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Asymptotically Optimal Allocation Policies for Transplant Queueing Systems Authors: Amin Khademi https://orcid.org/0000-0002-5281-8715
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1116-1140 |
| Abstrak | : | We consider the problem of dynamic allocation of organs to patients in a transplant system. The system is modeled as a multiclass bipartite matching system, in which patients may die/delist or move between classes due to changes of their health status. We study a stochastic queueing control problem (QCP) with the control process governing the allocation of each arriving organ, and the objective of maximizing the expected total life years, which consists of both pre- and posttransplant years, of the entire population in the system during a finite time horizon. We first construct a deterministic control problem, referred to as the fluid control problem (FCP), and show that it serves as a performance upper bound for the QCP. We next develop an asymptotic framework, in which large scaled overloaded transplant systems are considered, and show that the fluid scaled QCP attains the FCP upper bound asymptotically. We then propose a simple priority type policy for the QCP based on the optimal solution of the FCP, and establish its asymptotic optimality through a scaling limit theorem. Last, we conduct sensitivity analysis of the FCP with respect to the input parameters and functions to demonstrate the robustness of the proposed policy. |
A Generalized Rayleigh--Plesset Equation for Ions with Solvent Fluctuations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1098-1115 |
| Abstrak | : | We introduce a mathematical modeling framework for the conformational dynamics of charged molecules (i.e., solutes) in an aqueous solvent (i.e., water or salted water). The solvent is treated as an incompressible fluid, and its fluctuating motion is described by the Stokes equation with the Landau--Lifschitz stochastic stress. The motion of the solute-solvent interface (i.e., the dielectric boundary) is determined by the fluid velocity together with the balance of the viscous force, hydrostatic pressure, surface tension, solute-solvent van der Waals interaction force, and electrostatic force. The electrostatic interactions are described by the dielectric Poisson--Boltzmann theory. Within such a framework, we derive a generalized Rayleigh--Plesset equation, a nonlinear stochastic ordinary differential equation (SODE), for the radius of a spherical charged molecule, such as an ion. The spherical average of the stochastic stress leads to a multiplicative noise. We design and test numerical methods for solving the SODE and use the equation, together with explicit-solvent molecular dynamics simulations, to study the effective radius of a single ion. Potentially, our general modeling framework can be used to efficiently determine the solute-solvent interfacial structures and predict the free energies of more complex molecular systems. |
A Next-Generation Mathematical Model for Drug-Eluting Stents
| Pengarang | : | Yifan Wang |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1503-1529 |
| Abstrak | : | We propose a Navier--Stokes--Biot fluid-structure interaction (FSI) model to study the interaction between an incompressible, viscous Newtonian fluid and a poroelastic medium with permeability depending on the volumetric change of pore size. The FSI model is coupled to a set of advection-reaction-diffusion equations defined on moving domains so that we may study the interaction between the blood flow, a stented coronary artery, and time-dependent pharmacokinetics of drug absorption in drug-eluting stents. A monolithic approach is used to implement the proposed problem numerically within the context of finite element discretization. Nitsche's method is employed to enforce one of the coupling conditions at the moving fluid-poroelastic structure interface. Stability analysis is presented, providing conditions on Nitsche's penalty parameter under which the scheme is unconditionally stable. Using 3D simulations, five geometrically different metallic stent platforms are considered with two different pharmacokinetics to show how stent geometry and type of coating impact the biomechanical environment, the local hemodynamics, and the concentration of the pharmacological agents within the vascular wall and artery lumen. It is found that stent implantation changes the permeability properties of the arterial wall, as well as local hemodynamics, which may be responsible for the so-called edge effect, i.e., suboptimal reduction in restenosis rates near the edges of drug-eluting stents. To the best of our knowledge, this is the first study of drug-eluting stents that takes into account fluid-poroelastic structure interaction with permeability depending on the volumetric change of the pore size, coupled to an advection-reaction-diffusion model defined on moving domains. |
Optimal Chemotherapy for Brain Tumor Growth in a Reaction-Diffusion Model
| Pengarang | : | Mohsen Yousefnezhad |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1077-1097 |
| Abstrak | : | In this paper we address the question of determining optimal chemotherapy strategies to prevent the growth of brain tumor population. To do so, we consider a reaction-diffusion model which describes the diffusion and proliferation of tumor cells and a minimization problem corresponding to it. We shall establish that the optimization problem admits a solution and obtain a necessary condition for the minimizer. In a specific case, the optimizer is calculated explicitly, and we prove that it is unique. Then, a gradient-based efficient numerical algorithm is developed in order to determine the optimizer. Our results suggest a bang-bang chemotherapy strategy in a cycle which starts at the maximum dose and terminates with a rest period. Numerical simulations based upon our algorithm on a real brain image show that this is in line with the maximum tolerated dose (MTD), a standard chemotherapy protocol. |
The Inviscid Limit of Third-Order Linear and Nonlinear Acoustic Equations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1461-1482 |
| Abstrak | : | We analyze the behavior of third-order-in-time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson equation both in its Westervelt-type and in its Kuznetsov-type forms, that is, including general nonlinearities of quadratic type. As it turns out, sufficiently smooth solutions of these equations converge in the energy norm to the solutions of the corresponding inviscid models at a linear rate. Numerical experiments illustrate our theoretical findings. |
Toward Understanding the Boundary Propagation Speeds in Tumor Growth Models
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1052-1076 |
| Abstrak | : | At the continuous level, we consider two types of tumor growth models: the cell density model, based on the fluid mechanical construction, is more favorable for scientific interpretation and numerical simulations, and the free boundary model, as the incompressible limit of the former, is more tractable when investigating the boundary propagation. In this work, we aim to investigate the boundary propagation speeds in those models based on asymptotic analysis of the free boundary model and efficient numerical simulations of the cell density model. We derive, for the first time, some analytical solutions for the free boundary model with pressure jumps across the tumor boundary in multidimensions with finite tumor sizes. We further show that in the large radius limit, the analytical solutions to the free boundary model in one and multiple spatial dimensions converge to traveling wave solutions. The convergence rate in the propagation speeds are algebraic in multidimensions as opposed to the exponential convergence in one dimension. We also propose an accurate front capturing numerical scheme for the cell density model, and extensive numerical tests are provided to illustrate the analytical findings. |
Homogenization of Flow Through Periodic Networks
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1034-1051 |
| Abstrak | : | Using a discrete version of the method of multiple scales, the problem of fluid flow through an arbitrary periodic network of one-dimensional channels is homogenized to give Darcy flow. A simple, low-dimensional, linear algebra problem gives the effective permeability in terms of the channel conductances. The results are illustrated with an investigation of the statistical properties of effective permeability for a variety of random networks, as well as an evaluation of the correlation between permeability and network tortuosity. |
Surface Localization of Plasmons in Three Dimensions and Convexity
| Pengarang | : | Kazunori Ando |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1020-1033 |
| Abstrak | : | The Neumann--Poincare? operator defined on a smooth surface has a sequence of eigenvalues converging to zero, and the single-layer potentials of the corresponding eigenfunctions, called plasmons, decay to zero, i.e., are localized on the surface, as the index of the sequence ???? tends to infinity. We investigate quantitatively the surface localization of the plasmons in three dimensions. The results are threefold. We first prove that on smooth bounded domains of general shape, the sequence of plasmons converges to zero off the boundary surface almost surely at the rate faster than ????−1/2 as ???? →∞. We then prove that if the domain is strictly convex, then the convergence rate becomes ????−∞; namely, it is faster than ????−???? for any integer ????. As a consequence, we prove that cloaking by anomalous localized resonance does not occur on three-dimensional strictly convex smooth domains. We then look into the surface localization of the plasmons on the Clifford torus by numerical computations. The Clifford torus is taken as an example of nonconvex surfaces. The computational results show that the torus exhibits spectral properties completely different from strictly convex domains. In particular, they suggest that there is a subsequence of plasmons on the torus which has much slower decay than other entries of the sequence. |
Scale-Dependent and Self-Similar Rayleigh--Taylor and Richtmyer--Meshkov Dynamics Induced by Acceleration Varying with Length Scale
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1002-1019 |
| Abstrak | : | Rayleigh--Taylor (RT) and Richtmyer--Meshkov (RM) instabilities and RT/RM interfacial mixing are omnipresent in nature and technology and are often driven by variable acceleration. This work presents the detailed study of RT/RM dynamics induced by the acceleration varying as power-law with the length scale. We consider RT/RM dynamics within the framework of group theory and momentum model and find solutions for the scale-dependent dynamics and for the self-similar mixing. The effect of fluctuations on the self-similar mixing is also investigated by augmenting the momentum model with stochastic process. We find that the scale-dependent dynamics and the self-similar mixing can be RT type and RM type depending on the exponent of the acceleration power-law. For the scale-dependent dynamics the exponent value separating RT and RM subregimes approaches negative infinity, whereas for the self-similar mixing it is a finite negative value depending on the drag. Based on these results, we elaborate new theory benchmarks for future research. |
Stochastic Methods for Neutron Transport Equation III: Generational Many-to-One and Keff
| Pengarang | : | Alexander M. Cox |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 982-1001 |
| Abstrak | : | The neutron transport equation (NTE) describes the flux of neutrons over time through an inhomogeneous fissile medium. In the recent articles, [A. M. G. Cox et al., J. Stat. Phys., 176 (2019), pp. 425--455; E. Horton, A. E. Kyprianou, and D. Villemonais, Ann. Appl. Probab., 30 (2020), pp. 2573--2612] a probabilistic solution of the NTE is considered in order to demonstrate a Perron--Frobenius type growth of the solution via its projection onto an associated leading eigenfunction. In [S. C. Harris, E. Horton, and A. E. Kyprianou, Ann. Appl. Probab., 30 (2020), pp. 2815--2845; A. M. G. Cox et al., Monte Carlo Methods for the Neutron Transport Equation, https://arxiv.org/abs/2012.02864 (2020)], further analysis is performed to understand the implications of this growth both in the stochastic sense as well as from the perspective of Monte Carlo simulation. Such Monte Carlo simulations are prevalent in industrial applications, in particular where regulatory checks are needed in the process of reactor core design. In that setting, however, it turns out that a different notion of growth takes center stage, which is otherwise characterized by another eigenvalue problem. In that setting, the eigenvalue, sometimes called ????-effective (written ????????????????), has the physical interpretation as being the ratio of neutrons produced (during fission events) to the number lost (due to absorption in the reactor or leakage at the boundary) per typical fission event. In this article, we aim to supplement [J. Stat. Phys., 176 (2019), pp. 425--455; Ann. Appl. Probab., 30 (2020), pp. 2573--2612; Ann. Appl. Probab., 30 (2020), pp. 2815--2845; Monte Carlo Methods for the Neutron Transport Equation, https://arxiv.org/abs/2012.02864 (2020)] by developing the stochastic analysis of the NTE further to the setting where a rigorous probabilistic interpretation of ???????????????? is given, both in terms of a Perron--Frobenius type analysis as well as via classical operator analysis. To our knowledge, despite the fact that an extensive engineering literature and industrial Monte Carlo software are concentrated around the estimation of ???????????????? and its associated eigenfunction, we believe that our work is the first rigorous treatment in the probabilistic sense (which underpins some of the aforesaid Monte Carlo simulations). |