Anti-angiogenic therapy is a novel treatment approach for cancer that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this paper …
Anti-angiogenic therapy is a novel treatment approach for cancer that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this paper we consider a mathematical model where the stimulation term in the dynamics is proportional to the number of endothelial cells. This model is an example from a class of mathematical models for anti-angiogenic treatment that were developed and medically validated by Hahnfeldt, Panigrahy, Folkman and Hlatky [8]. The problem how to schedule a given amount of angiogenic inhibitors to achieve a maximum reduction in the primary cancer volume is considered as an optimal control problem and it is shown that optimal controls are bang-bang of the type 0a0 with 0 denoting a trajectory corresponding to no treatment and a a trajectory with treatment at maximum dose along which all inhibitors are being exhausted.
A mathematical model for the scheduling of angiogenic inhibitors that includes a pharmacokinetic equation is considered as an optimal control problem. When dosage and concentration of the inhibitor are identified, …
A mathematical model for the scheduling of angiogenic inhibitors that includes a pharmacokinetic equation is considered as an optimal control problem. When dosage and concentration of the inhibitor are identified, there exists an optimal singular arc of order 1 that forms the core of a synthesis of optimal controls. Under the standard pharmacokinetic linear model for the concentration this singular arc is preserved and still optimal, but its order increases to 2. This prevents concatenations of the singular arc with the constant bang controls and now the transitions to and from the singular arc are through chattering arcs. Optimal controls have the property that the associated concentration of inhibitors tracks the optimal singular arc for the reduced model without pharmacokinetic equations.
In this paper, optimization of antiangiogenic therapy for tumor management is considered as a nonlinear control problem. A new technique is developed to optimize antiangiogenic therapy which minimizes the volume …
In this paper, optimization of antiangiogenic therapy for tumor management is considered as a nonlinear control problem. A new technique is developed to optimize antiangiogenic therapy which minimizes the volume of a tumor and prevents it from growing using an optimum drug dose. To this end, an optimum desired trajectory is designed to minimize a performance index. Two controllers are then presented that drive the tumor volume to its optimum value. The first controller is proven to yield exponential results given exact model knowledge. The second controller is developed under the assumption of parameteric uncertainties in the system model. A least-squares estimation strategy based on a prediction error formulation and a Lyapunov-type stability analysis is developed to estimate the unknown parameters of the performance index. An adaptive controller is then designed to track the desired optimum trajectory. The proposed tumor minimization scheme is shown to minimize the tumor volume with an optimum drug dose despite the lack of knowledge of system parameters.
A biophysical tool is introduced that seeks to provide a theoretical basis for helping drug design teams assess the most promising drug targets and design optimal treatment strategies. The tool …
A biophysical tool is introduced that seeks to provide a theoretical basis for helping drug design teams assess the most promising drug targets and design optimal treatment strategies. The tool is grounded in a previously validated computational model of the feedback that occurs between a growing tumor and the evolving vasculature. In this paper, the model is particularly used to explore the therapeutic effectiveness of two drugs that target the tumor vasculature: angiogenesis inhibitors (AIs) and vascular disrupting agents (VDAs). Using sensitivity analyses, the impact of VDA dosing parameters is explored, as is the effects of administering a VDA with an AI. Further, a stochastic optimization scheme is utilized to identify an optimal dosing schedule for treatment with an AI and a chemotherapeutic. The treatment regimen identified can successfully halt simulated tumor growth, even after the cessation of therapy.
We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given …
We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.
Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumor growth, linking the relatively harmless avascular and the potentially fatal vascular growth phases of …
Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumor growth, linking the relatively harmless avascular and the potentially fatal vascular growth phases of the tumor. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration and proliferation; degradation of tissue; new capillary vessel formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is flow associated with the nascent network, subsequent growth evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix, and perfusion-related hemodynamic forces in a manner that may be described as both adaptive and dynamic. In this article, we first present a review of previous theoretical and computational models of angiogenesis and then indicate how recent developments in flow models are providing insight into antiangiogenic and chemotherapeutic drug treatment of solid tumors.
Cancer is a multistep process. Once a mutation of a single, normal cell has taken place, the transition from the transformed cell to a tumor and then to metastasis is …
Cancer is a multistep process. Once a mutation of a single, normal cell has taken place, the transition from the transformed cell to a tumor and then to metastasis is a complex process. In the case of a solid tumor (e.g., carcinoma) a crucial step in its development is the process of angiogenesis—the formation of a capillary network arising from pre-existing vasculature (as opposed to vasculogenesis; Vernon et al., 1992). Angiogenesis itself is a complex phenomenon, a dynamical system, involving the successful and timely interaction of many variables. Although on the one hand this complexity can make the task of the mathematical modeler difficult, on the other, it simultaneously provides the biomathematician with a full palette of rich colors to work with. In this chapter, we present several mathematical models describing the important aspects of tumor-related angiogenesis.
Angiogenesis is an important process in tumor growth. Because it is a process that allows the tumor to receive enough nutrients. Anti-angiogenesis therapy is one of the safe tumor treatment …
Angiogenesis is an important process in tumor growth. Because it is a process that allows the tumor to receive enough nutrients. Anti-angiogenesis therapy is one of the safe tumor treatment processes. It will demolish the angiogenic factors, resulting in tumor lack of the nutrient. The purposes of this study were to develop the new diffusion of nutrient models with tumor treatments to explain the nutrient concentration in each phase and analyze the growth. The simulation results indicate that in the treatment, the effectiveness of the inhibitor affects the concentration, tumor size and time to phase changes. In which the trend of concentration in each phase is different.
Разработана математическая модель роста опухоли в ткани с учетом ангиогенеза и антиангиогенной терапии. В модели учтены как конвективные потоки в ткани, так и собственная подвижность клеток опухоли. Считается, что клетка …
Разработана математическая модель роста опухоли в ткани с учетом ангиогенеза и антиангиогенной терапии. В модели учтены как конвективные потоки в ткани, так и собственная подвижность клеток опухоли. Считается, что клетка начинает мигрировать, если концентрация питательного вещества падает ниже критического уровня, и возвращается в состояние пролиферации в области с высокой концентрацией пищи. Злокачественные клетки, находящиеся в состоянии метаболического стресса, вырабатывают фактор роста эндотелия сосудов (VEGF), стимулируя опухолевый ангиогенез, что увеличивает приток питательных веществ. В работе моделируется антиангиогенный препарат, который необратимо связывается с VEGF, переводя его в неактивное состояние. Проведено численное исследование влияния концентрации и эффективности антиангиогенного препарата на скорость роста и структуру опухоли. Показано, что сама по себе противоопухолевая антиангиогенная терапия способна замедлить рост малоинвазивной опухоли, но не способна его полностью остановить.
A numerical analysis for a mathematical model of the tumor angiogenesis and tumor immune system interaction was performed in this paper. The optimal control was characterized related to chemotherapy treatment …
A numerical analysis for a mathematical model of the tumor angiogenesis and tumor immune system interaction was performed in this paper. The optimal control was characterized related to chemotherapy treatment administration. The model is expressed in terms of ordinary differential equations and describes the dependency of the tumor size, protein production and vessel density on the effective vessel density and effector-immune cells. The goal was to numerically simulate the model and determine the optimal drug dose that must be administrated in order to destroy the tumor cells and correlate it with the real biological systems. The results of the numerical simulation showed that the tumor will stop growing, and decrease in size, after applying the control. The results show a good correspondence with the real cases, thus proving the validity of the model.
Tumor-induced angiogenesis is the process by which unmetastasized tumors recruit red blood vessels by way of chemical stimuli to grow towards the tumor for vascularization and metastasis. We model the …
Tumor-induced angiogenesis is the process by which unmetastasized tumors recruit red blood vessels by way of chemical stimuli to grow towards the tumor for vascularization and metastasis. We model the process of tumor-induced angiogenesis at the tissue level using ordinary and partial differential equations (ODEs and PDEs) that have a source term. The source term is associated with a signal for growth factors from the tumor. We assume that the source term depends on time, and a parameter (time parameter). We use an explicit stabilized Runge-Kutta method to solve the partial differential equation. By introducing a source term into the PDE model, we extend the PDE model used by H. A. Harrington et al. Our results suggest that the time parameter could play some role in understanding angiogenesis.
Cancer arises when within a single cell multiple malfunctions of control systems occur, which are, broadly, the system that promote cell growth and the system that protect against erratic growth. …
Cancer arises when within a single cell multiple malfunctions of control systems occur, which are, broadly, the system that promote cell growth and the system that protect against erratic growth. Additional systems within the cell must be corrupted so that a cancer cell, to form a mass of any real size, produces substances that promote the growth of new blood vessels. Multiple mutations are required before a normal cell can become a cancer cell by corruption of multiple growth-promoting systems. We develop a simple mathematical model to describe the solid cancer growth dynamics inducing angiogenesis in the absence of cancer controlling mechanisms. The initial conditions supplied to the dynamical system consist of a perturbation in form of pulse: The origin of cancer cells from normal cells of an organ of human body. Thresholds of interacting parameters were obtained from the steady states analysis. The existence of two equilibrium points determine the strong dependency of dynamical trajectories on the initial conditions. The thresholds can be used to control cancer. Cancer can be settled in an organ if the following combination matches: better fitness of cancer cells, decrease in the efficiency of the repairing systems, increase in the capacity of sprouting from existing vascularization, and higher capacity of mounting up new vascularization. However, we show that cancer is rarely induced in organs (or tissues) displaying an efficient (numerically and functionally) reparative or regenerative mechanism.
Antiangiogenic therapy belongs to a modern cancer therapy and has several advantages over conventional cancer therapies. However, the cost of its angiogenic inhibitor is expensive. A strategy to determine the …
Antiangiogenic therapy belongs to a modern cancer therapy and has several advantages over conventional cancer therapies. However, the cost of its angiogenic inhibitor is expensive. A strategy to determine the least possible level of the inhibitor dose is proposed in the present work based on the nonlinear mathematical model of tumor growth dynamics under angiogenic inhibition. Specifically, inspired by optimal control design, a discrete Linear Quadratic Regulator (LQR) is carried out on the feedback linearization of the nonlinear model. In this scheme, the value of the input-weighting scalar in the LQR cost function is investigated since it correlates with the dose level of the angiogenic inhibitor. To verify the proposed strategy, simulations are performed with different values of the input-weighting scalar for a total treatment duration of 50 days. The maximum allowance of inhibitor dose level is set to 50 mg/kg based on the physiological limitation. It is found that the least level of inhibitor dose is acquired at certain value of the input-weighting scalar.
Modeling the tumor growth under angiogenic inhibition is an important step towards designing tumor treatment therapies based on mathematical tools. Our goal is to create a model for tumor growth …
Modeling the tumor growth under angiogenic inhibition is an important step towards designing tumor treatment therapies based on mathematical tools. Our goal is to create a model for tumor growth that describes the underlying physiological processes adequately while being as simple as possible. We propose a second-order model containing linear terms and one bilinear term modeling the dynamics of tumor volume and inhibitor level, and work out the parametric identification process for the model. The parametric identification of the model is done using measurements from experiments on C57Bl/6 mice with C38 colon adenocarcinoma treated with bevacizumab. The control group of the mice received one injection at the beginning of the experiment, these measurement data are used for parametric identification, while the case group of mice received injection at each day of the treatment, these measurements are used to validate the model. The validation showed that the proposed model is capable of describing the tumor growth dynamics.
Introduction: Approaches aiming to model the time course of tumor growth and tumor growth inhibition following a therapeutic intervention have recently been proposed for supporting decision making in oncology drug …
Introduction: Approaches aiming to model the time course of tumor growth and tumor growth inhibition following a therapeutic intervention have recently been proposed for supporting decision making in oncology drug development. When considered in a comprehensive model-based approach, tumor growth can be included in the cascade of quantitative and causally related markers that lead to the prediction of survival, the final clinical response. Areas covered: The authors examine articles dealing with the modeling of tumor growth and tumor growth inhibition in both preclinical and clinical settings. In addition, the authors review models describing how pharmacological markers can be used to predict tumor growth and models describing how tumor growth can be linked to survival endpoints. Expert opinion: Approaches and success stories of application of model-based drug development centered on tumor growth modeling are growing. It is also apparent that these approaches can answer practical questions on drug development more effectively than that in the past. For modeling purposes, some improvements are still needed related to study design and data quality. Further efforts are needed to encourage the mind shift from a simple description of data to the prediction of untested conditions that modeling approaches allow.
Several simple ordinary differential equation (ODE) models of tumor growth taking into account the development of its vascular network are discussed. Different biological aspects are considered from the simplest model …
Several simple ordinary differential equation (ODE) models of tumor growth taking into account the development of its vascular network are discussed. Different biological aspects are considered from the simplest model of Hahnfeldt et al. proposed in 1999 to a model which includes drug resistance of cancer cells to chemotherapy. Some of these models can be used in clinical oncology to optimize antiangiogenic and cytostatic drugs delivery so as to ensure maximum efficacy. Simple models of continuous and periodic protocols of combined therapy are implemented. Discussion on the dynamics of the models and their complexity is presented.
An optimal control problem for an abstract mathematical model for cancer chemotherapy is considered. The dynamics is for a single drug and includes pharmacodynamic (PD) and pharmacokinetic (PK) models. The …
An optimal control problem for an abstract mathematical model for cancer chemotherapy is considered. The dynamics is for a single drug and includes pharmacodynamic (PD) and pharmacokinetic (PK) models. The aim is to point out qualitative changes in the structures of optimal controls that occur as these pharmacometric models are varied. This concerns (i) changes in the PD-model for the effectiveness of the drug ( e.g. , between a linear log-kill term and a non-linear Michaelis-Menten type E max -model) and (ii) the question how the incorporation of a mathematical model for the pharmacokinetics of the drug effects optimal controls. The general results will be illustrated and discussed in the framework of a mathematical model for anti-angiogenic therapy.
This paper presents a smooth robust controller for anti-angiogenic treatment of a tumor growth. The proposed control algorithm directly aims to drive the carrying capacity of the vascular support network …
This paper presents a smooth robust controller for anti-angiogenic treatment of a tumor growth. The proposed control algorithm directly aims to drive the carrying capacity of the vascular support network to a desired trajectory that reduces overall tumor volume. For this aim, we present a closed-loop smooth robust controller that ensures the asymptotic tracking of a time-varying carrying capacity profile, despite the model uncertainties. A Lyapunov-based analysis approach is used to determine stability and performance results. Numerical simulation results are presented to demonstrate the performance and feasibility of the proposed approach. Simulation studies also examine some practical issues regarding the clinical application of the proposed dosing algorithm. Finally, the paper provides some basics for real-time implementation of the overall system.
Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability …
Healthy vasculature exhibits a hierarchical branching structure in which, on average, vessel radius and length change systematically with branching order. In contrast, tumor vasculature exhibits less hierarchy and more variability in its branching patterns. Although differences in vasculature have been highlighted in the literature, there has been very little quantification of these differences. Fractal analysis is a natural tool for comparing tumor and healthy vasculature, especially because it has already been used extensively to model healthy tissue. In this paper, we provide a fractal analysis of existing vascular data, and we present a new mathematical framework for predicting tumor growth trajectories by coupling: (1) the fractal geometric properties of tumor vascular networks, (2) metabolic properties of tumor cells and host vascular systems, and (3) spatial gradients in resources and metabolic states within the tumor. First, we provide a new analysis for how the mean and variation of scaling exponents for ratios of vessel radii and lengths in tumors differ from healthy tissue. Next, we use these characteristic exponents to predict metabolic rates for tumors. Finally, by combining this analysis with general growth equations based on energetics, we derive universal growth curves that enable us to compare tumor and ontogenetic growth. We also extend these growth equations to include necrotic, quiescent, and proliferative cell states and to predict novel growth dynamics that arise when tumors are treated with drugs. Taken together, this mathematical framework will help to anticipate and understand growth trajectories across tumor types and drug treatments.
This paper is concerned with analysis of two anticancer therapy models focused on sensitivity of therapy outcome with respect to model structure and parameters. Realistic periodic therapies are considered, combining …
This paper is concerned with analysis of two anticancer therapy models focused on sensitivity of therapy outcome with respect to model structure and parameters. Realistic periodic therapies are considered, combining cytotoxic and antiangiogenic agents, defined on a fixed time horizon. Tumor size at the end of therapy and average tumor size calculated over therapy horizon are chosen to represent therapy outcome. Sensitivity analysis has been performed numerically, concentrating on model parameters and structure at one hand, and on treatment protocol parameters, on the other. The results show that sensitivity of the therapy outcome highly depends on the model structure, helping to discern a good model. Moreover, it is possible to use this analysis to find a good protocol in case of heterogeneous tumors.
We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon …
We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.
<p style='text-indent:20px;'>We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the …
<p style='text-indent:20px;'>We review and discuss various aspects that the modeling of pharmacometric properties has on the structure of optimal solutions in mathematical models for cancer treatment. These include (i) the changes in the interpretation of the solutions as pharmacokinetic (PK) models are added, respectively deleted from the modeling and (ii) qualitative changes in the structures of optimal controls that occur as pharmacodynamic (PD) models are varied. The results will be illustrated with a sample of models for cancer treatment.
Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when …
Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.
A mathematical model for cancer chemotherapy of heterogeneous tumor populations is considered as an optimal control problem with the objective to minimize the tumor burden over a prescribed therapy horizon. …
A mathematical model for cancer chemotherapy of heterogeneous tumor populations is considered as an optimal control problem with the objective to minimize the tumor burden over a prescribed therapy horizon. While an upfront maximum tolerated dose (MTD) regimen with rest-period has been confirmed as mathematically optimal for models when the tumor population is homogeneous, in the presence of partially sensitive or even resistant cells, protocols that administer the therapeutic agents at lower dose rates described by so-called singular controls become a viable alternative. In this paper, the structure of protocols that follow an initial upfront maximum dose treatment with reduced dose rate singular controls is investigated. Such protocols reflect structures which in the medical literature sometimes are called chemo-switch protocols.
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We …
We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.
We consider an optimal control problem of the Mayer- type for a single-input, control affine, nonlinear system in s di- mension. In this paper, we analyze effects that a modeling …
We consider an optimal control problem of the Mayer- type for a single-input, control affine, nonlinear system in s di- mension. In this paper, we analyze effects that a modeling extension has on the optimality of singular controls when the control is re- placed with the output of a first-order, time-invariant linear system driven by a new control. This analysis is motivated by an opti- mal control problem for a novel cancer treatment method, tumor anti-angiogenesis, when such a linear differential equation, which represents the pharmacokinetics of the therapeutic agent, is added to the model. We show that formulas that define a singular control of order 1 and its associated singular arc carry over verbatim under this model extension, albeit with a different interpretation. But the intrinsic order of the singular control increases to 2. As a conse- quence, optimal concatenation sequences with the singular control change and the possibility of optimal chattering arcs arises.
A mathematical model for the scheduling of angiogenic inhibitors to control a vascularized tumor is considered as an optimal control problem. A complete synthesis of optimal solutions is given.
A mathematical model for the scheduling of angiogenic inhibitors to control a vascularized tumor is considered as an optimal control problem. A complete synthesis of optimal solutions is given.
The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the …
The problem of scheduling a given amount of angiogenic inhibitors is consideredas an optimal control problem with the objective of maximizing the achievable tumor reduction.For a dynamical model for the evolution of the carrying capacity of the vasculature formulatedin [15] optimal controls are computed for both a Gompertzian and logistic model of tumor growth.While optimal controls for the Gompertzian model typically contain a segment along whichthe control is singular, for the logistic model optimal controls are bang-bang with atmost two switchings.
Antiangiogenic therapy is a novel treatment approach in cancer therapy that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this …
Antiangiogenic therapy is a novel treatment approach in cancer therapy that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In this paper a mathematical model for antiangiogenic treatments based on a biologically validated model by Hahnfeldt et al. is analyzed as an optimal control problem and a full solution of the problem is given. Geometric methods from optimal control theory are utilized to arrive at the solution.
Journal Article Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment Get access Urszula Ledzewicz, Urszula Ledzewicz † Department of Mathematics and Statistics, Southern Illinois University …
Journal Article Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment Get access Urszula Ledzewicz, Urszula Ledzewicz † Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA †Corresponding author. Email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar John Marriott, John Marriott Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA Search for other works by this author on: Oxford Academic Google Scholar Helmut Maurer, Helmut Maurer Department of Mathematics and Statistics, Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms Universität Münster, Einsteinstrasse 62, D-48149 Münster, Germany Search for other works by this author on: Oxford Academic Google Scholar Heinz Schättler Heinz Schättler Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130-4899, USA Search for other works by this author on: Oxford Academic Google Scholar Mathematical Medicine and Biology: A Journal of the IMA, Volume 27, Issue 2, June 2010, Pages 157–179, https://doi.org/10.1093/imammb/dqp012 Published: 01 June 2010 Article history Received: 23 October 2008 Revision received: 06 February 2009 Accepted: 27 April 2009 Published: 01 June 2010
The effects of the angiogenic inhibitors endostatin, angiostatin, and TNP-470 on tumor growth dynamics are experimentally and theoretically investigated. On the basis of the data, we pose a quantitative theory …
The effects of the angiogenic inhibitors endostatin, angiostatin, and TNP-470 on tumor growth dynamics are experimentally and theoretically investigated. On the basis of the data, we pose a quantitative theory for tumor growth under angiogenic stimulator/inhibitor control that is both explanatory and clinically implementable. Our analysis offers a ranking of the relative effectiveness of these inhibitors. Additionally, it reveals the existence of an ultimate limitation to tumor size under angiogenic control, where opposing angiogenic stimuli come into dynamic balance, which can be modulated by antiangiogenic therapy. The competitive influences of angiogenically driven growth and inhibition underlying this framework may have ramifications for tissue size regulation in general.
Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying …
Mathematical models for cancer chemotherapy as optimal control problems are considered. Results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying assumption is made that drugs act instantaneously, are compared with more realistic models that include pharmacokinetic (PK) equations modelling the drug's plasma concentration and various pharmacodynamic (PD) models for the effect the concentrations have on cells.
In light of recent clinical developments, the importance of mathematical modeling in cancer prevention and treatment is discussed. An exist- ing model of cancer chemotherapy is reintroduced and placed within …
In light of recent clinical developments, the importance of mathematical modeling in cancer prevention and treatment is discussed. An exist- ing model of cancer chemotherapy is reintroduced and placed within current investigative frameworks regarding approaches to treatment optimization. Areas of commonality between the model predictions and the clinical findings are investigated as a way of further validating the model predictions and also establishing mathematical foundations for the clinical studies. The model predictions are used to propose additional ways that treatment optimization could enhance the clinical processes. Arising out of these, an expanded model of cancer is proposed and a treatment model is subsequently obtained. These models predict that malignant cells in the marrow and peripheral blood exhibit the tendency to evolve toward population levels that enable them to replace normal cells in these compartments in the untreated case. In the case of dose-dense treatment along with recombinant hematopoietic growth factors, the models predict a situation in which normal and abnormal cells in the marrow and peripheral blood are obliterated by drug action, while the normal cells regain their growth capabilities through growth-factor stimulation.
Linear Systems and the Time Optimal Control Problem. - Exercises. - Optimal Control for Nonlinear Systems. - Exercises. - Geometric Optimal Control. - Exercises. - Singular Trajectories and Feedback Classification. …
Linear Systems and the Time Optimal Control Problem. - Exercises. - Optimal Control for Nonlinear Systems. - Exercises. - Geometric Optimal Control. - Exercises. - Singular Trajectories and Feedback Classification. - Exercises. - Controllability, Higher Order Maximum Principle, Legendre-Clebsch and Goh Necessary Optimality Conditions. - Exercises. - The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C0-Optimality. - Time Minimal Control of Chemical Batch Reactors and Singular Trajectories. - Generic Properties of Singular Trajectories. - Exercises. - Singular Trajectories in Sub-Riemannian Geometry. - Exercises. - Micro-Local Resolution of the Singularity near a Singular Trajectory, Lagrangian Manifolds and Symplectic Stratifications. - Exercises. - Numerical Computations. - Conclusion and Perspectives. - Exercises. - References. - Index