Analysis of a complex physiology-directed model for inhibition of platelet aggregation by clopidogrel

Lambertus A. Peletier; Xi-Ling Jiang; Snehal Samant; Stephan Schmidt

doi:10.3934/dcds.2017039

Article Contents

2017, Volume 37, Issue 2: 945-961. Doi: 10.3934/dcds.2017039

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Analysis of a complex physiology-directed model for inhibition of platelet aggregation by clopidogrel

1.
Mathematical Institute, Leiden University, PB 9512,2300 RA Leiden, The Netherlands
2.
Department of Pharmaceutics, Center for Pharmacometrics and Systems Pharmacology, University of Florida at Lake Nona (Orlando), 6550 Sanger Road, Office 467, Orlando, FL 32827, USA

Author Bio: E-mail address: peletier@math.leidenuniv.nl; E-mail address: xiling.jiang@gmail.com; E-mail address: ssamant@ufl.edu; E-mail address: SSchmidt@cop.ufl.edu
* Corresponding author: Lambertus A. Peletier
* Corresponding author: Lambertus A. Peletier

Received: May 31, 2015

Revised: January 31, 2016

Published: May 2017

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Abstract

Clopidogrel is an anti-platelet compound that is widely used with aspirin to reduce the risk of cardiovascular incidents.In itself it is inactive; only after a biotransformation into its active metabolite clop-AM, does it inhibit platelet aggregation.Recently a system-pharmacological model has been proposed for the network of processes leading to reduced platelet aggregation.In this paper we present a mathematical analysis of this model and demonstrate how the complex pharmacokinetic modelcan be reduced to two simple coupled models, one for clopidogrel and one for clop-AM, yielding insight into the dynamicsof clop-AM and the impact of inter-individual differences on the level of inhibition.

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Mathematics Subject Classification: Primary:92C42, 92C45;Secondary:92C450.

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Figure 1. Schematic model of clopidogrel action on platelet aggregation: clopidogrel travels from the gut to the liver, where one fraction ($E_{CES1}$) is hydrolysed into an inactive metabolite, one fraction ($E_{CYP}$) is transformed into the active metabolite clop-AM and one fraction $F_H$ goes into systemic circulation. Clop-AM also goes into systemic circulation where it binds to receptors situated on the platelets and thus inhibits platelet aggregation. The compartments are numbered 1 to 6; the amounts of clopidogrel in the gut, the liver and plasma are denoted by, respectively, $A_1, A_2$ and $A_3$ and the amounts of clop-AM in the liver and in plasma by $A_4$ and $A_5$. The platelet reactivity in compartment 6 is denoted by $P$

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Figure 2. Graph of the clop-AM concentration in plasma ($C_5$) versus the clopidogrel concentration in the liver ($C_2$) based on equation (15) and parameter values given in Table 1

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Figure 4. Temporal behaviour of $A_1(t),\dots, A_5(t)$ according to the PK model with parameter values given by Table 1 after an iv bolus dose of 300 mg i.e., 931 $\mu$mol. In the left two panels values of $A_1-A_5$ are given on a linear scale, and in the right panel they are given on a logarithmic scale

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Figure 3. Graph of the response $P_{\rm ss}$ versus the clop-AM concentration in plasma ($C_5$) based on equation (18) and parameter values given in Table 2

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Figure 5. The relative maximal platelet aggregation $P(t)$ versus time according to equation (17), together with clop-AM concentration $C_5$ in blood plasma on two time scales: on the same scale of the PK graphs shown in Figure 4 and one on a time scale which is much larger. PK and PD parameters are taken from, respectively, Table 1 and Table 2, and the iv dose was 300 mg, i.e., 931 $\mu$mol

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Figure 6. Orbit in the $(x_4,x_5)$-plane (red) together with the null clines $\Gamma_4$ (blue) and $\Gamma_5$ (green) for PK parameter values from Table 1 and an iv bolus dose of 300 mg i.e., 931 $\mu$mol

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Table 1. PK parameter estimates

Parameter	Unit	Estimate	CV %
$Q_H$	L/h	50	0 (fixed)
$V_H$	L	1.5	0 (fixed)
$F_a$	-	0.5	0 (fixed)
$k_{a}$	1/h	9.28	7.63
$V_3$	L	61.3	24.3
$V_{max:CYP}$	$\mu$mol/h	314	23.2
$K_{m:CYP}$	$\mu$M	4.95	27.7
$CL_{int:CES1}$	L/h	19400	19.5
$CL_{50}$	L/h	3.86	11.5
$V_5$	L	3	0 (fixed)

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Table 2. PD parameter estimates

Parameter	Unit	Estimate	CV %
$k_{\rm in}$	1/h	0.00783	5.54
$k_{\rm out}$	1/h	0.00783	5.54
$k_{\rm irre}$	1/$\mu$M/h	4.06	4.14

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