# American Institute of Mathematical Sciences

February  2017, 37(2): 945-961. doi: 10.3934/dcds.2017039

## 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

* Corresponding author: Lambertus A. Peletier

Received  June 2015 Revised  February 2016 Published  November 2016

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.

Citation: Lambertus A. Peletier, Xi-Ling Jiang, Snehal Samant, Stephan Schmidt. Analysis of a complex physiology-directed model for inhibition of platelet aggregation by clopidogrel. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 945-961. doi: 10.3934/dcds.2017039
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##### References:
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 $P$
">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
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">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
">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
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">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
and an iv bolus dose of 300 mg i.e., 931 $\mu$mol">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
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)
 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)
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
 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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