Mathematical modeling of cancer treatments and the role of the immune system response to tumor invasion.

Doctor of Philosophy in Applied Mathematics.Despite the success of traditional cancer treatments, a definite cure to several cancers does not exist. Further, the traditional cancer treatments are highly toxic and have a relatively low efficacy. Current research and clinical trials have indicated that virotherapy, a procedure which uses replication-competent viruses to kill cancer cells, is less toxic and highly effective. Some recent studies suggest that the success of combating cancer lies in the understanding of tumour-immune interactions. However, the interaction dynamics of recent cancer treatments with the tumour and immune system response are still poorly understood. In this thesis we construct and analyse mathematical models in the form of ordinary and partial differential equations in order to explain tumour invasion dynamics and new forms of cancer treatment. We use these models to suggest possible measures needed in order to combat cancer. The thesis seeks to determine the most critical biological factors during tumour invasion, describe how the virus and immune system response influences the outcome of oncolytic virotherapy treatment, investigate how drug infusion methods determine the success of chemotherapy and virotherapy, and determine the efficacy of chemotherapy and virotherpy in depleting tumour cells from body tissue. We present a travelling wave analysis of a tumour-immune interaction model with immunotherapy. Here we aim to investigate the existence of travelling wave solutions of the model equations with and without immunotherapy and calculate the minimum wave speed with which tumour cells invade healthy tissue. This investigation highlights the properties which are most vital during tumour invasion. We use the geometric treatment of an apt-phase space to establish the intersection between stable and unstable manifolds. Numerical simulations are performed to support the analytical results. The analysis reveals that the main factors involved during tumour invasion include the tumour growth rate, resting immune cell growth rate, carrying capacity of the resting immune cells, resting cell supply, diffusion rate of the tumour cells, and the local kinetic interaction parameters. We also present a mathematical analysis of models that study tumour-immune-virus interactions using differential equations with spatial effects. The major aim is to investigate how virus and immune responses influence the outcome of oncolytic treatment. Stability analysis is carried out to determine the long term behaviour of the model solutions. Analytical traveling wave solutions are obtained using factorization of differential operators and numerical simulations are carried out using Runge-Kutta fourth order method and Crank-Nicholson methods. Our results show that the use of viruses as a means of cancer treatment can reduce the tumour cell concentration to a very low cancer dormant state or possibly eradicate all tumour cells in body tissue. The traveling wave solutions indicate an exponential increase and decrease in the immune cells density and tumour load in the long term respectively. A mathematical model of chemovirotherapy, a recent experimental treatment which combines virotherapy and chemotherapy, is constructed and analyzed. The aim is to compare the efficacy of three drug infusion methods and predict the outcome of oncolytic virotherapy-drug combination. A comparison of the efficacy of using each treatment individually, that is, chemotherapy and virotherapy, is presented. Analytical solutions of the model are obtained where possible and stability analysis is presented. Numerical solutions are obtained using the Runge- Kutta fourth order method. This study shows that chemovirotherapy may have a higher chance of reducing the tumour cell density in body tissue in a relatively short time. To the best of our knowledge, there has not been a mathematical study on the combination of both chemotherapy and virotherapy. Lastly, the chemovirotherapy model is extended to include spatial distribution characteristics, thus developing a model which describes avascular tumour growth under chemovirotherapy in a two dimensional spatial domain. Numerical investigation of the model solutions is carried out using a multi domain monomial based collocation method and pdepe, a finite element based method in Matlab. This study affirmed that chemovirotherapy may possibly eradicate all tumour cells in body tissue

Mathematical analysis of a tumour-immune interaction model : a moving boundary problem

A spatio-temporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is developed. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour swells to a dormant level.https://www.elsevier.com/locate/mbs2020-02-01hj2019Mathematics and Applied Mathematic

Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis

Oncolytic virotherapy (OV) has been emerging as a promising novel cancer treatment that may be further combined with the existing therapeutic modalities to enhance their effects. To investigate how OV could enhance chemotherapy, we propose an ODE based model describing the interactions between tumour cells, the immune response, and a treatment combination with chemotherapy and oncolytic viruses. Stability analysis of the model with constant chemotherapy treatment rates shows that without any form of treatment, a tumour would grow to its maximum size. It also demonstrates that chemotherapy alone is capable of clearing tumour cells provided that the drug efficacy is greater than the intrinsic tumour growth rate. Furthermore, OV alone may not be able to clear tumour cells from body tissue but would rather enhance chemotherapy if viruses with high viral potency are used. To assess the combined effect of OV and chemotherapy we use the forward sensitivity index to perform a sensitivity analysis, with respect to chemotherapy key parameters, of the virus basic reproductive number and the tumour endemic equilibrium. The results from this sensitivity analysis indicate the existence of a critical dose of chemotherapy above which no further significant reduction in the tumour population can be observed. Numerical simulations show that a successful combinational therapy of the chemotherapeutic drugs and viruses depends mostly on the virus burst size, infection rate, and the amount of drugs supplied. Optimal control analysis was performed, by means of Pontryagin's principle, to further refine predictions of the model with constant treatment rates by accounting for the treatment costs and sides effects.Comment: This is a preprint of a paper whose final and definite form is with 'Mathematical Biosciences and Engineering', ISSN 1551-0018 (print), ISSN 1547-1063 (online), available at [http://www.aimsciences.org/journal/1551-0018]. Submitted 27-March-2018; revised 04-July-2018; accepted for publication 10-July-201

Modelling the spatiotemporal dynamics of chemovirotherapy cancer treatment

Chemovirotherapy is a combination therapy with chemotherapy and oncolytic viruses. It is gaining more interest and attracting more attention in the clinical setting due to its effective therapy and potential synergistic interactions against cancer. In this paper, we develop and analyse a mathematical model in the form of parabolic non-linear partial differential equations to investigate the spatiotemporal dynamics of tumour cells under chemovirotherapy treatment. The proposed model consists of uninfected and infected tumour cells, a free virus, and a chemotherapeutic drug. The analysis of the model is carried out for both the temporal and spatiotemporal cases. Travelling wave solutions to the spatiotemporal model are used to determine the minimum wave speed of tumour invasion. A sensitivity analysis is performed on the model parameters to establish the key parameters that promote cancer remission during chemovirotherapy treatment. Model analysis of the temporal model suggests that virus burst size and virus infection rate determine the success of the virotherapy treatment, whereas travelling wave solutions to the spatiotemporal model show that tumour diffusivity and growth rate are critical during chemovirotherapy. Simulation results reveal that chemovirotherapy is more effective and a good alternative to either chemotherapy or virotherapy, which is in agreement with the recent experimental studies.University of Pretoria and DST/NRF SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.tandfonline.com/loi/tjbd20hj2017Mathematics and Applied Mathematic

Mathematical Analysis of a Mathematical Model of Chemovirotherapy: Effect of Drug Infusion Method

A mathematical model for the treatment of cancer using chemovirotherapy is developed with the aim of determining the efficacy of three drug infusion methods: constant, single bolus, and periodic treatments. The model is in the form of ODEs and is further extended into DDEs to account for delays as a result of the infection of tumor cells by the virus and chemotherapeutic drug responses. Analysis of the model is carried out for each of the three drug infusion methods. Analytic solutions are determined where possible and stability analysis of both steady state solutions for the ODEs and DDEs is presented. The results indicate that constant and periodic drug infusion methods are more efficient compared to a single bolus injection. Numerical simulations show that with a large virus burst size, irrespective of the drug infusion method, chemovirotherapy is highly effective compared to either treatments. The simulations further show that both delays increase the period within which a tumor can be cleared from body tissue

Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method

There has been considerable interest in seeking exact solutions of non-linear evolution equations that describe important physical and biological processes. Nonetheless, it is a difficult undertaking to determine closed form solutions of mathematical models that describe natural phenomena. This is because of their high non-linearity and the huge number of parameters of which they consist. In this article we determine, using the hyperbolic tangent (tanh) method, travelling wave solutions to non-linear evolution models of interest in biology and physics. These solutions have recognizable properties expected of other solutions and thus can be used to deduce properties of the general solutions

Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method

There has been considerable interest in seeking exact solutions of non-linear evolution equations that describe important physical and biological processes. Nonetheless, it is a difficult undertaking to determine closed form solutions of mathematical models that describe natural phenomena. This is because of their high non-linearity and the huge number of parameters of which they consist. In this article we determine, using the hyperbolic tangent (tanh) method, travelling wave solutions to non-linear evolution models of interest in biology and physics. These solutions have recognizable properties expected of other solutions and thus can be used to deduce properties of the general solutions

Determining COVID-19 Dynamics Using Physics Informed Neural Networks

The Physics Informed Neural Networks framework is applied to the understanding of the dynamics of COVID-19. To provide the governing system of equations used by the framework, the Susceptible–Infected–Recovered–Death mathematical model is used. This study focused on finding the patterns of the dynamics of the disease which involves predicting the infection rate, recovery rate and death rate; thus, predicting the active infections, total recovered, susceptible and deceased at any required time. The study used data that were collected on the dynamics of COVID-19 from the Kingdom of Eswatini between March 2020 and September 2021. The obtained results could be used for making future forecasts on COVID-19 in Eswatini