Model for tumour growth with treatment by continuous and pulsed chemotherapy

Peer reviewedPreprin

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

Μελέτη σχέσης κυτταρικής επικοινωνίας και ανοσίας

Στην παρούσα διπλωματική εργασία αναπτύσσονται κυτταρικά μοντέλα προσομοίωσης, με σκοπό την μελέτη βιολογικών φαινομένων όπως η κυτταρική επικοινωνία (μέσω ανταλλαγής χημικών ουσιών), η επίτευξη ανοσίας σε ένα σύστημα που εμπεριέχει κύτταρα με ανοσοποιητική δράση, η εμφάνιση ανοσολογικής απόκρισης και η συμπεριφορά της έναντι διαφορετικών νοσογόνων παραγόντων όπως τοξικές ουσίες, ενδοκυττάρια παράσιτα και εξωκυττάρια παράσιτα. Εξετάζεται επίσης η εμφάνιση πιο πολύπλοκων συμπεριφορών στο σύστημα όπως η ανοσολογική μνήμη και η αυτο-άνοση αντίδραση. Η σχεδίαση όλων των μοντέλων έγινε με βάση τις αρχές των εξατομικευμένων μοντέλων προσομοίωσης (Agent Based Models) και η γλώσσα προγραμματισμού που χρησιμοποιήθηκε για την υλοποίηση τους ήταν η JAVA.In the context of this thesis, cellular simulation models are developed, in order to study biological phenomena such as cellular communication (through exchange of chemical substances between cells), the occurrence of immunity inside a system that contains immune cells, the emergence of an immunological response and its behavior against various disease agents such as toxic substances, intracellular parasites and extracellular parasites. In addition, the emergence of more complex system behaviors is studied, like immunological memory and an autoimmune response. The creation of all the models was based on the principles of Agent Based Model design and the programming language that was used for the implementation of the models was JAVA

Using Fisher information approach in nonlinear dynamical systems

The aim of this thesis is to investigate nonlinear dynamical systems that exist in various fields such as engineering and science. Nonlinear dynamical systems permit the understanding and development of models of simple and complex phenomena. Specifically, this thesis includes an investigation of the following systems; the logistic model, the Gompertz model, predator-prey model, and three species model. In addition, we perform a comparison between the two most popular growth models; logistic and Gompertz models from the viewpoint of variability. The main focus is on the use of Fisher information as a measure of variability/sustainability which depends on the gradient of Probability Density Function (PDF). In this work, we present two case studies for each dynamical system. The first case study describes the analysis of these systems in their deterministic conditions whereas the second one presents the investigation of these systems in their indeterministic conditions (perturbed conditions), where the model parameters involve perturbations, elucidating the effects of these perturbations on the behaviour of the system. The variation in the model parameter values is considered in order to observe the behaviour of the different dynamical systems and detect dynamical changes in the behaviour of each species. Since Fisher information is considered as a measure of an intrinsic accuracy of the dynamical systems, therefore, we obtain Fisher information for different values of parameters in order to select the optimal parameter value where a peak of Fisher information is observed, which indicates to less variability in the behaviour of the system. Thus, the existence of Fisher information peak which linked to the narrowest PDF is investigated in the frame of time trace analysis. In summary, the main contribution of this work to the field is to assess the significance of Fisher information index for nonlinear dynamical systems including perturbations in the model parameters. Applying this measure to more complicated systems and comparing the results to other widely used measures would be of interest for future work