关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Mathematical models of tumor-immune interactions provide an analytic framework in which to address specific questions about tumor-immune dynamics. We present a new mathematical model that describes tumor-immune interactions, focusing on the role of natural killer (NK) and CD8+ T cells in tumor surveillance, with the goal of understanding the dynamics of immune-mediated tumor rejection. The model describes tumor-immune cell interactions using a system of differential equations. The functions describing tumor-immune growth, response, and interaction rates, as well as associated variables, are developed using a least-squares method combined with a numerical differential equations solver. Parameter estimates and model validations use data from published mouse and human studies. Specifically, CD8+ T-tumor and NK-tumor lysis data from chromium release assays as well as in vivo tumor growth data are used. A variable sensitivity analysis is done on the model. The new functional forms developed show that there is a clear distinction between the dynamics of NK and CD8+ T cells. Simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge are able to reproduce data from the published studies. A sensitivity analysis reveals that the variable to which the model is most sensitive is patient specific, and can be measured with a chromium release assay. The variable sensitivity analysis suggests that the model can predict which patients may positively respond to treatment. Computer simulations highlight the importance of CD8+ T-cell activation in cancer therapy.

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A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth

Deterministic and Stochastic Optimal Control

Dynamic optimization: The calculus of variations and optimal control in economics and management: Morton I. KAMIEN and Nancy L. SCHWARTZ Volume 4 in: Dynamic Economics: Theory and Applications, North-Holland, New York, 1981, xi + 331 pages, Dfl.90.00

A system of ordinary differential equations, which describes the interaction of HIV and T-cells in the immune system is utilized, and optimal controls representing drug treatment strategies of this model are explored. Two types of treatments are used, and existence and uniqueness results for the optimal control pair are established. The optimality system is derived and then solved numerically using an iterative method with a Runge–Kutta fourth order scheme. Copyright © 2002 John Wiley & Sons, Ltd.

Optimal control of an HIV immunology model

We examine an ordinary dierential system modeling the interaction of the HIV virus and the immune system of the human body. The optimal control representsapercentageeect thechemotherapyhasontheinteractionoftheCD4 + T cells withthevirus. WemaximizethebenetbasedontheTcell countandminimize the systemic cost based on the percentage of chemotherapy given. Existence of an optimal control is proven, and the optimal control is uniquely characterized in terms of the solution of the optimality system, which is the state system coupled with the adjoint system. In addition, numerical examples are given for illustration.

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Optimizing chemotherapy in an hiv model

Modeling Optimal Intervention Strategies for Cholera

We propose a mathematical model for the growth of cell-cycle-specific dose limiting bone marrow. In an attempt to determine effective methods of treatment without overdestruction of the bone marrow we implement optimal control theory. We design the control functional to maximize both the bone marrow mass and the dose over the treatment interval. Next we show that an optimal control exists for this problem, and then we characterize our optimal control in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. We show that the optimality system is unique for suitably small time intervals. Finally, we analyze the optimal control and the optimality system using numerical techniques. This allows us to suggest optimal methods of treatment that prevent excessive destruction of the bone marrow based on the specific weights in our objective functional.

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Optimal control applied to cell-cycle-specific cancer chemotherapy

One of the most challenging tasks in constructing a mathematical model of cancer treatment is the calculation of biological parameters from empirical data. This task becomes increasingly difficult if a model involves several cell populations and treatment modalities. A sophisticated model constructed by de Pillis et al., Mixed immunotherapy and chemotherapy of tumours: Modelling, applications and biological interpretations, J. Theor. Biol. 238 (2006), pp. 841‐862; involves tumour cells, specific and non-specific immune cells (natural killer (NK) cells, CD8 þ T cells and other lymphocytes) and employs chemotherapy and two types of immunotherapy (IL-2 supplementation and CD8 þ T-cell infusion) as treatment modalities. Despite the overall success of the aforementioned model, the problem of illustrating the effects of IL-2 on a growing tumour remains open. In this paper, we update the model of de Pillisetal. and then carefully identify appropriate values for the parameters of the new model according to recent empirical data. We determine new NK and tumour antigen-activated CD8 þ T-cell count equilibrium values; we complete IL-2 dynamics; and we modify the model in de Pillisetal. to allow for endogenous IL-2 production, IL-2-stimulated NK cell proliferation and IL-2-dependent CD8 þ T-cell self-regulations. Finally, we show that the potential patient-specific efficacy

/pdf/mathematical-model-creation-for-cancer-chemo-immunotherapy-5faemq04q7.pdf

Mathematical model creation for cancer chemo-immunotherapy

We investigate a mathematical model for the dynamics between tumor
cells, immune-effector cells, and the cytokine interleukin-2
(IL-2). In order to better determine under what circumstances the
tumor can be eliminated, we implement optimal control theory. We
design the control functional to maximize the effector cells and
interleukin-2 concentration and to minimize the tumor cells. Next,
we show that an optimal control exists for this problem. After
which, we characterize our unique optimal control in terms of the
solutions to the optimality system, which is the state system
coupled with the adjoint system. Finally, we analyze the optimal
control and optimality system using numerical techniques.

/pdf/optimal-control-applied-to-immunotherapy-1md7z28jrh.pdf

K. Renee Fister

Papers

Optimizing chemotherapy in an hiv model

Modeling Optimal Intervention Strategies for Cholera

Optimal control applied to cell-cycle-specific cancer chemotherapy

Mathematical model creation for cancer chemo-immunotherapy

Optimal control applied to immunotherapy