linear, models are currently used to describe drug distribution in a body, drug function, and drug elimination. Many of the processes involved in drug delivery occur on vastly different time scales. This fact and, in particular, the presence of fast forward and reverse drug binding to blood proteins, is used in this paper to produce the reduced models describing time dependent drug dynamics during intravenous drug delivery, i.e., when the drug is administered directly in patient's vein via catheter. In addition, the questions on whether the drug dosage must be adjusted in the presence of protein binding compared to the case of drugs which do not bind, as well as what happens when two administered drugs participate in competing protein binding reactions are addressed. The singularly perturbed models derived under natural assumptions are analyzed using the boundary function method approach.
Citation: Leonid Kalachev. Reduced models of drug delivery in the presence of fast protein binding[J]. Mathematics in Engineering, 2025, 7(2): 162-177. doi: 10.3934/mine.2025007
Drug dosage determination and potential drug interference when multiple medical compounds must be administered simultaneous is an important long-standing problem both in practical pharmacokinetics and in theoretical drug design modeling. Very simple, and mostly linear, models are currently used to describe drug distribution in a body, drug function, and drug elimination. Many of the processes involved in drug delivery occur on vastly different time scales. This fact and, in particular, the presence of fast forward and reverse drug binding to blood proteins, is used in this paper to produce the reduced models describing time dependent drug dynamics during intravenous drug delivery, i.e., when the drug is administered directly in patient's vein via catheter. In addition, the questions on whether the drug dosage must be adjusted in the presence of protein binding compared to the case of drugs which do not bind, as well as what happens when two administered drugs participate in competing protein binding reactions are addressed. The singularly perturbed models derived under natural assumptions are analyzed using the boundary function method approach.
| [1] | M. A. Hedaya, Basic pharmacokinetics, 3 Eds., New York: Routledge, 2023. https://doi.org/10.4324/9781003161523 |
| [2] | J. M. Ritter, R. Flower, G. Henderson, Y. K. Loke, D. MacEwan, H. P. Rang, Absorption and distribution of drugs, In: Rang and Dale's pharmacology, 9 Eds., Elsevier, 2020. |
| [3] | C. Palleria, A. Di Paolo, C. Giofrè, C. Caglioti, G. Leuzzi, A. Siniscalchi, et al., Pharmacokinetic drug-drug interaction and their implication in clinical management, J. Res. Med. Sci., 18 (2013), 601–610. |
| [4] |
X. Zheng, Z. Li, M. I. Podariu, D. S. Hage, Determination of rate constants and equilibrium constants for solution-phase drug-protein interactions by ultrafast affinity extraction, Anal. Chem., 86 (2014), 6454–6560. https://doi.org/10.1021/ac501031y doi: 10.1021/ac501031y
|
| [5] | R. E. O'Malley, Singular perturbation methods for ordinary differential equations, Vol. 89, New York: Springer-Verlag, 1991. https://doi.org/10.1007/978-1-4612-0977-5 |
| [6] | L. Edelstein-Keshet, Mathematical models in biology, Reprint of the 1988 original, Philadelphia: Society for Industrial and Applied Mathematics, 2005. |
| [7] | J. D. Murray, Mathematical biology, 2 Eds., Springer, 1993. https://doi.org/10.1007/978-3-662-08542-4 |
| [8] |
S. Schnell, P. K. Maini, A century of enzyme kinetics: reliability of the $K_M$ and $v_max$ estimates, Comm. Theoret. Biol., 8 (2003), 169–187. https://doi.org/10.1080/08948550390206768 doi: 10.1080/08948550390206768
|
| [9] | D. Shchepakin, L. Kalachev, M. Kavanaugh, Mathematical models in neuroscience: approaches to experimental design and reliable parameter determination, In: B. Sriraman, Handbook of the mathematics of the arts and sciences, Springer, 2021, 2319–2357. https://doi.org/10.1007/978-3-319-57072-3_134 |
| [10] | I. H. Segel, Enzyme kinetics: behavior and analysis of rapid equilibrium and steady state enzyme systems, New York: Wiley and Sons, 1975. |
| [11] |
L. A. Segel, M. Slemrod, The quasi-steady state assumption: a case study in perturbation, SIAM Review, 31 (1989), 446–477. https://doi.org/10.1137/1031091 doi: 10.1137/1031091
|
| [12] |
L. V. Kalachev, Reduced model of neurotransmitter transport in the presence of generic receptors and transporters, J. Phys.: Conf. Ser., 55 (2006), 114–129. https://doi.org/10.1088/1742-6596/55/1/011 doi: 10.1088/1742-6596/55/1/011
|
| [13] | A. B. Vasil'eva, V. F. Butuzov, L. V. Kalachev, The boundary function method for singular perturbation problems, Philadelphia: Society for Industrial and Applied Mathematics, 1995. https://doi.org/10.1137/1.9781611970784 |
| [14] |
H. Haario, L. Kalachev, M. Laine, Reduction and identification of dynamic models. Simple example: generic receptor model, Discrete Cont. Dyn. Syst., Ser. B, 18 (2013), 417–435. https://doi.org/10.3934/dcdsb.2013.18.417 doi: 10.3934/dcdsb.2013.18.417
|
| [15] |
K. G. Gurevich, Effect of blood protein concentrations on drug-dosing regimes: practical guidance, Theor. Biol. Med. Model., 10 (2013), 20. https://doi.org/10.1186/1742-4682-10-20 doi: 10.1186/1742-4682-10-20
|
| [16] |
J. A. Roberts, F. Pea, J. Lipman, The clinical relevance of plasma protein binding changes, Clin. Pharmacokinet., 52 (2013), 1–8. https://doi.org/10.1007/s40262-012-0018-5 doi: 10.1007/s40262-012-0018-5
|
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Leonid Kalachev. Reduced models of drug delivery in the presence of fast protein binding[J]. Mathematics in Engineering, 2025, 7(2): 162-177. doi: 10.3934/mine.2025007
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