By Sunčica Čanić
Stents are mesh-like tubes that interventional cardiologists use to prop open diseased arteries and restore normal blood flow (see Figure 1a). They were first introduced in the late 1980s to help reduce the rate of restenosis—re-narrowing of coronary arteries—commonly associated with angioplasty procedures employed to treat coronary artery disease. First-generation stents were bare metal stents (BMSs). BMSs are still in use with various improvements, including better construction materials and enhanced topological and geometric features. While BMSs do reduce restenosis rates following coronary angioplasty, they can cause severe intravascular problems—such as damaged endothelium and medial layer stretch leading to smooth muscle cell injury—that often result in complications like in-stent restenosis associated with smooth muscle cell proliferation into the arterial lumen. To further improve restenosis rates, next-generation stents were born in the late 1990s with the introduction of drug-eluting stents (DESs). DESs are coated with a polymer that incorporates an anti-proliferative drug. However, emerging reports in 2006 linked DESs with an increased risk of late-stage stent thrombosis arising from an inflammatory reaction to the polymer-based coating.
While DESs are currently state of the art in stent development, new designs are emerging that would circumvent the use of polymers. For example, Tejal Desai (University of California, San Francisco), is working on nanoengineered stents based on drug-free nanotechnology. The surface of such stents is nanoengineered in a way that promotes accelerated restoration of functional endothelium and provides a drug-free approach to keeping stents patent long term [8].
Figure 1. Stents restore normal blood flow in diseased arteries. 1a. A sketch of an implanted stent. 1b. A magnified view of a section of the multilayered arterial wall containing a stent strut. 1a is a public domain image, 1b courtesy of Sunčica Čanić.
Mathematical modeling and numerical simulations are indispensable tools in guiding optimal stent design [10]. The past two decades have yielded extensive studies on the mechanical properties of stents, biodegradable stents and coating, optimal strategies in anti-inflammatory drug delivery, and stent impact on local and global blood flow and vascular walls. See the works of Paolo Zunino, Francesco Migliavacca, Joao S. Soares, Kumbakonam Rajagopal, Wei Wu, Dario Gastaldi, James E. Moore, Lucas H. Timmins, Alison Marsden, Sean McGinty, Christopher McCormick, Dimitrios Kiousis, Christian Gasser, and Gerhard A. Holzapfel, to name a few.
As models became more sophisticated, deeper mathematical questions had to be addressed to continue advancing the field. This spurred the development of new mathematics in the area of fluid-structure interaction (FSI) involving elastic, viscoelastic, poroelastic, and mesh-like structures to capture the interaction between time-dependent blood flow and stented vascular tissue.
Our group has been working on modeling, mathematical analysis, and computations of FSI between blood flow and vascular walls treated with stents. While the creation of computational methods for biological FSI has been an active research area for the past 40 years (see, for example, the works of Charles Peskin, Jean-Frédéric Gerbeau, Boyce E. Griffith et al., Alfio Quarteroni et al., Thomas J.R. Hughes et al., Yuri Bazilevs et al., and Shawn C. Shadden et al.,) the mathematical analysis of solutions for this class of problems is still under development.
The problem consists of coupling the Navier-Stokes equations for an incompressible, viscous fluid that model blood flow to a system of partial differential equations (PDEs) modeling the elastodynamics of an elastic structure. Equations of linearly and nonlinearly elastic membranes or shells have been used to model the thin intimal layer, while equations of finite elasticity assuming linear or certain nonlinear hyperelastic models have been used to model the thick media/adventitia layer.
Researchers have proposed several approaches for modeling stents, mostly based on three-dimensional (3D) approximations. However, such approximations are computationally expensive because stent components (stent struts) are slender bodies (see Figure 2). In 2010, we developed a reduced, one-dimensional (1D) stent net model in collaboration with Josip Tabača [9]. Our model uses a network of Antman-Cosserat 1D curved rods [4], which approximates the full 3D model with high accuracy and provides significant computational savings. The resulting stent net model is a system of 1D hyperbolic balance laws defined on a graph domain [5]. Coupling this model to arterial walls and fluid flow is very challenging from a mathematical standpoint since it involves coupling PDEs of different dimensions [5]. Analysis of these problems is currently an active research area (see the works of Tabača, Zunino, Kent-Andre Mardal, Jan M. Nordbotten, and Marie E. Rognes).
Figure 2. 2a. A patient’s angiogram showing a curved coronary artery. 2b. Four different stents analyzed in [3]. 2a courtesy of David Paniagua, 2b courtesy of Boston Scientific and Abbott Vascular.
In collaboration with medical specialists David Paniagua and David Fish of the Texas Heart Institute, we used our model to suggest optimal design of a stent for transcatheter aortic valve replacement (U.S. patent US9125739 B2) that is currently utilized in medical interventions involving Colibri Heart Valve.
The coupled FSI problem is difficult to study. In addition to nonlinearity in the fluid equations (and possibly the structure models), coupling across the deformed fluid-structure interface gives rise to a strong geometric nonlinearity. This is because fluid and structure have comparable densities in biological problems, and their interaction results in significant energy exchange. The structure’s motion is substantially affected by the fluid mass it displaces as it moves, since both fluid and structure are “equally heavy.” This is known as the added mass effect. Energy imbalance and stability issues arise when the added mass effect is not factored into the design of numerical schemes and in the mathematical analysis of the problem, as it keeps the frequency of structure oscillations under control.
Mathematical analysis of solutions to the coupled fluid-stent-artery interaction problem has practical relevance. It provides detailed understanding of how the energy of the problem depends on the parameters in the problem, it gives insight into possible singularities that may be associated with tissue damage (see Figure 3a), and it offers information about wave reflections due to the presence of stents, which have been known to increase the overall blood pressure and workload on the heart. Moreover, constructive existence proofs can motivate the design of a computational scheme for solving the underlying FSI problem. In a series of manuscripts with Boris Muha over the last five years, we have developed constructive existence proofs to study a class of nonlinear, moving boundary problems involving blood flow and various structures used in modeling vascular walls with and without a stent [1, 6]. The proofs are based on semi-discretization of the coupled problem in time and use of an operator splitting strategy to separate the fluid (parabolic) and structure (hyperbolic) subproblems. This approach defines a sequence of approximate solutions whose subsequences converge to a weak solution of the coupled problem as the time-discretization step goes to zero. As the problem is nonlinear, a compactness argument must be used to allow passing to the limit in nonlinear terms. However, classical compactness results cannot be directly applied since approximate problems are defined on different (fluid) domains that move in time. This led us to develop a generalization of the Aubin-Lions-Simon compactness lemma for problems on moving domains [7]. Our result explained that the reason the semi-discretized approximations converge to a weak solution is the regularization by fluid viscosity, which is transferred to the vascular wall via the no-slip condition. Namely, the oscillations in the structure (vascular wall) “feel” the fluid viscous damping through an operator similar to the square root of the negative Laplacian—obtained via the Dirichlet-Neumann operator—which acts on the structure velocity and keeps the amplitude of structure oscillations under control.
Figure 3. 3a. The intimal layer with four different stents showing possible damage to the intima. 3b. Stress experienced by the intima; red designates excessive elastic deformation and blue represents resistance to deformation. The colors indicate deviation from normal stress values (green). Cypher-like stents have the best performance. Figure courtesy of Sunčica Čanić research group.
The algorithm developed in the constructive existence proof served as a foundation for a class of partitioned, loosely-coupled numerical schemes (devised with Martina Bukač and Roland Glowinski [2]). We successfully used these schemes to study various questions related to the performance of stents inserted in curved coronary arteries and moving with the heart’s contractions. Our numerical simulations with Yifan Wang [3] revealed that the Cypher-like stent outperforms the Palmaz-like, Xience-like, and Express-like stents based on injury to the intimal layer and tissue distortion in the media layer measured by von Mises stress (see Figure 3). Additionally, we found that open-cell design—with every other horizontal stent strut missing—and stents with sinusoidal horizontal struts have significantly less overall bending rigidity while maintaining the radial stiffness necessary to keep arteries open. These traits are preferred for curved coronary arteries.
References
[1] Bukač, M, Čanić, S., & Muha, B. (2016). A nonlinear fluid-structure interaction problem in compliant arteries treated with vascular stents. Appl. Math. Opt., 73, 433-473.
[2] Bukač, M., Čanić, S., Muha, B., & Glowinski, R. (2016). An operator splitting approach to the solution of fluid-structure interaction problems in hemodynamics. In R. Glowinski, S.J. Osher, & W. Yin (Eds.), Springer Series in Scientific Computation: Splitting Methods in Communication, Imaging, Science, and Engineering (pp. 731-772). New York, NY: Springer.
[3] Bukač, M., Čanić, S., Tambača, J., & Wang, Y. (2019). Fluid-structure interaction between pulsatile blood flow and a curved stented coronary artery on a beating heart: a four stent computational study. Comp. Meth. Appl. Mech. Eng. Accepted.
[4] Čanić, S., & Tambača, J. (2012). Cardiovascular stents as PDE nets: 1D vs. 3D. IMA J. Appl. Math., 77(6), 748-779.
[5] Čanić, S., Galović, M., Ljulj, M., & Tambača, J. (2017). A dimension-reduction based coupled model of mesh-reinforced shells. SIAM J. Appl. Math., 77(2), 744-769.
[6] Muha, B., & Čanić, S. (2013). Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls. Arch. Ration. Mech. Anal., 207(3), 919-968.
[7] Muha, B., & Čanić, S. (2019). A generalization of the Aubin-Lions-Simon Compactness Lemma to problems on moving domains. J. Diff. Eq. In print.
[8] Nuhn, H., Blanco, C.E., & Desai, T.A. (2017). Nanoengineered Stent Surface to Reduce In-Stent Restenosis in Vivo. ACS Appl. Mater. Interfaces, 9, 19677-19686.
[9] Tambača, J., Kosor, M., Čanić, S., & Paniagua, D. (2010). Mathematical Modeling of Endovascular Stents. SIAM J. Appl. Math., 70(6), 1922-1952.
[10] Zunino, P., Tambača, J., Cutri, E., Čanić, S., Formaggia, L., & Magliavacca, F. (2016). Integrated stent models based on dimension reduction. Review and future perspectives. Ann. Biomed. Eng., 44(2), 604-617.
Sunčica Čanić is a professor in the Department of Mathematics at the University of California, Berkeley. She has been recognized for her contributions to the modeling and analysis of partial differential equations motivated by applications in the life sciences.