Jorge Tiago

Existence Results for Optimal Control Problems with Some Special Nonlinear Dependence on State and Control

We present a general approach to prove existence of solutions for optimal control problems not based on typical convexity conditions, which quite often are very hard, if not impossible, to check. By taking advantage of several relaxations of the problem, we isolate an assumption which guarantees the existence of solutions of the original optimal control problem. To show the validity of this crucial hypothesis through various means and in various contexts is the main goal of this paper. In each such situation, we end up with some existence result. In particular, we would like to stress a general result that takes advantage of the particular structure of both the cost functional and the state equation. One main motivation for our work here comes from a model for guidance and control of ocean vehicles. Some explicit existence results and comparison examples are given.

Mathematical Modeling of Atherosclerotic Plaque Formation Coupled with a Non-Newtonian Model of Blood Flow

We deal with a mathematical model of atherosclerosis plaque formation, which describes the early formation of atherosclerotic lesions. The model assumes that the inflammatory process starts with the penetration of low-density lipoproteins cholesterol in the intima, and that penetration will occur in the area of lower shear stress. Using a system of reaction-diffusion equations, we first provide a one-dimensional model of lesion growth. Then we perform numerical simulations on an idealized two-dimensional geometry of the carotid artery bifurcation before and after the formation of the atherosclerotic plaque. For that purpose, we consider the blood as an incompressible non-Newtonian fluid with shear-thinning viscosity. We also present a study of the wall shear stress and blood velocity behavior in a geometry with one plaque and also with two plaques in different positions.

A data assimilation approach for non-Newtonian blood flow simulations in 3D geometries

Abstract Blood flow simulations can play an important role in medical training and diagnostic predictions associated to several pathologies of the cardiovascular system. The main challenge, at the present stage, is to obtain reliable numerical simulations in the particular districts of the cardiovascular system that we are interested in. Here, we propose a Data Assimilation procedure, in the form of a non linear optimal control problem of Dirichlet type, to reconstruct the blood flow profile from known data, available in certain parts of the computational domain. This method will allow us to obtain the boundary conditions, not fully determined by the physics of the model, in order to recover more accurate simulations. To solve the control problem we propose a Discretize then Optimize (DO) approach, based on a stabilized finite element method. Numerical simulations on 3D geometries are performed to validate this procedure. In particular, we consider some idealized geometries of interest, and real geometries such as a saccular aneurysm and a bypass. We assume blood as an homogeneous fluid with non-Newtonian inelastic shear-thinning behavior. The results show that, even in the presence of noisy data, accuracy can be improved using the optimal control approach.

Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery

The inflammatory process of atherosclerosis leads to the formation of an atheromatous plaque in the intima of the blood vessel. The plaque rupture may result from the interaction between the blood and the plaque. In each cardiac cycle, blood interacts with the vessel, considered as a compliant nonlinear hyperelastic. A three dimensional idealized fluid-structure interaction (FSI) model is constructed to perform the blood-plaque and blood-vessel wall interaction studies. An absorbing boundary condition (BC) is imposed directly on the outflow in order to cope with the spurious reflexions due to the truncation of the computational domain. The difference between the Newtonian and non-Newtonian effects is highlighted. It is shown that the von Mises and wall shear stresses are significantly affected according to the rigidity of the wall. The numerical results have shown that the risk of plaque rupture is higher in the case of a moving wall, while in the case of a fixed wall the risk of progression of the atheromatous plaque is higher.

A velocity tracking approach for the Data Assimilation problem in blood flow simulations

Several advances have been made in data assimilation techniques applied to blood flow modeling. Typically, idealized boundary conditions, only verified in straight parts of the vessel, are assumed. We present a general approach, on the basis of a Dirichlet boundary control problem, that may potentially be used in different parts of the arterial system. The relevance of this method appears when computational reconstructions of the 3D domains, prone to be considered sufficiently extended, are either not possible, or desirable, because of computational costs. On the basis of taking a fully unknown velocity profile as the control, the approach uses a discretize then optimize methodology to solve the control problem numerically. The methodology is applied to a realistic 3D geometry representing a brain aneurysm. The results show that this data assimilation approach may be preferable to a pressure control strategy and that it can significantly improve the accuracy associated to typical solutions obtained using idealized velocity profiles.

Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis

Atherosclerosis is a chronic inflammatory disease that occurs mainly in large and medium-sized elastic and muscular arteries. This pathology is essentially caused by the high concentration of low-density-lipoprotein (LDL) in the blood. It can lead to coronary heart disease and stroke, which are the cause of around 17.3 million deaths per year in the world. Mathematical modeling and numerical simulations are important tools for a better understanding of atherosclerosis and subsequent development of more effective treatment and prevention strategies. The atherosclerosis inflammatory process can be described by a model consisting of a system of three reaction-diffusion equations (representing the concentrations of oxidized LDL, macrophages and cytokines inside the arterial wall) with non-linear Neumann boundary conditions. In this work we prove the existence, uniqueness and boundedness of global solutions, using the monotone iterative method. Numerical simulations are performed in a rectangle representing the intima, to illustrate the mathematical results and the atherosclerosis inflammatory process.

Computational modeling of megakaryocytic differentiation of umbilical cord blood-derived stem/progenitor cells

Abstract Quantifying the effect of exogenous parameters regulating megakaryopoiesis would enhance the design of robust and efficient protocols to produce platelets. We developed a computational model based on time-dependent ordinary differential equations (ODEs) which decoupled expansion and differentiation kinetics of cells using a subpopulation dynamic model. The model described umbilical cord blood (UCB)-derived cells behavior in response to the external stimuli during expansion and megakaryocytic differentiation ex vivo. We observed that the rate of expansion of Mk progenitors and production of mature Mks were higher when TPO was included in the expansion stage and cytokines were added during differentiation stage. Our computational approach suggests that the Mk progenitors were an important intermediate population that their dynamic should be optimized in order to establish an efficient protocol. This model provides important insights into dynamics of cell subpopulations during megakaryopoiesis process and could potentially contribute toward the rational design of cell-based therapy bioprocesses.

On the optimal control of a class of non-Newtonian fluids

We consider optimal control problems of systems governed by stationary, incompressible generalized Navier–Stokes equations with shear dependent viscosity in a two-dimensional or three-dimensional domain. We study a general class of viscosity functions with shear-thinning and shear-thickening behavior. We prove an existence result for such class of optimal control problems.

Boundary Control Problems in Hemodynamics

Blood flow simulations can be improved by integrating known data into the numerical modeling approach. Data Assimilation techniques based on a variational formulation play an important role in this issue. We propose a non-linear optimal control problem to reconstruct the blood flow profile from partial observations of known data in different geometries. Blood flow is assumed to behave as a homogeneous fluid with non-Newtonian inelastic shear-thinning behavior or, to simplify, blood flow is governed by the Navier-Stokes equations. Using a Discretize then Optimize (DO) approach, we solve a non-linear optimal control problem and present numerical results that indicate its robustness with respect to different idealized geometries and measured data. Blood flow in real vessels will also be considered, including the discussion of particular clinical cases.

Numerical modeling of cardiovascular physiology Study of dynamic changes during autonomic reflexes

In this paper, we present a computational software to analyze simultaneous left ventricle (LV) pressure and volume measurements that takes full advantage of the single-beat method. LV pressure and volume data may be combined to construct pressure-volume (PV) loops, which enables the extraction of valuable parameters for LV and arterial physiologic function assessment, in particular LV systolic and diastolic performance, mechanical energies and efficiency. Using this software, we analyzed instantaneous and dynamic hemodynamic changes in vivo in rabbits during acute cardiovascular perturbations produced by 1) cardiac baroreflex, 2) arterial chemoreflex, and 3) von Bezold-Jarisch reflex. This paper reports results of this analysis.