Paolo Zunino

Mathematical and Numerical Modeling of Solute Dynamics in Blood Flow and Arterial Walls

The numerical modeling of solutes absorption processes by the arterial wall is of paramount interest for the understanding of the relationships between the local features of blood flow, the nourishing of the inner arterial wall by the blood solutes, and the pathologies that can appear when this process is for some reason perturbed. In the present work, two models for the solutes dynamics are investigated. In the first model, which is essentially based on the one introduced by Rappitsch and Perktold [J. Biomech. Engrg., 118 (1996), pp. 511--519] and Rappitsch, Perktold, and Pernkopf [Internat. J. Numer. Methods Fluids, 25 (1997), pp. 847--857], the Navier--Stokes equations for an incompressible fluid, describing the blood velocity and pressure fields, are coupled with an advection-diffusion equation for the solute concentration. The wellposedness of this model is discussed. The second model considers also the solutes dynamics inside the arterial wall, described by a pure diffusion equation. Actually, this is a heterogeneous model, coupling different equations in different parts of the domain at hand. Its wellposedness is proven. Moreover, in view of the numerical study, an iterative finite element method by subdomains is proposed and its convergence properties are analyzed. Finally, several numerical results comparing the different models in situations of physiologic interest are illustrated.

Expansion and drug elution model of a coronary stent

The present study illustrates a possible methodology to investigate drug elution from an expanded coronary stent. Models based on finite element method have been built including the presence of the atherosclerotic plaque, the artery and the coronary stent. These models take into account the mechanical effects of the stent expansion as well as the effect of drug transport from the expanded stent into the arterial wall. Results allow to quantify the stress field in the vascular wall, the tissue prolapse within the stent struts, as well as the drug concentration at any location and time inside the arterial wall, together with several related quantities as the drug dose and the drug residence times.

A Domain Decomposition Method Based on Weighted Interior Penalties for Advection-Diffusion-Reaction Problems

We propose a domain decomposition method for advection-diffusion-reaction equations based on Nitsches transmission conditions. The advection-dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries. We prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method. The convergence of the resulting domain decomposition method is proved, and this result holds true uniformly with respect to the diffusion parameter. The numerical scheme that we propose here can thus be applied straightforwardly to diffusion-dominated, advection-dominated, and hyperbolic problems. Some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method, and we compare our method with some other domain decomposition techniques for advection-diffusion equations.

A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks.

We introduce a general class of mixture models suitable to describe water-dependent degradation and erosion of biodegradable polymers in conjunction with drug release. The ability to predict and quantify degradation and erosion has direct impact in a variety of biomedical applications and is a useful design tool for biodegradable implants and tissue engineering scaffolds. The model is based on a finite number of constituents describing the polydisperse polymeric system, each representing chains of an average size, and two additional constituents, water and drug. Hydrolytic degradation of individual chains occurs at the molecular level and mixture constituents diffuse individually accordingly to Ficks 1st law at the bulk level - such analysis confers a multi-scale aspect to the resulting reaction-diffusion system. A shift between two different types of behavior, each identified to surface or bulk erosion, is observed with the variation of a single non-dimensional parameter measuring the relative importance of the mechanisms of reaction and diffusion. Mass loss follows a sigmoid decrease in bulk eroding polymers, whereas decreases linearly in surface eroding polymers. Polydispersity influences degradation and erosion of bulk eroding polymers and drug release from unstable surface eroding matrices is dramatically enhanced in an erosion-controlled release.

A multiphysics/multiscale 2D numerical simulation of scaffold-based cartilage regeneration under interstitial perfusion in a bioreactor.

In vitro tissue engineering is investigated as a potential source of functional tissue constructs for cartilage repair, as well as a model system for controlled studies of cartilage development and function. Among the different kinds of devices for the cultivation of 3D cartilage cell colonies, we consider here polymeric scaffold-based perfusion bioreactors, where an interstitial fluid supplies nutrients and oxygen to the growing biomass. At the same time, the fluid-induced shear acts as a physiologically relevant stimulus for the metabolic activity of cells, provided that the shear stress level is appropriately tuned. In this complex environment, mathematical and computational modeling can help in the optimal design of the bioreactor configuration. In this perspective, we propose a computational model for the simulation of the biomass growth, under given inlet and geometrical conditions, where nutrient concentration, fluid dynamic field and cell growth are consistently coupled. The biomass growth model is calibrated with respect to the shear stress dependence on experimental data using a simplified short-time analysis in which the nutrient concentration and the fluid-induced shear stress are assumed constant in time and uniform in space. Volume averaging techniques are used to derive effective parameters that allow to upscale the microscopic structural properties to the macroscopic level. The biomass growth predictions obtained in this way are significant for long times of culture.

Multiscale Boundary Conditions for Drug Release from Cardiovascular Stents

In this study, we focus on a specific application, the modeling and simulation of drug release from cardiovascular drug eluting stents. In particular, we analyze the drug release dynamics from the stent coating, where the drug is initially stored, to the surrounding arterial wall. The main challenge consists of accounting for multiple space scales. To this purpose, we derive suitable boundary conditions accounting for the smaller scales on the macroscopic model. This approach, applied to drug delivery, significantly cuts down the computational cost of the numerical simulations and allows us to consider a realistic problem setting.

A Domain Decomposition Method for Advection-Diffusion Processes with Application to Blood Solutes

In the present paper we consider a heterogeneous model for the dynamics of a blood solute both in the vascular lumen and inside the arterial wall. In the lumen, we consider an advection-diffusion equation, where the convective field is provided by the velocity of blood, which is in turn obtained by solving the Navier--Stokes equations. Inside the arterial wall we consider a pure diffusive dynamics. Since the endothelial layer at the interface between the lumen and the wall acts as a permeable membrane, whose permeability depends on the shear rate exerted by the blood, the solute concentration is discontinuous across this membrane. A possible approach for the numerical study of this kind of problem is inspired by domain decomposition techniques. In particular, we introduce a splitting in the computation and alternate the solution of the advection-diffusion equation in the lumen with that of the diffusion equation in the wall. We set up an efficient iterative method, based on a suitable reformulation of the problem in terms of a Steklov--Poincare interface equation. This formulation is a nonstandard one because of the concentration discontinuity at the lumen-wall interface and plays a key role in the proof of convergence of our method. In particular, we prove that the convergence rate performed by the proposed method is independent of the finite element space discretization and provides a criterion for the selection of an acceleration parameter. nSeveral numerical results, referred to as biomedical applications, support our theoretical conclusions and illustrate the efficiency of this algorithm.

A computational model of drug delivery through microcirculation to compare different tumor treatments.

Starting from the fundamental laws of filtration and transport in biological tissues, we develop a computational model to capture the interplay between blood perfusion, fluid exchange with the interstitial volume, mass transport in the capillary bed, through the capillary walls and into the surrounding tissue. These phenomena are accounted at the microscale level, where capillaries and interstitial volume are viewed as two separate regions. The capillaries are described as a network of vessels carrying blood flow. We apply the model to study drug delivery to tumors. The model can be adapted to compare various treatment options. In particular, we consider delivery using drug bolus injection and nanoparticle injection into the blood stream. The computational approach is suitable for a systematic quantification of the treatment performance, enabling the analysis of interstitial drug concentration levels, metabolization rates and cell surviving fractions. Our study suggests that for the treatment based on bolus injection, the drug dose is not optimally delivered to the tumor interstitial volume. Using nanoparticles as intermediate drug carriers overrides the shortcomings of the previous delivery approach. This work shows that the proposed theoretical and computational framework represents a promising tool to compare the efficacy of different cancer treatments.

Drug delivery patterns for different stenting techniques in coronary bifurcations: a comparative computational study

The treatment of coronary bifurcation lesions represents a challenge for the interventional cardiologists due to the lower rate of procedural success and the higher risk of restenosis. The advent of drug-eluting stents (DES) has dramatically reduced restenosis and consequently the request for re-intervention. The aim of the present work is to provide further insight about the effectiveness of DES by means of a computational study that combines virtual stent implantation, fluid dynamics and drug release for different stenting protocols currently used in the treatment of a coronary artery bifurcation. An explicit dynamic finite element model is developed in order to obtain realistic configurations of the implanted devices used to perform fluid dynamics analysis by means of a previously developed finite element method coupling the blood flow and the intramural plasma filtration in rigid arteries. To efficiently model the drug release, a multiscale strategy is adopted, ranging from lumped parameter model accounting for drug release to fully 3-D models for drug transport to the artery. Differences in drug delivery to the artery are evaluated with respect to local drug dosage. This model allowed to compare alternative stenting configurations (namely the Provisional Side Branch, the Culotte and the Inverted Culotte techniques), thus suggesting guidelines in the treatment of coronary bifurcation lesions and addressing clinical issues such as the effectiveness of drug delivery to lesions in the side branch, as well as the influence of incomplete strut apposition and overlapping stents.

Model Reduction Strategies Enable Computational Analysis of Controlled Drug Release from Cardiovascular Stents

Medicated cardiovascular stents, also called drug eluting stents (DES), represent a relevant application of controlled drug release mechanisms. Modeling of drug release from DES also represents a challenging problem for theoretical and computational analysis. In particular, the study of drug release involves models with singular behavior, arising, for instance, in the analysis of drug release in the small diffusion regime. Moreover, the application to realistic stent configurations requires one to account for complex designs of the device. To efficiently obtain satisfactory simulations of DES we rely on a multiscale strategy, based on lumped parameter (0D) models to account for drug release, one dimensional (1D) models to efficiently handle complex stent patterns and fully three-dimensional (3D) models for drug transfer in the artery, including the lumen and the arterial wall. The application of these advanced mathematical models makes it possible to perform a computational analysis of the fluid dynamics ...