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2011

Volume 71, Issue 6, pp. 1849-2358

† Special Section on Controlled Drug Delivery

Well-posedness and Long-time Behavior for a Nonstandard Viscous Cahn–Hilliard System

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, and Jürgen Sprekels

SIAM J. Appl. Math. 71, pp. 1849-1870 (22 pages)

Online Publication Date: November 03, 2011

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We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $\omega$-limit set.

Subband Diffusion Models for Quantum Transport in a Strong Force Regime

C. Ringhofer

SIAM J. Appl. Math. 71, pp. 1871-1895 (25 pages)

Online Publication Date: November 15, 2011

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We derive semiclassical approximations to quantum transport models in thin slabs with applications to SOI (silicon oxide on insulator)-type semiconductor devices via a subband approach. In the regime considered, the forces acting on the particles across the slab are much larger than the forces in the lateral direction of the slab. In a semiclassical limit the transport picture can be described on large time scales by a system of subband convection-diffusion equations with an interband collision operator, modeling the transfer of mass (charge) between the different eigenspaces and driving the system towards a local Maxwellian equilibrium.

A Periodic Epidemic Model with Age Structure in a Patchy Environment

Xiuxiang Liu and Xiao-Qiang Zhao

SIAM J. Appl. Math. 71, pp. 1896-1917 (22 pages)

Online Publication Date: November 15, 2011

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In this paper, a periodic epidemic model with age structure in a patchy environment is introduced. We investigate its global dynamics in term of the basic reproduction number $\mathcal{R}_0$, and show that there exists at least one positive periodic state and the disease persists when $\mathcal{R}_0>1$ while the disease will die out if $\mathcal{R}_0<1$. Some numerical examples are given to confirm our analytic results and to show that the age and spatial heterogeneities are important factors for the global dynamics.

A Structured Population Model of Cell Differentiation

Marie Doumic, Anna Marciniak-Czochra, Benoît Perthame, and Jorge P. Zubelli

SIAM J. Appl. Math. 71, pp. 1918-1940 (23 pages) | Cited 2 times

Online Publication Date: November 15, 2011

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We introduce and analyze several aspects of a new model for cell differentiation. It assumes that differentiation of progenitor cells is a continuous process. From the mathematical point of view, it is based on partial differential equations of transport type. Specifically, it consists of a structured population equation with a nonlinear feedback loop. This models the signaling process due to cytokines, which regulate the differentiation and proliferation process. We compare the continuous model to its discrete counterpart, a multicompartmental model of a discrete collection of cell subpopulations recently proposed by Marciniak-Czochra et al. [Stem Cells Dev., 18 (2009), pp. 377–386] to investigate the dynamics of the hematopoietic system. We obtain uniform bounds for the solutions, characterize steady state solutions, and analyze their linearized stability. We show how persistence or extinction might occur according to values of parameters that characterize the stem cells' self-renewal. We also perform numerical simulations and discuss the qualitative behavior of the continuous model vis à vis the discrete one.

Evolution and Breakup of Viscous Rotating Drops

M. A. Fontelos, V. J. García-Garrido, and U. Kindelán

SIAM J. Appl. Math. 71, pp. 1941-1964 (24 pages)

Online Publication Date: November 22, 2011

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We study the evolution of a viscous fluid drop rotating about a fixed axis at constant angular velocity $\Omega$ or constant angular momentum $L$ surrounded by another viscous fluid. The problem is considered in the limit of large Ekman number and small Reynolds number. The analysis is carried out by combining asymptotic analysis and full numerical simulation by means of the boundary element method. We pay special attention to the stability/instability of equilibrium shapes and the possible formation of singularities representing a change in the topology of the fluid domain. When the evolution is at constant $\Omega$, depending on its value, drops can take the form of a flat film whose thickness goes to zero in finite time or an elongated filament that extends indefinitely. When evolution takes place at constant $L$ and axial symmetry is imposed, thin films surrounded by a toroidal rim can develop, but the film thickness does not vanish in finite time. When axial symmetry is not imposed and $L$ is sufficiently large, drops break axial symmetry and, depending on the value of $L$, reach an equilibrium configuration with a 2-fold symmetry or break up into several drops with a 2- or 3-fold symmetry. The mechanism of breakup is also described.

Approximate Steady State Models for Magnetic Resonance Elastography

Yu Jiang, Hiroshi Fujiwara, and Gen Nakamura

SIAM J. Appl. Math. 71, pp. 1965-1989 (25 pages)

Online Publication Date: December 08, 2011

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MRE (magnetic resonance elastography) is a new diagnostic modality for measuring the stiffness of soft tissues in a living body. MRE itself measures only the displacements of waves propagating in the tissues, but the MRE-measured data can be linked to the stiffness through a proper model for wave propagation in the tissues and by solving some inverse problem under this model. Soft tissues of a living body are usually considered to be viscoelastic and nearly incompressible. In this paper we model the soft tissues as an isotropic Kelvin–Voigt material and interpret the meaning of nearly incompressible by an asymptotic analysis. As a consequence of this analysis, we show that the so-called modified Stokes system approximates very well the isotropic nearly incompressible Kelvin–Voigt model, and this explains why we can see only transverse waves inside soft tissues even if we inject longitudinal vibrations into the tissues from their surfaces. Further, by comparing the results of numerical simulations with MRE-measured data, we show that the modified Stokes system is a proper model.

On the Rayleigh–Taylor Instability for Incompressible, Inviscid Magnetohydrodynamic Flows

Ran Duan, Fei Jiang, and Song Jiang

SIAM J. Appl. Math. 71, pp. 1990-2013 (24 pages)

Online Publication Date: December 08, 2011

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We study the Rayleigh–Taylor instability for two incompressible, immiscible, inviscid magnetohydrodynamic (MHD) fluids with zero resistivity, evolving with a free interface in the presence of a uniform gravitational field. We first construct the Rayleigh–Taylor steady-state solution with a denser fluid lying above the light one. Then, we turn to an analysis of the equations obtained from linearizing around such a steady state. By solving a system of ordinary differential equations, we construct the normal mode solutions to the linearized problem that grow exponentially in time. A Fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the piecewise Sobolev space $H^k$, thus leading to an ill-posedness result for the linearized problem in the sense of Hadamard. Using these pathological solutions, we can then demonstrate the ill-posedness of the original nonlinear problem.

Well-posedness of a Compressible Gas-liquid Model with a Friction Term Important for Well Control Operations

Helmer André Friis and Steinar Evje

SIAM J. Appl. Math. 71, pp. 2014-2047 (34 pages)

Online Publication Date: December 08, 2011

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In this work we continue our investigations of a compressible gas-liquid model with special focus on inclusion of external frictional forces in the momentum balance. The model is often used for multiphase well flow modeling, which is important for different well control operations. The frictional forces have a major impact on the pressure gradient, which determines the pressure distribution along the wellbore. Compression and decompression of gas in turn strongly depend on the pressure level along the wellbore. A precise understanding of these mechanisms is important since gas-kick scenarios and blow-out behavior are strongly linked to decompression effects. This work is a continuation of our recent work [SIAM J. Appl. Math., 71 (2011), pp. 409–442]. The novelty of the present work lies in the facts that (i) we consider a full momentum equation, whereas a simplified one was used in the first work, (ii) the gas and liquid masses vanish at the boundaries making the analysis more involved, and (iii) special care must be given to the frictional term to make sure that it is balanced with other terms such that a well-defined model is obtained. The analysis ensures that global existence of weak solutions is obtained under suitable assumptions on initial data (e.g., decay rate at the boundaries for gas and liquid mass) and parameters that determine growth rate of mass terms associated with, respectively, the wall friction term and viscous term.

Mathematical Analysis of the Junction of Two Acoustic Open Waveguides

Anne-Sophie Bonnet-Ben Dhia, Benjamin Goursaud, and Christophe Hazard

SIAM J. Appl. Math. 71, pp. 2048-2071 (24 pages)

Online Publication Date: December 08, 2011

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The present paper concerns the scattering of a time-harmonic acoustic wave by the junction of two open uniform waveguides, where the junction is limited to a bounded region. We consider a two-dimensional problem for which wave propagation is described by the scalar Helmholtz equation. The main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave. We use here modal radiation conditions which extend the classical conditions used for closed waveguides. They are based on the generalized Fourier transforms which diagonalize the transverse contributions of the Helmholtz operator on both sides of the junction. We prove the existence and uniqueness of the solution, which seems to be the first result in this context. The originality lies in the proof of uniqueness, which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized Fourier variable.

Nonexistence of the von Neumann Reflection Configuration for the Triple Point Paradox

Geng Lai and Wancheng Sheng

SIAM J. Appl. Math. 71, pp. 2072-2092 (21 pages)

Online Publication Date: December 08, 2011

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In an attempt to resolve the von Neumann triple point paradox, a reflection, called von Neumann reflection (vNR), was reported by Colella and Henderson [J. Fluid Mech., 213 (1990), pp. 71–94] in investigating numerically the weak shock reflection. They reported that in this type of reflection configuration the incident shock and the Mach stem are seen to be smoothly merged and the reflected disturbance is not a shock but a curved band of self-similar compression wave. One should like to know whether or not the vNR configuration is a mathematically possible flow pattern. In this paper we prove that the reflection configuration in which the Mach shock is smoothly merged into the incident shock at a point and the wave behind this point is smooth is a mathematically impossible flow pattern for the two-dimensional (2D) self-similar potential flow equation and the 2D self-similar Euler equations.

Dielectric Boundary Force in Molecular Solvation with the Poisson–Boltzmann Free Energy: A Shape Derivative Approach

Bo Li, Xiaoliang Cheng, and Zhengfang Zhang

SIAM J. Appl. Math. 71, pp. 2093-2111 (19 pages)

Online Publication Date: December 08, 2011

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In an implicit-solvent description of molecular solvation, the electrostatic free energy is given through the electrostatic potential. This potential solves a boundary-value problem of the Poisson–Boltzmann equation in which the dielectric coefficient changes across the solute-solvent interface—the dielectric boundary. The dielectric boundary force acting on such a boundary is the negative first variation of the electrostatic free energy with respect to the location change of the boundary. In this work, the concept of shape derivative is used to define such variations, and formulas of the dielectric boundary force are derived. It is shown that such a force is always in the direction toward the charged solute molecules.

Microwave Imaging by Elastic Deformation

Habib Ammari, Yves Capdeboscq, Frédéric de Gournay, Anna Rozanova-Pierrat, and Faouzi Triki

SIAM J. Appl. Math. 71, pp. 2112-2130 (19 pages)

Online Publication Date: December 08, 2011

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In this paper, we show that by using microwave measurements at different frequencies and ultrasound localized perturbations to create local changes in the medium it is possible to extend the method developed by Ammari et al. in [SIAM J. Appl. Math., 68 (2008), pp. 1557–1573] to problems of the form $\nabla\cdot(a\nabla u)+k^{2}qu=0$ in $\Omega$, $u=\varphi$ on $\partial\Omega$, and to reliably reconstruct both the real-valued functions $a$ and $q$ from the internal energies $a|\nabla u|^2$ and $q|u|^2$.

Improved Current-Voltage Approximations for Currents Exceeding the Diffusion Limit

Ehud Yariv

SIAM J. Appl. Math. 71, pp. 2131-2150 (20 pages)

Online Publication Date: December 08, 2011

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Direct ionic current flows through a binary electrolyte solution bounded by two reactive electrodes which are connected to a voltage source. At sufficiently large voltages, the classical boundary-layer structure of the one-dimensional transport problem breaks down, resulting in currents exceeding Nernst's diffusion limit. While the asymptotic structure at these high voltages is well understood [B. Zaltzman and I. Rubinstein, J. Fluid Mech., 579 (2007), pp. 173–226], the direct calculation of the pertinent current-voltage relation for the electrochemical cell is thwarted by the appearance of nonconverging integrals. We describe a systematic regularization procedure which allows this calculation.

Gravitational Overturning in Stratified Particulate Flows

Yuri D. Sobral and E. John Hinch

SIAM J. Appl. Math. 71, pp. 2151-2167 (17 pages)

Online Publication Date: December 13, 2011

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With the motivation of understanding the formation of bubbles in fluidized beds, we investigate the stability of stratified particulate flows to transverse disturbances, leading to gravitational overturning. We consider a one-fluid model in which particles are responsible for the stratification of the flow but do not slip relative to the fluid and do not diffuse. A linear stability analysis and a numerical simulation of the governing equations are performed in order to determine and characterize the instability of the flow. We observe that stratified flows are unstable to transverse disturbances and that the instability is driven by a tilt-and-slide mechanism that creates ascending regions of low concentration of particles and descending regions of high concentration of particles. This mechanism might be related to the formation of bubbles in fluidized beds.

Two Adjoint-Based Optimization Approaches for a Free Surface Stokes Flow

Sabine Repke, Nicole Marheineke, and René Pinnau

SIAM J. Appl. Math. 71, pp. 2168-2184 (17 pages)

Online Publication Date: December 13, 2011

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This work deals with the optimal control of a free surface Stokes flow which responds to an applied outer pressure. Typical applications are fiber spinning or thin film manufacturing. We present and discuss two adjoint-based optimization approaches that differ in the treatment of the free boundary as either a state or control variable. In both cases the free boundary is modeled as the graph of a function. The PDE-constrained optimization problems are numerically solved by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, where the gradient of the reduced cost function is expressed in terms of adjoint variables. Numerical results for both strategies are finally compared with respect to accuracy and efficiency.

The Radial Volume Derivative and the Critical Boundary Displacement for Cavitation

Pablo V. Negrón-Marrero and Jeyabal Sivaloganathan

SIAM J. Appl. Math. 71, pp. 2185-2204 (20 pages)

Online Publication Date: December 15, 2011

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We study the displacement boundary value problem of minimizing the total energy $E(\mathbf{u})$ stored in a nonlinearly elastic body occupying a spherical domain $B$ in its reference configuration over (possibly discontinuous) radial deformations $\mathbf{u}$ of the body subject to affine boundary data $\mathbf{u}(\mathbf{x})=\lambda\mathbf{x}$ for $\mathbf{x}\in \partial B$. For a given value of $\lambda$, we define what we call the radial volume derivative at $\lambda$, denoted by $G(\lambda )$, which measures the stability or instability of the underlying homogeneous deformation $\mathbf{u}^h_\lambda(\mathbf{x})\equiv \lambda\mathbf{x}$ to the formation of holes. We give conditions under which the critical boundary displacement $\lambda_{\mathrm{crit}}$ for radial cavitation is the unique solution of $G(\lambda )=0$. Moreover, we prove that the radial volume derivative $G(\lambda )$ can be approximated by the corresponding volume derivative for a punctured ball $B_\epsilon$, containing a pre-existing cavity of radius $\epsilon >0$ in its reference state, in the limit $\epsilon\rightarrow 0$ and we use this to propose a numerical scheme to determine $\lambda_{\mathrm{crit}}$. We illustrate these general results with analytical and numerical examples.

Confined Elastic Curves

Patrick W. Dondl, Luca Mugnai, and Matthias Röger

SIAM J. Appl. Math. 71, pp. 2205-2226 (22 pages)

Online Publication Date: December 15, 2011

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We consider the problem of minimizing Euler's elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values $+1$ on the inside and $-1$ on the outside of the curve. The outer container now becomes just the domain of the phase field. Diffuse approximations of the elastica energy and the curve length are well known; implementing the topological constraint thus becomes the main difficulty here. We propose a solution based on a diffuse approximation of the winding number, present a proof that one can approximate a given sharp interface using a sequence of phase fields, and show some numerical results using finite elements based on subdivision surfaces.
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Special Section on Controlled Drug Delivery

Alexander Nepomnyashchy and Vladimir Volpert, Guest Editors

SIAM J. Appl. Math. 71, pp. 2227-2228 (2 pages)

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Controlled drug delivery has been attracting a great deal of attention in the medical community for years as an efficient way of providing treatment for a wide class of diseases. The common principle on which various drug delivery devices are based is mass transfer of the given drug toward particular organs, in which either the mass transfer rate, or place, or both are prescribed according to certain medical protocols. Much progress has been achieved in the design and development of various controlled drug delivery systems, and many people routinely take medicine designed for controlled release. Mathematical modeling of drug delivery systems is very important because a successful model can provide a better understanding and a quantitative description of the physical, chemical and biological processes governing the performance of the systems. On the basis of this description, better controlled drug delivery systems can be designed.
We hope that this collection of papers will result in an increased attention of applied mathematicians to this class of important mass transfer and control problems. We also hope that practitioners will become more aware of the mathematical results that can be useful.
The papers differ in mathematical sophistication, ranging from models described by systems of ODEs to quite complex PDE problems. They consider quite a few different aspects of drug delivery thus introducing a significant portion of the field. Each paper has an extensive introduction that should make the paper understandable to an applied mathematician who is not an expert in drug delivery.
A discussion of controlled drug delivery in cancer immunotherapy is presented in “Controlled Drug Delivery in Cancer Immunotherapy: Stability, Optimization, and Monte Carlo Analysis" by Minelli, Topputo, and Bernelli-Zazzera, who formulate and solve an optimal control problem and show that the control policy is effective even when the patient's initial conditions are uncertain. A hybrid model of cell population dynamics, where cells are discrete elements whose dynamics depend on continuous intracellular and extracellular processes, is developed in “Hybrid Model of Erythropoiesis and Leukemia Treatment with Cytosine Arabinoside" by Kurbatova, Bernard, Bessonov, Crauste, Demin, Dumontet, Fischer, and Volpert, to simulate the evolution of immature red blood cells in the bone marrow. The model is used to study normal and leukemic red blood cell production and treatment of leukemia. In “Quadratic Models to Fit Experimental Data of Paclitaxel Release Kinetics from Biodegradable Polymers," Blanchet, Delfour, and Garon validate three ODE models against experimental data in order to better understand drug release kinetics.
A model for drug diffusion from a spherical polymeric drug delivery device is considered in “Asymptotic and Numerical Results for a Model of Solvent-Dependent Drug Diffusion through Polymeric Spheres" by McCue, Hsieh, Moroney, and Nelson. Here the solvent diffuses into the polymer, which transitions from a glassy to a rubbery state, thus resulting in a moving boundary problem. In “Model Reduction Strategies Enable Computational Analysis of Controlled Drug Release from Cardiovascular Stents," D'Angelo, Zunino, Porpora, Morlacchi, and Migliavacca deal with drug-eluting stents and consider a hierarchy of mathematical models ranging from a lumped ODE model to fully three-dimensional models for drug transfer in the artery. A gene delivery of nucleic acid to the cell nucleus problem is discussed in “Modeling the Early Steps of Cytoplasmic Trafficking in Viral Infection and Gene Delivery," in which Amoruso, Lagache, and Holcman focus on plasmid DNA and virus cytoplasmic trafficking.
The editors thank all the authors and reviewers for their contributions to this collection.

Controlled Drug Delivery in Cancer Immunotherapy: Stability, Optimization, and Monte Carlo Analysis

Andrea Minelli, Francesco Topputo, and Franco Bernelli-Zazzera

SIAM J. Appl. Math. 71, pp. 2229-2245 (17 pages)

Online Publication Date: December 20, 2011

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A discussion on controlled drug delivery in cancer immunotherapy is presented in this paper. A fifth-order model is adopted to describe the dynamics of the tumor–immune interaction. Natural equilibrium points of this system are sought, and their stability is analyzed. An optimal control problem is stated and solved numerically. Both continuous and discrete controls are treated, and their implications on the therapy protocol are discussed. The robustness of the optimal therapies is assessed a posteriori with a Monte Carlo analysis. This shows that the control policy is effective even when the initial patient conditions are affected by uncertainties.

Hybrid Model of Erythropoiesis and Leukemia Treatment with Cytosine Arabinoside

Polina Kurbatova, Samuel Bernard, Nikolai Bessonov, Fabien Crauste, Ivan Demin, Charles Dumontet, Stephan Fischer, and Vitaly Volpert

SIAM J. Appl. Math. 71, pp. 2246-2268 (23 pages)

Online Publication Date: December 20, 2011

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A hybrid model of cell population dynamics, where cells are discrete elements whose dynamics depend on continuous intracellular and extracellular processes, is developed to simulate the evolution of immature red blood cells in the bone marrow. Cell differentiation, self-renewal, or apoptosis are determined by an intracellular network, based on two proteins, Erk and Fas, and described by ordinary differential equations and by local extracellular regulation performed by Fas-ligand, a protein produced by mature cells whose concentration evolution is represented by a partial differential equation. The model is used to study normal and leukemic red blood cell production (erythropoiesis) and treatment of leukemia. Normal cells are assumed to have a circadian rhythm that influences their cell cycle progression, whereas leukemic cells are assumed to escape circadian rhythms. We consider a treatment based on periodic administration of cytosine arabinoside (Ara-C), an anticancer agent targeting cells in DNA synthesis. A detailed pharmacodynamic/pharmacokinetic model of Ara-C is proposed and used to simulate the treatment. Influence of the period of treatment and delivery time on the outcome of the treatment is investigated and stress the relevance of considering chronotherapeutic treatments to treat leukemia.

Quadratic Models to Fit Experimental Data of Paclitaxel Release Kinetics from Biodegradable Polymers

Guillaume Blanchet, Michel C. Delfour, and André Garon

SIAM J. Appl. Math. 71, pp. 2269-2286 (18 pages)

Online Publication Date: December 20, 2011

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In order to achieve prescribed drug release kinetics some authors have been investigating biphasic and possibly multiphasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao and Venkatraman [J. Control. Release, 130 (2008), pp. 9–14] and Lao, Venkatraman, and Peppas [Eur. J. Pharm. Biopharm., 70 (2008), pp. 796–803]. The present paper validates three mathematical ordinary differential equation models against their experimental data: a quadratic one for the release of paclitaxel, a combination of this model with the idea of partition coefficients of Lao, Venkatraman, and Peppas [Eur. J. Pharm. Biopharm., 70 (2008), pp. 796–803] for polymer blends, and the Verlhurst population model with a log kill law coupled with the drug release model that describes the evolution of smooth muscle cells in the presence of paclitaxel released from three neat polymers.

Asymptotic and Numerical Results for a Model of Solvent-Dependent Drug Diffusion through Polymeric Spheres

Scott W. McCue, Mike Hsieh, Timothy J. Moroney, and Mark I. Nelson

SIAM J. Appl. Math. 71, pp. 2287-2311 (25 pages)

Online Publication Date: December 20, 2011

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A model for drug diffusion from a spherical polymeric drug delivery device is considered. The model contains two key features. The first is that solvent diffuses into the polymer, which then transitions from a glassy to a rubbery state. The interface between the two states of polymer is modeled as a moving boundary, whose speed is governed by a kinetic law; the same moving boundary problem arises in the one-phase limit of a Stefan problem with kinetic undercooling. The second feature is that drug diffuses only through the rubbery region, with a nonlinear diffusion coefficient that depends on the concentration of solvent. We analyze the model using both formal asymptotics and numerical computation, the latter by applying a front-fixing scheme with a finite volume method. Previous results are extended and comparisons are made with linear models that work well under certain parameter regimes. Finally, a model for a multilayered drug delivery device is suggested, which allows for more flexible control of drug release.

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

Carlo D'Angelo, Paolo Zunino, Azzurra Porpora, Stefano Morlacchi, and Francesco Migliavacca

SIAM J. Appl. Math. 71, pp. 2312-2333 (22 pages)

Online Publication Date: December 20, 2011

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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 and drug release for a medicated stent implanted into a coronary bifurcation, a treatment where clinical complications still have to be fully understood.

Modeling the Early Steps of Cytoplasmic Trafficking in Viral Infection and Gene Delivery

C. Amoruso, T. Lagache, and D. Holcman

SIAM J. Appl. Math. 71, pp. 2334-2358 (25 pages)

Online Publication Date: December 20, 2011

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Delivery of nucleic acid to the cell nucleus is a fundamental step in gene therapy. In this review of modeling drug and gene delivery, we focus on the particular stage of plasmid DNA or virus cytoplasmic trafficking. A challenging problem is to quantify the success of this limiting stage. We present some models and simulations of plasmid trafficking and of the limiting phase of DNA-polycation escape from an endosome and discuss virus cytoplasmic trafficking. The models can be used to assess the success of viral escape from endosomes, to quantify the early step of viral-cell infection, and to propose new simulation tools for designing new hybrid-viruses as synthetic vectors.
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