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:
[1]

World Health Organisation, Global Status Report on Noncommunicable Diseases 2010, Report of the World Health Organisation, 2011. Available from: http://www.who.int/nmh/publications/ncd_report_full_en.pdf.

[2]

A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, Tumour cords and their response to anticancer agents, in Modelling and Simulation in Science, Engineering, and Technology (ed. N. Bellomo), Birkhauser, (2008), 183–206.

[3]

F. BozsakJ. M. Chomaz and A. I. Barakat, Modeling the transport of drugs eluted from stents: Physical phenomena driving drug distribution in the arterial wall, Biomechanics and Modeling in Mechanobiology, 13 (2014), 327-347. doi: 10.1007/s10237-013-0546-4.

[4]

F. BozsakD. Gonzalez-RodriguezZ. SternbergerP. BelitzT. BewleyJ. M. Chomaz and A. I. Barakat, Optimization of drug delivery by drug-eluting stents, PLoS One, 10 (2015), e0130182. doi: 10.1371/journal.pone.0130182.

[5]

A. CaiazzoD. EvansJ.-L. FalconeJ. HegewaldE. LorenzB. StahlD. WangJ. BernsdorfB. ChopardJ. GunnR. HoseM. KrafczykP. LawfordR. SmallwoodD. Walker and A. Hoekstra, A Complex Automata approach for in-stent restenosis: Two-dimensional multiscale modelling and simulations, Journal of Computational Science, 2 (2011), 9-17. doi: 10.1016/j.jocs.2010.09.002.

[6]

Y. S. ChatzizisisA. U. CoskunM. JonasE. R. EdelmanC. L. Feldman and P. H. Stone, Role of endothelial shear stress in the natural history of coronary atherosclerosis and vascular remodeling: molecular, cellular, and vascular behavior, Journal of the American College of Cardiology, 49 (2007), 2379-2393. doi: 10.1016/j.jacc.2007.02.059.

[7]

D. E. DrachmanE. R. EdelmanP. SeifertA. R. GroothuisD. A. BornsteinK. R. KamathM. PalasisD. YangS. H. Nott and C. Rogers, Neointimal thickening after stent delivery of paclitaxel: Change in composition and arrest of growth over six months, Journal of the American College of Cardiology, 36 (2000), 2325-2332. doi: 10.1016/S0735-1097(00)01020-2.

[8]

J. A. FerreiraJ. Naghipoor and P. de Oliveira, Analytical and numerical study of a coupled cardiovascular drug delivery model, Journal of Computational and Applied Mathematics, 275 (2015), 433-446. doi: 10.1016/j.cam.2014.04.021.

[9]

F. J. H. GijsenF. MigliavaccaS. SchievanoL. SocciL. PetriniA. ThuryJ. J. WentzelA. F. W. van der SteenP. W. S. Serruys and G. Dubini, Simulation of stent deployment in a realistic human coronary artery, BioMedical Engineering OnLine, 7 (2008), 1-23. doi: 10.1186/1475-925X-7-23.

[10]

W. KhanS. FarahA. Nyska and A. J. Domb, Carrier free rapamycin loaded drug eluting stent: In vitro and in vivo evaluation, Journal of Controlled Release, 168 (2013), 70-76. doi: 10.1016/j.jconrel.2013.02.012.

[11]

W. KhanS. Farah and A. J. Domb, Drug eluting stents: developments and current status, Journal of Controlled Release, 161 (2012), 703-712. doi: 10.1016/j.jconrel.2012.02.010.

[12]

B. M. MazzagJ. S. Tamaresis and A. I. Barakat, A model for shear stress sensing and transduction in vascular endothelial cells, Biophysical Journal, 84 (2003), 4087-4101. doi: 10.1016/S0006-3495(03)75134-0.

[13]

S. McGintyS. McKeeR. Wadsworth and C. McCormick, Modeling arterial wall drug concentrations following the insertion of a drug-eluting stent, SIAM Journal on Applied Mathematics, 73 (2013), 2004-2028. doi: 10.1137/12089065X.

[14]

S. McGintyT. T. N. VoM. MeereS. McKee and C. McCormick, Some design considerations for polymer-free drug-eluting stents: A mathematical approach, Acta Biomaterialia, 18 (2015), 213-225. doi: 10.1016/j.actbio.2015.02.006.

[15]

S. McGintyM. WheelS. McKee and C. McCormick, Does anisotropy promote spatial uniformity of stent-delivered drug distribution in arterial tissue?, International Journal of Heat and Mass Transfer, 90 (2015), 266-279. doi: 10.1016/j.ijheatmasstransfer.2015.06.061.

[16]

J. C. Panetta, A mathematical model of breast and ovarian cancer treated with paclitaxel, Mathematical Biosciences, 146 (1997), 89-113. doi: 10.1016/S0025-5564(97)00077-1.

[17]

S. PantN. W. BressloffA. I. Forrester and N. Curzen, The influence of strut-connectors in stented vessels: A comparison of pulsatile flow through five coronary stents, Annals of Biomedical Engineering, 38 (2010), 1893-1907. doi: 10.1007/s10439-010-9962-0.

[18]

G. Pontrelli and F. de Monte, Modelling of mass convection-diffusion in stent-based drug delivery, XXV Congresso Nazionale UIT sulla Trasmissione del Calore, (2007).

[19]

H. TahirI. NiculescuC. Bona-CasasR. M. H. Merks and A. G. Hoekstra, An in silico study on the role of smooth muscle cell migration in neointimal formation after coronary stenting, Journal of the Royal Society Interface, 12 (2015), 20150358. doi: 10.1098/rsif.2015.0358.

[20]

A. R. TzafririA. GroothuisG. S. Price and E. R. Edelman, Stent elution rate determines drug deposition and receptor-mediated effects, Journal of Controlled Release, 161 (2012), 918-926. doi: 10.1016/j.jconrel.2012.05.039.

[21]

T. T. N. VoA. Peddle and W. Lee, Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅰ: Drug transport, Mathematical Biosciences and Engineering, 14 (2017), 491-509. doi: 10.3934/mbe.2017030.

[22]

T. T. N. VoR. YangY. Rochev and M. Meere, A mathematical model for drug delivery, Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry, 17 (2012), 521-528.

[23]

J. WattS. KennedyC. McCormickE. O. AgbaniA. McPhadenA. MullenP. CzudajB. BehnischR. M. Wadsworth and K. G. Oldroyd, Soccinobucal-eluting stents increase neointimal thickening and peri-strut inflammation in a porcine coronary model, Catheterization and Cardiovascular Interventions, 81 (2013), 698-708. doi: 10.1002/ccd.24473.

[24]

T. K. YeungC. GermondX. Chen and Z. Wang, The mode of action of taxol: Apoptosis at low concentration and necrosis at high concentration, Biochemical and Biophysical Research Communications, 263 (1999), 398-404. doi: 10.1006/bbrc.1999.1375.

[25]

P. ZuninoC. D'AngeloL. PetriniC. VergaraC. Capelli and F. Migliavacca, Numerical simulation of drug eluting coronary stents: Mechanics, fluid dynamics, and drug release, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 3633-3644. doi: 10.1016/j.cma.2008.07.019.

show all references

References:
[1]

World Health Organisation, Global Status Report on Noncommunicable Diseases 2010, Report of the World Health Organisation, 2011. Available from: http://www.who.int/nmh/publications/ncd_report_full_en.pdf.

[2]

A. Bertuzzi, A. Fasano, A. Gandolfi and C. Sinisgalli, Tumour cords and their response to anticancer agents, in Modelling and Simulation in Science, Engineering, and Technology (ed. N. Bellomo), Birkhauser, (2008), 183–206.

[3]

F. BozsakJ. M. Chomaz and A. I. Barakat, Modeling the transport of drugs eluted from stents: Physical phenomena driving drug distribution in the arterial wall, Biomechanics and Modeling in Mechanobiology, 13 (2014), 327-347. doi: 10.1007/s10237-013-0546-4.

[4]

F. BozsakD. Gonzalez-RodriguezZ. SternbergerP. BelitzT. BewleyJ. M. Chomaz and A. I. Barakat, Optimization of drug delivery by drug-eluting stents, PLoS One, 10 (2015), e0130182. doi: 10.1371/journal.pone.0130182.

[5]

A. CaiazzoD. EvansJ.-L. FalconeJ. HegewaldE. LorenzB. StahlD. WangJ. BernsdorfB. ChopardJ. GunnR. HoseM. KrafczykP. LawfordR. SmallwoodD. Walker and A. Hoekstra, A Complex Automata approach for in-stent restenosis: Two-dimensional multiscale modelling and simulations, Journal of Computational Science, 2 (2011), 9-17. doi: 10.1016/j.jocs.2010.09.002.

[6]

Y. S. ChatzizisisA. U. CoskunM. JonasE. R. EdelmanC. L. Feldman and P. H. Stone, Role of endothelial shear stress in the natural history of coronary atherosclerosis and vascular remodeling: molecular, cellular, and vascular behavior, Journal of the American College of Cardiology, 49 (2007), 2379-2393. doi: 10.1016/j.jacc.2007.02.059.

[7]

D. E. DrachmanE. R. EdelmanP. SeifertA. R. GroothuisD. A. BornsteinK. R. KamathM. PalasisD. YangS. H. Nott and C. Rogers, Neointimal thickening after stent delivery of paclitaxel: Change in composition and arrest of growth over six months, Journal of the American College of Cardiology, 36 (2000), 2325-2332. doi: 10.1016/S0735-1097(00)01020-2.

[8]

J. A. FerreiraJ. Naghipoor and P. de Oliveira, Analytical and numerical study of a coupled cardiovascular drug delivery model, Journal of Computational and Applied Mathematics, 275 (2015), 433-446. doi: 10.1016/j.cam.2014.04.021.

[9]

F. J. H. GijsenF. MigliavaccaS. SchievanoL. SocciL. PetriniA. ThuryJ. J. WentzelA. F. W. van der SteenP. W. S. Serruys and G. Dubini, Simulation of stent deployment in a realistic human coronary artery, BioMedical Engineering OnLine, 7 (2008), 1-23. doi: 10.1186/1475-925X-7-23.

[10]

W. KhanS. FarahA. Nyska and A. J. Domb, Carrier free rapamycin loaded drug eluting stent: In vitro and in vivo evaluation, Journal of Controlled Release, 168 (2013), 70-76. doi: 10.1016/j.jconrel.2013.02.012.

[11]

W. KhanS. Farah and A. J. Domb, Drug eluting stents: developments and current status, Journal of Controlled Release, 161 (2012), 703-712. doi: 10.1016/j.jconrel.2012.02.010.

[12]

B. M. MazzagJ. S. Tamaresis and A. I. Barakat, A model for shear stress sensing and transduction in vascular endothelial cells, Biophysical Journal, 84 (2003), 4087-4101. doi: 10.1016/S0006-3495(03)75134-0.

[13]

S. McGintyS. McKeeR. Wadsworth and C. McCormick, Modeling arterial wall drug concentrations following the insertion of a drug-eluting stent, SIAM Journal on Applied Mathematics, 73 (2013), 2004-2028. doi: 10.1137/12089065X.

[14]

S. McGintyT. T. N. VoM. MeereS. McKee and C. McCormick, Some design considerations for polymer-free drug-eluting stents: A mathematical approach, Acta Biomaterialia, 18 (2015), 213-225. doi: 10.1016/j.actbio.2015.02.006.

[15]

S. McGintyM. WheelS. McKee and C. McCormick, Does anisotropy promote spatial uniformity of stent-delivered drug distribution in arterial tissue?, International Journal of Heat and Mass Transfer, 90 (2015), 266-279. doi: 10.1016/j.ijheatmasstransfer.2015.06.061.

[16]

J. C. Panetta, A mathematical model of breast and ovarian cancer treated with paclitaxel, Mathematical Biosciences, 146 (1997), 89-113. doi: 10.1016/S0025-5564(97)00077-1.

[17]

S. PantN. W. BressloffA. I. Forrester and N. Curzen, The influence of strut-connectors in stented vessels: A comparison of pulsatile flow through five coronary stents, Annals of Biomedical Engineering, 38 (2010), 1893-1907. doi: 10.1007/s10439-010-9962-0.

[18]

G. Pontrelli and F. de Monte, Modelling of mass convection-diffusion in stent-based drug delivery, XXV Congresso Nazionale UIT sulla Trasmissione del Calore, (2007).

[19]

H. TahirI. NiculescuC. Bona-CasasR. M. H. Merks and A. G. Hoekstra, An in silico study on the role of smooth muscle cell migration in neointimal formation after coronary stenting, Journal of the Royal Society Interface, 12 (2015), 20150358. doi: 10.1098/rsif.2015.0358.

[20]

A. R. TzafririA. GroothuisG. S. Price and E. R. Edelman, Stent elution rate determines drug deposition and receptor-mediated effects, Journal of Controlled Release, 161 (2012), 918-926. doi: 10.1016/j.jconrel.2012.05.039.

[21]

T. T. N. VoA. Peddle and W. Lee, Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅰ: Drug transport, Mathematical Biosciences and Engineering, 14 (2017), 491-509. doi: 10.3934/mbe.2017030.

[22]

T. T. N. VoR. YangY. Rochev and M. Meere, A mathematical model for drug delivery, Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry, 17 (2012), 521-528.

[23]

J. WattS. KennedyC. McCormickE. O. AgbaniA. McPhadenA. MullenP. CzudajB. BehnischR. M. Wadsworth and K. G. Oldroyd, Soccinobucal-eluting stents increase neointimal thickening and peri-strut inflammation in a porcine coronary model, Catheterization and Cardiovascular Interventions, 81 (2013), 698-708. doi: 10.1002/ccd.24473.

[24]

T. K. YeungC. GermondX. Chen and Z. Wang, The mode of action of taxol: Apoptosis at low concentration and necrosis at high concentration, Biochemical and Biophysical Research Communications, 263 (1999), 398-404. doi: 10.1006/bbrc.1999.1375.

[25]

P. ZuninoC. D'AngeloL. PetriniC. VergaraC. Capelli and F. Migliavacca, Numerical simulation of drug eluting coronary stents: Mechanics, fluid dynamics, and drug release, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 3633-3644. doi: 10.1016/j.cma.2008.07.019.

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.
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 $\gamma$.
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.
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)).
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).
Figure 6.  Example of drug effectiveness, $\mu$.
Figure 7.  Phase plot of proliferative cell fraction, $\mu_{k}>0$.
Figure 8.  Example of modified $\psi$ value, considering drug effects.
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 $
$ \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 $
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 $L_{2}(t) = L_{m}(P_{m}(1 - e^{-t}) + 1)^{\zeta}$
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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