# American Institute of Mathematical Sciences

October  2018, 15(5): 1117-1135. doi: 10.3934/mbe.2018050

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

 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 27, 2017 Accepted  March 2018 Published  May 2018

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.

Citation: Adam Peddle, William Lee, Tuoi Vo. Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅱ: Cell proliferation. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1117-1135. doi: 10.3934/mbe.2018050
##### References:

show all references

##### References:
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.
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 $\gamma$.
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.
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)).
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).
Example of drug effectiveness, $\mu$.
Phase plot of proliferative cell fraction, $\mu_{k}>0$.
Example of modified $\psi$ value, considering drug effects.
The equations of state for the various models considered herein
 Full System $\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma - \alpha - \lambda_{P} -\mu_{P})P + (\beta + \eta - \mu_{Q})Q$ $\dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = (\alpha)P - (\beta + \eta + \lambda_{Q} - \mu_{Q})Q$ Reduced System $\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma' - \alpha - \mu_{P})P + (\psi - \mu_{Q})Q$ $\dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = \alpha P - (\psi + \lambda_{Q} - \mu_{Q})Q$ Growth-Inhibiting Model $\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma' - \alpha - \mu_{P})P + \psi Q$ $\dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = \alpha P - \psi Q$ Transition-Blocking Model $\dfrac{\partial P}{\partial x} + \dfrac{\partial}{\partial t}(uP) = (\gamma' - \alpha)P + (\psi - \mu_{Q})Q$ $\dfrac{\partial Q}{\partial x} + \dfrac{\partial}{\partial t}(uQ) = \alpha P - (\psi - \mu_{Q})Q$
 Full System $\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma - \alpha - \lambda_{P} -\mu_{P})P + (\beta + \eta - \mu_{Q})Q$ $\dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = (\alpha)P - (\beta + \eta + \lambda_{Q} - \mu_{Q})Q$ Reduced System $\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma' - \alpha - \mu_{P})P + (\psi - \mu_{Q})Q$ $\dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = \alpha P - (\psi + \lambda_{Q} - \mu_{Q})Q$ Growth-Inhibiting Model $\dfrac{\partial P}{\partial t} + \dfrac{\partial}{\partial x}(uP) = (\gamma' - \alpha - \mu_{P})P + \psi Q$ $\dfrac{\partial Q}{\partial t} + \dfrac{\partial}{\partial x}(uQ) = \alpha P - \psi Q$ Transition-Blocking Model $\dfrac{\partial P}{\partial x} + \dfrac{\partial}{\partial t}(uP) = (\gamma' - \alpha)P + (\psi - \mu_{Q})Q$ $\dfrac{\partial Q}{\partial x} + \dfrac{\partial}{\partial t}(uQ) = \alpha P - (\psi - \mu_{Q})Q$
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 $L_{2}(t) = L_{m}(P_{m}(1 - e^{-t}) + 1)^{\zeta}$
 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 $L_{2}(t) = L_{m}(P_{m}(1 - e^{-t}) + 1)^{\zeta}$
 [1] Tuoi Vo, William Lee, Adam Peddle, Martin Meere. Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅰ: Drug transport. Mathematical Biosciences & Engineering, 2017, 14 (2) : 491-509. doi: 10.3934/mbe.2017030 [2] John Boscoh H. Njagarah, Farai Nyabadza. Modelling the role of drug barons on the prevalence of drug epidemics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 843-860. doi: 10.3934/mbe.2013.10.843 [3] Mario Grassi, Giuseppe Pontrelli, Luciano Teresi, Gabriele Grassi, Lorenzo Comel, Alessio Ferluga, Luigi Galasso. Novel design of drug delivery in stented arteries: A numerical comparative study. Mathematical Biosciences & Engineering, 2009, 6 (3) : 493-508. doi: 10.3934/mbe.2009.6.493 [4] Shalela Mohd Mahali, Song Wang, Xia Lou. Determination of effective diffusion coefficients of drug delivery devices by a state observer approach. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1119-1136. doi: 10.3934/dcdsb.2011.16.1119 [5] Shalela Mohd--Mahali, Song Wang, Xia Lou, Sungging Pintowantoro. Numerical methods for estimating effective diffusion coefficients of three-dimensional drug delivery systems. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 377-393. doi: 10.3934/naco.2012.2.377 [6] Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014 [7] Song Wang, Xia Lou. An optimization approach to the estimation of effective drug diffusivity: From a planar disc into a finite external volume. Journal of Industrial & Management Optimization, 2009, 5 (1) : 127-140. doi: 10.3934/jimo.2009.5.127 [8] Zhenzhen Chen, Sze-Bi Hsu, Ya-Tang Yang. The continuous morbidostat: A chemostat with controlled drug application to select for drug resistance mutants. Communications on Pure & Applied Analysis, 2020, 19 (1) : 203-220. doi: 10.3934/cpaa.2020011 [9] Lambertus A. Peletier. Modeling drug-protein dynamics. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 191-207. doi: 10.3934/dcdss.2012.5.191 [10] Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 [11] Brandon Lindley, Qi Wang, Tianyu Zhang. A multicomponent model for biofilm-drug interaction. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 417-456. doi: 10.3934/dcdsb.2011.15.417 [12] Cristian Tomasetti, Doron Levy. An elementary approach to modeling drug resistance in cancer. Mathematical Biosciences & Engineering, 2010, 7 (4) : 905-918. doi: 10.3934/mbe.2010.7.905 [13] Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami Radunskaya, Ian Tucker. Predicting the drug release kinetics of matrix tablets. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 261-277. doi: 10.3934/dcdsb.2009.12.261 [14] Avner Friedman, Najat Ziyadi, Khalid Boushaba. A model of drug resistance with infection by health care workers. Mathematical Biosciences & Engineering, 2010, 7 (4) : 779-792. doi: 10.3934/mbe.2010.7.779 [15] Deborah C. Markham, Ruth E. Baker, Philip K. Maini. Modelling collective cell behaviour. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123 [16] Yangjin Kim, Soyeon Roh. A hybrid model for cell proliferation and migration in glioblastoma. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 969-1015. doi: 10.3934/dcdsb.2013.18.969 [17] Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 601-616. doi: 10.3934/mbe.2008.5.601 [18] Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227 [19] Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803 [20] Andrzej Swierniak, Jaroslaw Smieja. Analysis and Optimization of Drug Resistant an Phase-Specific Cancer. Mathematical Biosciences & Engineering, 2005, 2 (3) : 657-670. doi: 10.3934/mbe.2005.2.657

2018 Impact Factor: 1.313