Search

We study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the effect of optimal chemotherapy doses on solid tumor during treatment and cancer recurrence after the end of treatment. First, an optimal control problem is designed. Then, by changing the control range, the objective function, and adding a control to the previous problem, three new optimal control problems are constructed. We pursue two critical ideas for the success of the drug delivery system to the patient: minimizing the normalized global population density of cancer cells during treatment and at the end of treatment and minimizing side effects during treatment. The optimal control problem is considered with two linear and quadratic objective functions. These models are solved by a well‐established numerical method with high accuracy and a low discretization cost. The numerical results have confirmed that reducing the total dose of the cytotoxic drugs during the treatment period leads to a rapid recurrence of cancer shortly after the end of the treatment period. [ABSTRACT FROM AUTHOR]

In this paper, a free boundary model of tumor growth with drug application including the heterogeneity or different types of tumor cells (caused by mutations and different values of drug and nutrient concentrations inside the tumor) is studied. Heterogeneity is included in the model by a variable −1≤y≤1$$ -1\le y\le 1 $$. It is assumed that converting from mutation y1$$ {y}_1 $$ to mutation y2$$ {y}_2 $$ happens with probability P(y1,y2)$$ P\left({y}_1,{y}_2\right) $$. A Caputo time fractional‐order hyperbolic equation describes the evolution of tumor cells depending on y$$ y $$. It also includes two Caputo time fractional‐order parabolic equations describing the diffusions of nutrients (e.g., oxygen and glucose) and drug concentrations. Instead of integer‐order time derivatives, the fractional ones are considered. In this study, it is aimed to employ the least squares curve fitting method to fit the order of fractional derivatives, coefficient, and rates of the model. For this purpose, using the mathematical model, we have considered the boundary densities of the tumor cells of different types and near‐boundary concentrations of drug and nutrient as the functions of the unknown orders, coefficients and rates (unknown variables). Due to the complexity of the problem, we have obtained the functions numerically. For this, using a change of variable, we have changed the free boundary problem to a problem with fixed domain. Thus, Riemann–Liouville fractional‐order integrals are added to the problem. In the spatial domain, the problem is discretized using the collocation method. In the temporal domain, the fractional derivatives and integrals (with order α$$ \alpha $$) are approximated in mesh points (with step size t∗$$ {t}^{\ast } $$) applying a method with error O((t∗)2−α)$$ O\left({\left({t}^{\ast}\right)}^{2-\alpha}\right) $$. Then, the unknown variables are obtained by fitting the functions for unknown variables to the data. In order to obtain the variables, a quadratic objective function is considered. The discretized function is substituted into the objective function, then the objective function is minimized using the trust‐region reflective algorithm. Finally, some numerical examples are presented to verify the efficiency of the method. In the examples, we have added noises generated from Gaussian distributions to the data and the effects of noise on the fitted coefficients, and rates are illustrated using some figures and tables. The noisy data are also plotted to have a clear vision of the effects of noises on the data. It is shown that despite the noise, the prediction of the radius of the tumor is acceptable. [ABSTRACT FROM AUTHOR]

In this paper, a game of competition between two populations is designed. Populations are considered as two players competing against each other to increase their sizes. Players have been provided with costly two-component strategies. Population densities are described by two coupled nonlinear stochastic second-order parabolic equations. We have added stochastic terms to the deterministic model to consider random perturbations. First, we have considered the stochastic payoff functions and proved the game has a unique stochastic Nash equilibrium. Then, we have shown the stochastic Nash equilibrium is also a Nash equilibrium for the game with deterministic payoff functions defined by expectation. Cooperative game is also studied and stochastic Pareto efficient strategies are obtained. Then, the existence and uniqueness analyses of the deterministic Nash equilibrium and deterministic Pareto efficient strategies for the game with deterministic payoff functions are provided. The explicit forms of Nash equilibriums and Pareto efficient strategies are presented. Results about the losses the players experience when they don't cooperate are given. Finally, some numerical experiments are included to illustrate what we have claimed in the statements of theorems and show the effects of the Nash equilibrium and Pareto efficient strategy on the payoffs of the players. [ABSTRACT FROM AUTHOR]

In this investigation, an optimal control problem for a stochastic mathematical model of population competition is studied. We have considered the stochastic model of population competition by adding the stochastic terms to the deterministic model to take into account the random perturbations and uncertainties caused by the environment to have more reliable model. The model has formulated the population densities competing against each other to be saved from destruction, attract more members and so on, using two nonlinear stochastic parabolic equations. Four factors including the status and the growth rates of the populations are considered as the control variables (which can be controlled by the members of the populations or policy makers who make decisions for the populations for particular purposes) to control the evolution of the population densities. Then, the optimal control problem for the stochastic model of population competition is studied. Employing the tangent-normal cone techniques, the Ekeland variational principle and other theorems proved throughout the paper, we have shown there exists unique stochastic optimal control. We have also presented the exact form of the optimal control in terms of stochastic adjoint states. [ABSTRACT FROM AUTHOR]

In this study, a combination of spectral and fixed point methods is used to solve an optimal control problem for a model of tumour growth. The growth of tumour is modelled using three first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the optimal control problem, four control variables are employed to control the concentration of nutrient and drug on the boundary and inside the tumour. Since the problem is nonlinear, applying the fixed point method, in each step of iteration, the problem is changed to a linear one and the parabolic equations are solved using the spectral method. The convergence and stability of method are proven. Some examples are considered to illustrate the efficiency of method. Finally, some figures are provided to reflect the effects of control on the densities of tumourcells. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]