One-compartment stochastic pharmacokinetic model
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Mar 8, 2023
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Abstract
In this work, we consider a pharmacokinetic (PK) model with first-order drug absorption and first-order elimination that represent the concentration of drugs in the body, including both the absorption and elimination parts, and we also add a random factor to describe the variability between patients and the environment. Using Itô’s lemma and the Laplace transform, we obtain the solutions in integral form for a single and constant dosage regimen in time. Moreover, formulas for the expected value and the variance for each case of study are presented, which allows the statistical assessment of the proposed models, as well as predicting the ideal path of drug concentration and its uncertainty. These results are important in the long-term analysis of drug concentration and the persistence of therapeutic level. Further, a numerical method for the solution of the stochastic differential equation (SDE) is introduced
and developed.
Keywords
Stochastic differential equations, Itô's Lemma, Analytic solutions, PK model
References
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[25] Schnider TW, Minto CF, Gambus PL, Andresen C, Goodale DB, Shafer SL, Youngs EJ. The Influence of Method of Administration and covariates on the pharmacokinetics of propofol in Adult Volunteers. Anesthesiology, 88(5): 1170–1182, 1998.
doi: 10.1097/00000542-199805000-00006
doi: 10.1371/journal.pone.0193074
[2] Qianning L, Qingsong S. A Stochastic Analysis of the One Compartment Pharmacokinetic Model Considering Optimal Controls. IEEE Access, 8: 181825–181834, 2020.
doi: 10.1109/access.2020.3028741
[3] Gibaldi M, Perrier D. Pharmacokinetics (Drugs and the Pharmaceutical Sciences: a Seriesof Textbooks and Monographs). Informa Healthcare, 2007.
[4] Ferrante L, Bompadre S, Leone L. A stochastic compartmental model with long lasting infusion. Biometrical journal, 45: 182–194, 2003.
doi: 10.1002/bimj.200390004
[5] Ramanathan M. An application of itô’s lemma in population pharmacokinetics and pharmacodynamics. Pharmaceutical Research, 16(4): 584–586, 1999.
doi: 10.1023/a%3A1011910800110
[6] Liu Z, Yang Y. Uncertain pharmacokinetic model based on uncertain differential equation.Applied Mathematics and Computation, 404: 126118, 2021.
doi: 10.1016/j.amc.2021.126118
[7] Liu B. Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, 3:3–10, 2009.
[8] Ramanathan M. A method for estimating pharmacokinetic risks of concentration dependentdrug interactions from preclinical data. Drug Metabolism and disposition, 27: 1479–1487, 1999.
[9] Ditlevsen S, De Gaetano A. Stochastic deterministic uptake of dodecanedioic acid by isolated rat liver. Bulletin of Mathematical Biology, 67: 547–561, 2005.
doi: 10.1016/j.bulm.2004.09.005
[10] Donnet S, Samson A. A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models. Advanced Drug Delivery Reviews, 65(7): 929–939, 2013.
doi: 10.1016/j.addr.2013.03.005
[11] Liu Z, Yang Y. Pharmacokinetic model based on multifactor uncertain differential equation, Applied Mathematics and Computation, 392: 1–16, 2021. doi: 10.1016/j.amc.2020.125722
[12] Picchini U, Ditlevsen S, De gaetano A. Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations. Journal of Mathematical Biology, 53: 771–796, 2006.
doi: 10.1007/s00285-006-0032-z
[13] Saqlain M, Alam M, Rönnegård L, Westin J. Investigating Stochastic Differential Equations Modelling for Levodopa Infusion in Patients with Parkinson’s Disease. European Journal of Drug Metabolism and Pharmacokinetics, 45: 41–49, 2020.
doi: 10.1007/s13318-019-00580-w
[14] Tornoe CW, Jacobse JL, Madsen H. Grey-box pharmacokinetic/pharmacodynamic modelling of a euglycaemic clamp study. Jourrnal of Mathematical Biology. 48(6): 591–604, 2003.
doi: 10.1007/s00285-003-0257-z
[15] Wagner J, Nelson E. Per Cent Absorbed Time Plots Derived from Blood Level and/or Urinary Excretion Data. Journal of Pharmaceutical Sciences, 5(2): 610–611, 1963.
doi: 10.1002/jps.2600520629
[16] Sweeney GD. Variability in the human drug response. Thrombosis Research, 29: 3–15, 1983.
doi: 10.1016/0049-3848(83)90353-5
[17] Karniadakis G, Zhang Z. Numerical methods for stochastic partial differential equations with white noise, Applied mathematical sciences, 196, 2017.
[18] Oksendal B. Stochastic differential equations: an introduction with applications. Springer-Verlag, 2007.
[19] Kloeden P, Platen E, Numerical solution of stochastic differential equations. Springer-Verlag, 1992.
[20] Xiao Y, Zhang L, Fang Y. Numerical solution stability of general stochastic differential equation, Journal of Interdisciplinary Mathematics, 21(6): 1471–1479, 2018.
doi: 10.1080/09720502.2018.1513822
[21] Boeckmann A, Sheiner L, Beal S., NONMEM Users Guide-Part V: Introductory Guide. NONMEM Project Group. University of California, San Francisco, 1994.
[22] Fisz M. Probability theory and mathematical statistics. Krieger Publication Company, 1980.
[23] Bialer M. A simple method for determining whether absorption and elimination constants rates are equal in the one-compartment open model with first-order processes. Journal of Pharmacokinetics and Pharmacodynamics, 8(1): 111–113, 1980.
doi: 10.1007/BF01059453
[24] Patel IH. Concentration ratio method to determine the constant rate for the special case when ka = ke. Journal of Pharmaceutical Sciences, 73(6): 859–861, 1984.
doi: 10.1002/jps.2600730648
[25] Schnider TW, Minto CF, Gambus PL, Andresen C, Goodale DB, Shafer SL, Youngs EJ. The Influence of Method of Administration and covariates on the pharmacokinetics of propofol in Adult Volunteers. Anesthesiology, 88(5): 1170–1182, 1998.
doi: 10.1097/00000542-199805000-00006
How to Cite
Cano Macias, R., Jiménez Moscoso, J. A., & Ruiz Vera, J. M. (2023). One-compartment stochastic pharmacokinetic model. Universitas Scientiarum, 28(1), 23–41. https://doi.org/10.11144/Javeriana.SC281.ocsp
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Mathematics and Statistics
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