This paper presents a three-dimensional (3D) quadratic partial differential equation model of the drug release from a thin film of biodegradable polymer to a surrounding medium. Its very innovative ...feature is to go directly from the experimental normalized release curves of Lao and Venkatraman J. Control. Release, 130 (2008), pp. 9–14 to a flux condition at the interface between the polymer and the medium that only requires the identification of the two parameters of the highly accurate ordinary differential equation model of Blanchet, Delfour, and Garon SIAM J. Appl. Math., 71 (2011), pp. 2269–2286. In the context of drug eluting stents, it is a practical and economical tool to theoretically and numerically simulate the 3D release of drug from the thin polymer film to the integrated wall and lumen of the blood vessel for evaluation and design. This approach avoids resorting to time-dependent or nonlinear diffusion in the polymer.
The central result of this paper is a new nonlinear equation which describes the evolution of the oriented distance function $b_\Omega$ of a set $\Omega$ with thin boundary under the influence of a ...velocity field. We relate it to equations and constructions used in the context of level set methods. We further introduce a new moving narrow-band method which not only can be readilyimplemented to solve our evolution equation, but could also be used for equations of motion by curvatures. In the process we review and sharpen the characterization of smooth sets and manifolds and sets of positive reach (e.g., local semiconvexity in an extended sense of the oriented distance function of the closure of the set). For W2,p}-Sobolev domains a new characterization and a compactness theorem are given in terms of the Laplacian of the oriented distance function rather than its whole Hessian matrix.
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CEKLJ, DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
Stents are used in interventional cardiology to keep a diseased vessel open. New stents are coated with a medicinal agent to prevent early reclosure due to the proliferation of smooth muscle cells. ...It is recognised that it is the dose of the agent that effectively controls the growth. This paper focusses on the asymptotic behaviour of the dose for general families of coated stents under a fixed ratio between the coated region of the stent and the targeted region of the vessel and set therapeutic bounds on the dose. It generalises the results of Delfour, Garon and Longo for stents made of a sequence of thin equally spaced rings to stents with an arbitrary pattern. It gives the equation of the asymptotic dose for a normal tiling of the target region using the theory of tilings, patterns and motifs on a cylinder.
"This book is a most welcome addition to the literature of this field, where it serves the need for a modern treatment on topics that only very recently have found a satisfactory solution... Many ...readers will appreciate the concise exposition. ""Presents, or refers to, the most recent and updated results in the field. For this reason, it should serve as an excellent asset to anyone pursuing a research career in the field."" - Mathematical Reviews (reviews of Volumes I and II of the First Edition) The quadratic cost optimal control problem for systems described by linear ordinary differential equations occupies a central role in the study of control systems both from a theoretical and design point of view. The study of this problem over an infinite time horizon shows the beautiful interplay between optimality and the qualitative properties of systems such as controllability, observability, stabilizability, and detectability. This theory is far more difficult for infinite dimensional systems such as those with time delays and distributed parameter systems. This reorganized, revised, and expanded edition of a two-volume set is a self-contained account of quadratic cost optimal control for a large class of infinite dimensional systems. The book is structured into five parts. Part I reviews basic optimal control and game theory of finite dimensional systems, which serves as an introduction to the book. Part II deals with time evolution of some generic controlled infinite dimensional systems and contains a fairly complete account of semigroup theory. It incorporates interpolation theory and exhibits the role of semigroup theory in delay differential and partial differential equations. Part III studies the generic qualitative properties of controlled systems. Parts IV and V examine the optimal control of systems when performance is measured via a quadratic cost. Boundary control of parabolic and hyperbolic systems and exact controllability are also covered. New material and original features of the Second Edition: * Part I on finite dimensional controlled dynamical systems contains new material: an expanded chapter on the control of linear systems including a glimpse into H-infinity theory and dissipative systems, and a new chapter on linear quadratic two-person zero-sum differential games. * A unique chapter on semigroup theory and interpolation of linear operators brings together advanced concepts and techniques that are usually treated independently. * The material on delay systems and structural operators is not available elsewhere in book form. Control of infinite dimensional systems has a wide range and growing number of challenging applications. This book is a key reference for anyone working on these applications, which arise from new phenomenological studies, new technological developments, and more stringent design requirements. It will be useful for mathematicians, graduate students, and engineers interested in the field and in the underlying conceptual ideas of systems and control."
A general compactness theorem for shape/geometric analysis and optimization is given for a family of subsets verifying the uniform fat segment property in a bounded open holdall with or without ...constraints on the De Giorgi or the gamma-density perimeter of Bucur and Zolesio. The uniform fat segment property is shown to be equivalent to the uniform cusp property introduced in with a continuous non-negative cusp function. This equivalence remains true for cusp functions that are only continuous at the origin. The equivalence of sets verifying a segment property with their C0-graph representation is further sharpened for sets with a compact boundary. Our C0-graph characterization is shown to be equivalent to both the uniform cusp property and the uniform segment property. It is used to formulate sufficient conditions on the local graphs of a family of subsets of a bounded open holdall to get compactness. A first condition assumes that the local graphs are bounded above by a cusp function; a second condition which requires that the local graphs be equicontinuous turns out to be equivalent to the first one. The respective solutions of the Laplacian with homogeneous Dirichlet or Neumann boundary condition are shown to be continuous with respect to domains in that family.
The object of this paper is twofold. We first present constructions which induce topologies on subsets of a fixed domain or hold-all
D in
R
N
by using set parameterized functions in an appropriate ...function space. Second, we study the role of the family of
oriented distance functions (also known as algebraic or signed distance functions) in the analysis of shape optimization problems. They play an important role in the introduction of topologies which retain the classical geometric properties associated with sets: convexity, exterior normals, mean curvature,
C
k
boundaries, etc.
The notion of dose that comes from the biologists has been introduced by Delfour et al. (2005 SIAM J. Appl. Math. 65(3):858-881) in the context of coated stents to control restenosis. Assuming a ...stationary velocity profile of the blood flow in the lumen, it leads to a time-independent equation for the dose that considerably simplifies the analysis and the design problem. Under stable conditions the blood flow is pulsative, that is the velocity field can be assumed to be periodic. So it is necessary to justify the replacement of the periodic field by its time average over the pulsation period. In this paper, firstly we introduce the new unfolded dose and its equations without a priori constraint on the size of the period. So it can be used in biochemical problems where the period is large compared to the time constants of the system. Secondly, we show that, as the period goes to zero, the velocity field can be replaced by its average over the period. Numerical tests on a one-dimensional example are included to illustrate the theory.
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