A mixed radiotherapy and chemotherapy model for treatment of cancer with metastasis

In this paper, a mathematical model of cancer treatment, in the form of a system of ordinary differential equations, by chemotherapy and radiotherapy where there is metastasis from a primary to a secondary site has been proposed and analyzed. The interaction between immune cells and cancer cells has been examined, and the chemotherapy agent has been considered as a predator on both normal and cancer cells. The metastasis may be time delayed. For better investigation of the treatment process and based on physical investigation, the immanent effects of inputs on cancer dynamic have been investigated. It is supposed that the interaction between NK cells and tumor cells changes during the chemotherapy. This novel approach is useful not only to gain a broad understanding of the specific system dynamics but also to guide the development of combination therapies. The analysis is carried out both analytically (where possible) and numerically. By considering such immanent effects, the tumor‐free equilibrium point will be stable at the end of treatment, and the tumor can not recur again, and the patient will totally recover. So, the present analysis suggests that a proper treatment method should change the dynamics of the cancer instead of only reducing the population of cancer cells. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells

Abstract

Virotherapy is an effective strategy in cancer treatment. It eliminates tumor cells without harming the healthy cells. In this article, a deterministic mathematical model to understand the dynamics of tumor cells in response to virotherapy is formulated and analyzed by incorporating cytotoxic T lymphocytes (CTLs). The basic reproduction number and the immune response reproduction number are computed and different equilibria of the proposed model are found. The local stability of different equilibria is discussed in detail. Further, the proposed model is extended to stochastic model. Numerical simulation is performed for both deterministic and stochastic models. It is observed that when both the reproduction numbers are greater than one, which corresponds to existence of unique nontrivial equilibrium point, dynamics of deterministic and stochastic models are almost same. The deterministic model shows a very complex dynamics when one or both the reproduction numbers are below one. The system exhibits both backward bifurcation and Hopf-bifurcation for suitable sets of parameters and in this situation it is not easy to predict the dynamics of cancer cells and virus particles. The existence of backward bifurcation demonstrates the fact that partial success of virotherapy can be achieved even if the immune response reproduction number is less than one.     

Medium range optimization of copper extraction planning under uncertainty in future copper prices

Deterministic mine planning models along a time horizon have proved to be very effective in supporting decisions on sequencing the extraction of material in copper mines. Some of these models have been developed for, and used successfully by CODELCO, the Chilean state copper company. In this paper, we wish to consider the uncertainty in a very volatile parameter of the problem, namely, the copper price along a given time horizon. We represent the uncertainty by a multistage scenario tree. The resulting stochastic model is then converted into a mixed 0–1 Deterministic Equivalent Model using a compact representation. We first introduce the stochastic model that maximizes the expected profit along the time horizon over all scenarios (i.e., as in a risk neutral environment). We then present several approaches for risk management in a risk averse environment. Specifically, we consider the maximization of the Value-at-Risk and several variants of the Conditional Value-at-Risk (one of them is new), the maximization of the expected profit minus the weighted probability of having an undesirable scenario in the solution provided by the model, and the maximization of the expected profit subject to stochastic dominance constraints recourse-integer for a set of profiles given by the pairs of target profits and bounds on either the probability of failure or the expected profit shortfall. We present an extensive computational experience on the actual problem, by comparing the risk neutral approach, the tested risk averse strategies and the performance of the traditional deterministic approach that uses the expected value of the uncertain parameters. The results clearly show the advantage of using the risk neutral strategy over the traditional deterministic approach, as well as the advantage of using any risk averse strategy over the risk neutral one.     

Markovian modeling for dependent interrecurrence times in bladder cancer

A methodology to model a process in which repeated events occur is presented. The context is the evolution of non-muscle-invasive bladder carcinoma (NMIBC), characterized by recurrent relapses. It is based on the statistical flowgraph approach, a technique specifically suited for semi-Markov processes. A very useful feature of the flowgraph framework is that it naturally incorporates the management of censored data. However, this approach presents two difficulties with the process to be modeled. On one hand, the management of covariates is not straightforward. However, it is of great interest to know how the characteristics of a certain patient influence the evolution of the disease. On the other hand, repeated events on the same subject are generally not independent, in which case the semi-Markov framework is not sufficient because the semi-Markov assumption implies independence among waiting time distributions. We solve this issue by extending the flowgraph methodology using the Markovian arrival process (MAP), which does successfully model the dependence between events. Along the way, we provide a procedure to consider covariates and censored times in MAPs, a pending task needed in this field. In short, we have managed to extend the flowgraph methodology beyond the semi-Markovian framework, simplifying the incorporation of covariates and keeping the management of censored times. All of which has allowed us to build a multistate model of the evolution of NMIBC. The developed model allows us to compute the Survival function for any evolution of a patient with specific clinic-pathological characteristics in this primary tumor.     

KCC theory and its application in a tumor growth model

In this study, we consider the stability of tumor model by using the standard differential geometric method that is known as Kosambi‐Cartan‐Chern (KCC) theory or Jacobi stability analysis. In the KCC theory, we describe the time evolution of tumor model in geometric terms. We obtain nonlinear connection, Berwald connection and KCC invariants. The second KCC invariant gives the Jacobi stability properties of tumor model. We found that the equilibrium points are Jacobi unstable for positive coordinates. We also discussed the time evolution of components of deviation tensor and the behavior of deviation vector near the equilibrium points.     

Mathematical modelling of tongue deformation during swallow in patients with head and neck cancer

ABSTRACT

Cancer localized to the tongue is often characterized by increased stiffness in the affected region. This stiffness affects swallow in a manner that is difficult to quantify in patients. A biomechanical model was developed to simulate the spatio-temporal deformation of the tongue during the pharyngeal phase of swallow in patients with cancer of the tongue base. The model involves finite element analysis (FEA) of a three-dimensional (3D) model of the tongue reconstructed from magnetic resonance images (MRI). The tongue tissue is assumed to be hyper-elastic. In order to examine the effects of tissue change (increased stiffness) due to the presence of cancer localized to the tongue base, various sections of the 3D geometry are modified to exhibit different elastic properties. Three cases are considered, representing the normal tongue, a tongue with early-stage cancer, and tongue with late-stage cancer. Early- and late-stage cancers are differentiated by the degree of stiffness within the base of tongue tissue. Analysis of the model suggests that healthy tongue has a maximum deformation of 9.38 mm, whereas tongues having mild cancer and severe cancer have a maximum deformation of 8.65 and 6.17 mm, respectively. Biomechanical modelling is a useful tool to explain and estimate swallowing abnormalities associated with tongue cancer and post-treatment characteristics.     

An analysis of a mathematical model describing acid‐mediated tumor invasion

We present a mathematical analysis of a reaction‐diffusion model describing acid‐mediated tumor invasion. The model describes the spatial distribution and temporal evolution of tumor cells, normal cells, and excess lactic acid concentration. The model assumes that tumor‐induced alteration of microenvironmental pH provides a simple but complete mechanism for cancer invasion. We provide results on the existence and uniqueness of a solution considering Neumann boundary condition and comments about the same results with Dirichlet boundary conditions. We also provide numerical simulations to the solutions considering both boundary conditions.     

考虑大数据营销和风险规避的绿色供应链决策与协调

通过建立考虑大数据营销及零售商风险规避的博弈模型,对绿色供应链定价、产品绿色度及利润进行比较分析。研究发现：无论集中决策、双方风险中性分散决策还是仅零售商风险规避分散决策,考虑大数据营销时的供应链整体期望利润和产品绿色度较高,且大数据营销效率因子对产品绿色度的增加有正向作用;双方风险中性分散决策下,一定条件下,两部定价契约能够有效协调供应链整体利润,实现帕累托改进;仅零售商风险规避分散决策下,零售商的风险规避行为会降低其对大数据营销的投入,一定条件下,两部定价契约也能够实现供应链整体期望利润的帕累托改进。     

Evolutionary dynamics of cancer:From epigenetic regulation to cell population dynamics——mathematical model framework,applications,and open problems

Predictive modeling of the evolutionary dynamics of cancer is a challenging issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer, including plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulation of stem cell regeneration is essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a G0 phase cell cycle model, which leads to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity;connects the heterogeneity with cell-to-cell variance in cellular behaviors(for example, proliferation, apoptosis, and differentiation/senescence);and can be extended to include gene mutation-induced tumor development. Hybrid computational models are developed based on the mathematical model framework and are applied to the processes of inflammationinduced tumorigenesis and tumor relapse after chimeric antigen receptor(CAR)-T cell therapy. Finally, we propose several mathematical problems related to the proposed differential-integral equation. Solutions to these problems are crucial for understanding the evolutionary dynamics of cancer.     

Turing and Hopf Bifurcation in a Diffusive Tumor-immune Model

In order to understand the effect of the diffusion reaction on the interaction between tumor cells and immune cells, we establish a tumor-immune reaction diffusion model with homogeneous Neumann boundary conditions. Firstly, we investigate the existence condition and the stability condition of the coexistence equilibrium solution. Secondly, we obtain the sufficient and necessary conditions for the occurrence of Turing bifurcation and Hopf bifurcation. Thirdly, we perform some numerical simulations to illustrate the complex spatiotemporal patterns near the bifurcation curves. Finally, we explain spatiotemporal patterns in the diffusion action of tumor cells and immune cells.     

Stability analysis of mathematical models for nonlinear growth kinetics of breast cancer stem cells

Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to help in designing experiments to develop novel therapeutic strategies against cancer. In this paper, by using theory of functional and ordinary differential equations, we study the existence and stability of nonlinear growth kinetics of breast cancer stem cells. First, we provide a sufficient condition for the existence and uniqueness of the solution for nonlinear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a nontrivial steady‐state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions that allow for this criteria to be satisfied. Next, we apply these theorems to a special case of the system of functional differential equations that has been used to model nonlinear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Copyright © 2017 John Wiley & Sons, Ltd.     

From a cellular automaton model of tumor–immune interactions to its macroscopic dynamical equation: A drift–diffusion data analysis approach

Several models of tumor growth have been developed from various perspectives and for multiple scales. Due to the complexity of interactions, how the macroscopic dynamics formed by such interactions at the microscopic level is a difficult problem. In this paper, we focus on reconstructing a model from the output of an experimental model. This is carried out by the data analysis approach. We simulate the growth process of tumor with immune competition by using cellular automata technique adapted from previous studies. We employ an analysis of data given by the simulation output to derive an evolution equation of macroscopic dynamics of tumor growth. In a numerical example we show that the dynamics of tumor at stationary state can be described by an Ornstein–Uhlenbeck process. We show further how the result can be linked to the stochastic Gompertz model.     

General phase transition models for vehicular traffic with point constraints on the flow

In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one‐lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free‐flow phase and by a system of 2 conservation laws in the congested phase. The free‐flow phase is described by a one‐dimensional fundamental diagram corresponding to a Newell‐Daganzo type flux. The congestion phase is described by a two‐dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced.     

Optimal Timing to Initiate Medical Treatment for a Disease Evolving as a Semi-Markov Process

In this paper, we consider the problem of the optimal timing to initiate a medical treatment. In the absence of treatment, we model the disease evolution as a semi-Markov process. The optimal time to initiate the treatment is a stopping time, which maximizes the total expected reward for the patient. We propose a stochastic dynamic programming formulation to find this stopping time. Under some plausible conditions, we show that the maximum total expected reward at the start of a health state will be smaller when the patient is in a more severe state. We then prove that the optimal policy for initializing the treatment is determined by a time threshold for each given health state. That is, in each health state, the treatment should be planned to start, when the patient’s duration time in the health state reaches (or exceeds, in the case of a late observation of the patient’s health status) a certain threshold level. We also present numerical examples to illustrate our model and to provide managerial insights.     

A MODEL OF CHINOOK SALMON POPULATION DYNAMICS INCORPORATING SIZE‐SELECTIVE EXPLOITATION AND INHERITANCE OF POLYGENIC CORRELATED TRAITS

Abstract Concern regarding the potential for selective fisheries to degrade desirable characteristics of exploited fish populations is growing worldwide. Although the occurrence of fishery‐induced evolution in a wild population has not been irrefutably documented, considerable theoretical and empirical evidence for that possibility exists. Environmental conditions influence survival and growth in many species and may mask comparatively subtle trends induced by selective exploitation, especially given the evolutionarily short time series of data available from many fisheries. Modeling may be the most efficient investigative tool under such conditions. Motivated by public concern that large‐mesh gillnet fisheries may be altering Chinook salmon in western Alaska, we constructed a stochastic model of the population dynamics of Chinook salmon. The model contained several individually based components and incorporated size‐selective exploitation, assortative mating, size‐dependent female fecundity, density‐dependent survival, and the heritability of size and age. Substantial reductions in mean size and age were observed under all scenarios. Concurrently reducing directional selection and increasing spawning abundance was most effective in stimulating population recovery. Use of this model has potential to improve our ability to investigate the consequences of selective exploitation and aid development of improved management strategies to more effectively sustain fish and fisheries into the future.     

Inventory competition for newsvendors under the objective of profit satisficing

Inventory competition for newsvendors (NVs) has been studied extensively under the objective of expected profit maximization which is based on risk neutrality. In this paper, we study this classic problem under the objective of profit satisficing which is based on downside-risk aversion. Consistent with prior literature, we consider two possible scenarios. In the first scenario, each NV’s demand depends on the stocking levels of all NVs other than herself. In this scenario, we show that there is a unique Nash equilibrium where all NVs optimally order as if they were independent. In the second scenario, each NV’s demand depends on the stocking levels of all NVs including herself. We prove the existence of Nash equilibrium for both additive and multiplicative forms of demands. As a special case, we also study symmetrical NVs under the proportional allocation model. We show that at equilibrium, if the number of NVs exceeds a threshold, the market becomes highly competitive.     

The eye and the fist: Optimizing search and interdiction

Interdiction operations involving search, identification, and interception of suspected objects are of great interest and high operational importance to military and naval forces as well as nation’s coast guards and border patrols. The interdiction scenario discussed in this paper includes an area of interest with multiple neutral and hostile objects moving through this area, and an interdiction force, consisting of an airborne sensor and an intercepting surface vessel or ground vehicle, whose objectives are to search, identify, track, and intercept hostile objects within a given time frame. The main contributions of this paper are addressing both airborne sensor and surface vessel simultaneously, developing a stochastic dynamic-programming model for optimizing their employment, and deriving operational insight. In addition, the search and identification process of the airborne sensor addresses both physical (appearance) and behavioral (movement pattern) signatures of a potentially hostile object. As the model is computationally intractable for real-world scenarios, we propose a simple heuristic policy, which is shown, using a bounding technique, to be quite effective. Based on a numerical case study of maritime interdiction operations, which includes several representative scenarios, we show that the expected number of intercepted hostile objects, following the heuristic decision policy, is at least 60% of the number of hostile objects intercepted following an optimal decision policy.     

Application of collocation method for solving a parabolic‐hyperbolic free boundary problem which models the growth of tumor with drug application

In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first‐order hyperbolic equations are given that describe the evolution of cells and other two second‐order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd.     

Evaluation and default time for companies with uncertain cash flows

In this study, we propose a modelling framework for evaluating companies financed by random liabilities, such as insurance companies or commercial banks. In this approach, earnings and costs are driven by double exponential jump–diffusion processes and bankruptcy is declared when the income falls below a default threshold, which is proportional to the charges. A change of numeraire, under the Esscher risk neutral measure, is used to reduce the dimension. A closed form expression for the value of equity is obtained in terms of the expected present value operators, with and without disinvestment delay. In both cases, we determine the default threshold that maximizes the shareholder’s equity. Subsequently, the probabilities of default are obtained by inverting the Laplace transform of the bankruptcy time. In numerical applications of the proposed model, we apply a procedure for calibration based on market and accounting data to explain the behaviour of shares for two real-world examples of insurance companies.     

A ruin model with random income and dependence between claim sizes and claim intervals

In this paper,we consider a generalization of the classical ruin model,where the income is random and the distribution of the time between two claim occurrences depends on the previous claim size.This model is more appropriate than the classical ruin model.Explicit expression for the generating function of the Gerber-Shiu expected discounted penalty function are derived.A similar model is discussed.Finally,the result are showed by two examples.