| Citation: |
Jing Zhang, Jiao Jiang, Xiaotian Wu. MATHEMATICAL SOLUTION OF A PHARMACOKINETIC MODEL WITH SIMULTANEOUS FIRST-ORDER AND HILL-TYPE ELIMINATION[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 623-643. doi: 10.11948/20210365
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MATHEMATICAL SOLUTION OF A PHARMACOKINETIC MODEL WITH SIMULTANEOUS FIRST-ORDER AND HILL-TYPE ELIMINATION
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Abstract
Mathematical studies on pharmacokinetic models are essentially important for drug development and optimal dose design. Considering the interaction between drug molecules and their receptors, the elimination of drug molecules can exhibit Hill-type kinetics. In this paper, motivated by the recombinant human granulocyte colony-stimulating factor (G-CSF) and the transcendent Lambert $W$ function, we have studied the mathematical solutions of a one-compartment nonlinear pharmacokinetic model with simultaneous first-order and Hill-type $(n=2)$ elimination for the case of intravenous bolus administration. By introducing three well-defined transcendental functions depending on three different scenarios, we have established the closed-form precise solutions of time course of drug concentration, which is a method to calculate drug concentrations at any time point. As a result, we also have derived the explicit expressions of some key pharmacokinetic surrogates such as the elimination half-life $t_{1/2}$ and total drug exposure (i.e. area under the concentration curve (AUC)), which are found as dose-dependent. Finally, a case study of a G-CSF drug is quantitatively illustrated to delineate our theoretical results, including the elimination half-life and AUC for different dosages. Our findings can provide an effective guidance for drugs with simultaneous first-order and Hill-type $(n=2)$ elimination in clinical pharmacology.
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References
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Jing Zhang, Jiao Jiang, Xiaotian Wu. MATHEMATICAL SOLUTION OF A PHARMACOKINETIC MODEL WITH SIMULTANEOUS FIRST-ORDER AND HILL-TYPE ELIMINATION[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 623-643. doi: 10.11948/20210365
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Figure 1.
Time courses of drug concentration of model (2.1) for different three cases, where
$ V_{m}=1\, \mu g/(L\cdot h);\ K_{D}=10\, \mu g/L;\ V_{d}=1\, L;\ D=300\, \mu g/L $ $ K_{ren}=0.04\, h^{-1}, \ 0.05\, h^{-1}, \ 0.09\, h^{-1} $ $ 0<r<1, \ r=1, \ r>1 $ -
Figure 2.
The relationship between elimination half-life (
$ t_{1/2} $ $ 0<r<1 $ $ K_{ren}=0.153/h, V_{m}=0.5\mu g/(L\cdot h), K_{D}=1.44\mu g/L, V_{d}=1.788L, {\rm(a)} D=20\mu g; {\rm(b)} D=3.5\mu g; {\rm(c)} D=0.4\mu g $ -
Figure 3.
The relationship between AUC and dose
$ D $ $ V_{m}=1\mu g/(L\cdot h) $ $ K_{D}=10\mu g/L $ $ V_{d}=1L $ $ K_{ren}=0.04/h, \ 0.05/h, \ 0.09/h $ $ 0<r<1 $ $ r=1 $ $ r>1 $ -
Figure 4.
The relationship between AUC and nonlinear parameters
$ K_{D} $ $ V_{m} $ $ r>1 $ $ K_{ren}=0.429/h, V_{d}=1L, D=20\mu g $ -
Figure 5.
The dominant role of two parallel, linear elimination and Hill-type elimination (
$ n=2 $ -
Figure 6.
The range of drug concentration induced by model (1.2) and model (2.1)(
$ 0<r<1 $ $ K_{ren}=0.04/h, V_{m}=1\mu g/(L\cdot h), K_{D}=10\mu g/L, V_{d}=1L, D=20\mu g $ -
Figure 7.
The relationship between
$ t_{1/2} $ $ 0<r<1 $ $ K_{ren}=0.04/h, V_{m}=1\mu g/(L\cdot h), K_{D}=10\mu g/L, V_{d}=1L $ $ D=20\mu g $ -
Figure 8.
Three real branch images corresponding to defined functions. The abscissa is the independent variable
$ x $ $ f(x), g(x), h(x) $