Fractional model of cancer immunotherapy and its optimal control

Cancer is one of the most serious illnesses in all of the world. Although most of the cancer patients are treated with chemotherapy, radiotherapy and surgery, wide research is conducted related to experimental and theoretical immunology. In recent years, the research on cancer immunotherapy has led to major medical advances. Cancer immunotherapy refers to the stimulation of immune system to deal with cancer cells. In medical practice, it is mainly achieved by using effector cells such as activated T-cells and Interleukin-2 (IL-2), which is the main cytokine responsible for lymphocyte activation, growth and differentiation. A well-known mathematical model, named as Kirschner-Panetta (KP) model, represents richly the dynamics of the interaction between cancer cells, IL-2 and the effector cells. The dynamics of the KP model is described and the solution to which is approximated by using polynomial approximation based methods such as Adomian decomposition method and differential transform method. The rich nonlinearity of the KP model causes these approaches to become so complicated in order to deal with the representation of polynomial approximations. It is illustrated that the approximated polynomials are in good agreement with the solution obtained by common numerical approaches. In the KP model, the growth of the tumour cells can be expressed by a linear function or any limited-growth function such as logistic equation, in which the cancer population possesses an upper bound mentioned as carrying capacity. Effector cells and IL-2 construct two external sources of medical treatment to stimulate immune system to eradicate cancer cells. Since the main goal in immunotherapy is to remove the tumour cells with the least probable medication side effects, an advanced version of the model may include a time dependent external sources of medical treatment, meaning that the external sources of medical treatment could be considered as control functions of time and therefore the optimum use of medical sources can be evaluated in order to achieve the optimal measure of an objective function. With this sense of direction, two distinct strategies are explored. The first one is to only consider the external source of effector cells as the control function to formulate an optimal control problem. It is shown under which circumstances, the tumour is eliminated. The approach in the formulation of the optimal control is the Pontryagin maximum principal. Furthermore the optimal control problem will be dealt with using particle swarm optimization (PSO). It is shown that the obtained results are significantly better than those obtained by previous researchers. The second strategy is to formulate an optimal control problem by considering both the two external sources as the controls. To our knowledge, it is the first time to present a multiple therapeutic protocol for the KP model. Some MATLAB routines are develop to solve the optimal control problems based on Pontryagin maximum principal and also the PSO. As known, fractional differential equations are more appropriate to describe the persistent memory of physical phenomena. Thus, the fractional KP model is defined in the sense of Caputo differentiation operator. An effective method for numerical treatment of the model is described, namely Predictor-Corrector method of Adams-Bashforth-Moulton type. A robust MATLAB routine is coded based on the mentioned approach and the solution obtained will be compared with those of the classical KP model. The code is prepared in such a way to be able to deal with systems of fractional differential equations, in which each equation has its own fractional order (i.e. multi-order systems of fractional differential equations). The theorems for existence of solutions and the stability analysis of the fractional KP model are represented. In this regard, a frequently used method of solving fractional differential equations (FDEs) is described in details, namely multi-step generalized differential transform method (MSGDTM), then it is illustrated that the method neglects the persistent memory property and takes the incorrect approach in dealing with numerical solutions of FDEs and therefore it is unfit to be used in differential equations governed by fractional differentiation operators. The sigmoidal behavior of the solution to the logistic equation caused it to be one of the most versatile models in natural sciences and therefore the fractional logistic equation would be a relevant problem to be dealt with. Thus, a power series of Mittag-Leffer functions is introduced, the behaviour of which is in good agreement with the solution to fractional logistic equation (FLE), and then a fractional integro-differential equation is represented and proved to be satisfied with the power series of Mittag-Leffler function. The obtained fractional integro-differential equation is named as modified fractional differential equation (MFDL) and possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in the thesis, may be appropriately applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics. Inverse problems to FDEs occur in many branches of science. Such problems have been investigated, for instance, in fractional diffusion equation and inverse boundary value problem for semi- linear fractional telegraph equation. The determination of the order of fractional differential equations is an issue, which has been analyzed and discussed in, for instance, fractional diffusion equations. Thus, fractional order estimation has been conducted for some classes of linear fractional differential equations, by introducing the relationship between the fractional order and the asymptotic behaviour of the solutions to linear fractional differential equations. Fractional optimal control problems, in which the system and (or) the objective function are described based on fractional derivatives, are much more complicated to be solved by using a robust and reliable numerical approach. Thus, a MATLAB routine is provided to solve the optimal control for fractional KP model and the obtained solutions are compared with those of classical KP model. It is shown that the results for fractional optimal control problems are better than classical optimal control problem in the sense of the amount of drug administration

A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method

In this note, two numerical methods of solving fractional differential equations (FDEs) are briefly described, namely predictor-corrector approach of Adams-Bashforth-Moulton type and multi-step generalized differential transform method (MSGDTM), and then a demonstrating example is given to compare the results of the methods. It is shown that the MSGDTM, which is an enhancement of the generalized differential transform method, neglects the effect of non-local structure of fractional differentiation operators and fails to accurately solve the FDEs over large domains.Comment: 12 pages, 2 figure

Mathematical modeling of blood flow through an eccentric catheterized artery

The mathematical model of blood flow through a catheterized stenosed artery is considered. A catheter is a tube, which is used in medicine for patients who are bedridden and whose blood pressure needs to be measured and monitored continuously. An example is the use of catheter during X-ray angiography or coronary balloon angioplasty in cardiac patients. Inserting a catheter in an artery will alter some characteristics of blood flow. This project investigates the effect on blood flow characteristics such as the velocity, the wall shear stress, the resistance impedance and the streamlines when a catheter is inserted into a stenosed artery. The catheter and the artery are assumed to be in a co-axial and eccentric position while blood is assumed to be Newtonian. The governing Navier-Stokes equations are solved analytically using perturbation method. The results show that a catheter placed in an eccentric position does alter the blood flow characteristics such that the axial velocity and the wall shear stress distribution are higher while the resistance impedance values are lower compared to their values in an artery where the catheter is concentrically placed. It is also found that under the same situation, the position of trapping moves closer to the wall of the stenosis while the size of the trapped bolus increases

Determination of order in linear fractional differential equations

The order of fractional differential equations (FDEs) has been proved to be of great importance in an accurate simulation of the system under study. In this paper, the orders of some classes of linear FDEs are determined by using the asymptotic behaviour of their solutions. Specifically, it is demonstrated that the decay rate of the solutions is influenced by the order of fractional derivatives. Numerical investigations are conducted into the proven formulas

Modified Fractional Logistic Equation

In the article [B.J.West, Exact solution to fractional logistic equation, Physica A: Statistical Mechanics and its Applications 429 (2015) 103-108], the author has obtained a function as the solution to fractional logistic equation (FLE). As demonstrated later in [I. Area, J. Losada, J. J. Nieto, A note on the fractional logistic equation, Physica A: Statistical Mechanics and its Applications 444 (2016) 182-187], this function (West function) is not the solution to FLE, but nevertheless as shown by West, it is in good agreement with the numerical solution to FLE. The West function indicates a compelling feature, in which the exponentials are substituted by Mittag-Leffler functions. In this paper, a modified fractional logistic equation (MFLE) is introduced, to which the West function is a solution. The proposed fractional integro-differential equation possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in this article, may be applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics.Comment: 16 pages, 3 figure

How much BiGAN and CycleGAN-learned hidden features are effective for COVID-19 detection from CT images? A comparative study

Bidirectional generative adversarial networks (BiGANs) and cycle generative adversarial networks (CycleGANs) are two emerging machine learning models that, up to now, have been used as generative models, i.e., to generate output data sampled from a target probability distribution. However, these models are also equipped with encoding modules, which, after weakly supervised training, could be, in principle, exploited for the extraction of hidden features from the input data. At the present time, how these extracted features could be effectively exploited for classification tasks is still an unexplored field. Hence, motivated by this consideration, in this paper, we develop and numerically test the performance of a novel inference engine that relies on the exploitation of BiGAN and CycleGAN-learned hidden features for the detection of COVID-19 disease from other lung diseases in computer tomography (CT) scans. In this respect, the main contributions of the paper are twofold. First, we develop a kernel density estimation (KDE)-based inference method, which, in the training phase, leverages the hidden features extracted by BiGANs and CycleGANs for estimating the (a priori unknown) probability density function (PDF) of the CT scans of COVID-19 patients and, then, in the inference phase, uses it as a target COVID-PDF for the detection of COVID diseases. As a second major contribution, we numerically evaluate and compare the classification accuracies of the implemented BiGAN and CycleGAN models against the ones of some state-of-the-art methods, which rely on the unsupervised training of convolutional autoencoders (CAEs) for attaining feature extraction. The performance comparisons are carried out by considering a spectrum of different training loss functions and distance metrics. The obtained classification accuracies of the proposed CycleGAN-based (resp., BiGAN-based) models outperform the corresponding ones of the considered benchmark CAE-based models of about 16% (resp., 14%)

Metaheuristics and Pontryagin's minimum principle for optimal therapeutic protocols in cancer immunotherapy: a case study and methods comparison

In this paper, the performance appropriateness of population-based metaheuristics for immunotherapy protocols is investigated on a comparative basis while the goal is to stimulate the immune system to defend against cancer. For this purpose, genetic algorithm and particle swarm optimization are employed and compared with modern method of Pontryagin's minimum principle (PMP). To this end, a well-known mathematical model of cell-based cancer immunotherapy is described and examined to formulate the optimal control problem in which the objective is the annihilation of tumour cells by using the minimum amount of cultured immune cells. In this regard, the main aims are: (i) to introduce a single-objective optimization problem and to design the considered metaheuristics in order to appropriately deal with it; (ii) to use the PMP in order to obtain the necessary conditions for optimality, i.e. the governing boundary value problem; (iii) to measure the results obtained by using the proposed metaheuristics against those results obtained by using an indirect approach called forward-backward sweep method; and finally (iv) to produce a set of optimal treatment strategies by formulating the problem in a bi-objective form and demonstrating its advantages over single-objective optimization problem. A set of obtained results conforms the performance capabilities of the considered metaheuristics