Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅱ: Cell proliferation

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doi:10.3934/mbe.2018050

Article Contents

2018, Volume 15, Issue 5: 1117-1135. Doi: 10.3934/mbe.2018050

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Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅱ: Cell proliferation

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1.
College of Engineering, Mathematics, and Physical Sciences, University of Exeter, Exeter, Devon, EX4 4QF, United Kingdom
2.
Department of Mathematics, University of Portsmouth, Winston Churchill Ave, Portsmouth PO1 2UP, United Kingdom
3.
Mathematics Applications Consortium for Science and Industry, University of Limerick, Castletroy, Co. Limerick, Ireland

Received: April 26, 2017

Accepted: February 28, 2018

Published: October 2018

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Abstract

The aim of a drug eluting stent is to prevent restenosis of arteries following percutaneous balloon angioplasty. A long term goal of research in this area is to use modelling to optimise the design of these stents to maximise their efficiency. A key obstacle to implementing this is the lack of a mathematical model of the biology of restenosis. Here we investigate whether mathematical models of cancer biology can be adapted to model the biology of restenosis and the effect of drug elution. We show that relatively simple, rate kinetic models give a good description of available data of restenosis in animal experiments, and its modification by drug elution.

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Mathematics Subject Classification: Primary: 93M60, 92C50; Secondary: 93A30, 92C35.

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Figure 1. Experimental data: neointimal thickness in rabbit iliac artery. Reproduced from [7]. Each data point denotes the mean value across several experiments, with 23 rabbits in total for each the bare-metal and the drug-eluting stents. The lines joining the dots are a linear interpolation. In practice, the behaviour of the intima for times between these data points has not been experimentally determined, and it is a major point of this work to better elucidate the dynamics over the entire post-implantation time.

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Figure 2. The modelled cell cycle is shown. Rates of change from $Q$ to $P$ and back are denoted $\beta$ and $\alpha$ respectively. Loss rates from the two phases are denoted by $\lambda_{P}$ and $\lambda_{Q}$. Finally, the growth rate in the proliferative phase is denoted with $\gamma$.

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Figure 3. Phase plot of proliferative cell fraction. $\frac{\partial P}{\partial t}$ vs. $P$ is shown for both the healthy and inflamed arteries. The distance between these solutions may be used to estimate the net transition rate, $\psi$. This distance is shown on the plot with a double-headed arrow.

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Figure 4. Proliferative fraction, $P$, in response to drug-free stent implantation. Note the presence of two temporal domains. On the first, depicted with a dashed line, there is an inflammatory response to the implantation of a stent (cf. $\bar{P}_{1}$, equation (22)). On the second, indicated with a dotted line, the vasculature is returning to its normal state (cf. $\bar{P}_{2}$, equation (27)).

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Figure 5. The increase in the intimal thickness, $L(t)$, in response to drug-free stent implantation. As with Figure 4, note the presence of two temporal domains. On the first, the increase in thickness corresponding to the inflamed response given in equation (31) is shown with a dashed line. The second corresponds to the return to steady state corresponding to equation (33).

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Figure 6. Example of drug effectiveness, $\mu$.

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Figure 7. Phase plot of proliferative cell fraction, $\mu_{k}>0$.

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Figure 8. Example of modified $\psi$ value, considering drug effects.

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Table 1. The equations of state for the various models considered herein

Full System	$ \dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma - \alpha - \lambda_{P} -\mu_{P})P + (\beta + \eta - \mu_{Q})Q $
Full System	$ \dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = (\alpha)P - (\beta + \eta + \lambda_{Q} - \mu_{Q})Q $
Reduced System	$\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma' - \alpha - \mu_{P})P + (\psi - \mu_{Q})Q $
Reduced System	$ \dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = \alpha P - (\psi + \lambda_{Q} - \mu_{Q})Q $
Growth-Inhibiting Model	$\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma' - \alpha - \mu_{P})P + \psi Q $
Growth-Inhibiting Model	$ \dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = \alpha P - \psi Q $
Transition-Blocking Model	$\dfrac{\partial P}{\partial x} + \dfrac{\partial}{\partial t}(uP) = (\gamma' - \alpha)P + (\psi - \mu_{Q})Q $
Transition-Blocking Model	$ \dfrac{\partial Q}{\partial x} + \dfrac{\partial}{\partial t}(uQ) = \alpha P - (\psi - \mu_{Q})Q $

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Table 2. The equations describing the thickness of the intimal layer over the course of inflammation and return to normal.

Inflammatory Phase	$L_{1}(t) = L_{0}\exp\left ({\zeta \frac{(1 + \rho)t + e^{-(\rho + 1)t} - 1}{1 - \rho}}\right )$
Post-inflammatory Phase	$L_{2}(t) = L_{m}(P_{m}(1 - e^{-t}) + 1)^{\zeta}$

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