Model for tumour growth with treatment by continuous and pulsed chemotherapy

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Oscillatory dynamics in a model of vascular tumour growth -- implications for chemotherapy

Background\ud \ud Investigations of solid tumours suggest that vessel occlusion may occur when increased pressure from the tumour mass is exerted on the vessel walls. Since immature vessels are frequently found in tumours and may be particularly sensitive, such occlusion may impair tumour blood flow and have a negative impact on therapeutic outcome. In order to study the effects that occlusion may have on tumour growth patterns and therapeutic response, in this paper we develop and investigate a continuum model of vascular tumour growth.\ud Results\ud \ud By analysing a spatially uniform submodel, we identify regions of parameter space in which the combination of tumour cell proliferation and vessel occlusion give rise to sustained temporal oscillations in the tumour cell population and in the vessel density. Alternatively, if the vessels are assumed to be less prone to collapse, stable steady state solutions are observed. When spatial effects are considered, the pattern of tumour invasion depends on the dynamics of the spatially uniform submodel. If the submodel predicts a stable steady state, then steady travelling waves are observed in the full model, and the system evolves to the same stable steady state behind the invading front. When the submodel yields oscillatory behaviour, the full model produces periodic travelling waves. The stability of the waves (which can be predicted by approximating the system as one of λ-ω type) dictates whether the waves develop into regular or irregular spatio-temporal oscillations. Simulations of chemotherapy reveal that treatment outcome depends crucially on the underlying tumour growth dynamics. In particular, if the dynamics are oscillatory, then therapeutic efficacy is difficult to assess since the fluctuations in the size of the tumour cell population are enhanced, compared to untreated controls.\ud Conclusions\ud \ud We have developed a mathematical model of vascular tumour growth formulated as a system of partial differential equations (PDEs). Employing a combination of numerical and analytical techniques, we demonstrate how the spatio-temporal dynamics of the untreated tumour may influence its response to chemotherapy.\ud Reviewers\ud \ud This manuscript was reviewed by Professor Zvia Agur and Professor Marek Kimmel

Applications of Mathematical Modelling in Oncolytic Virotherapy and Immunotherapy

Cancer is a devastating disease that touches almost everyone and finding effective treatments presents a highly complex problem, requiring extensive multidisciplinary research. Mathematical modelling can provide insight into both cancer formation and treatment. A range of techniques are developed in this thesis to investigate two promising therapies: oncolytic virotherapy, and combined oncolytic virotherapy and immunotherapy. Oncolytic virotherapy endeavours to eradicate cancer cells by exploiting the aptitude of virus-induced cell death. Building on this premise, combined oncolytic virotherapy and immunotherapy aims to harness and stimulate the immune system's inherent ability to recognise and destroy cancerous cells. Using deterministic and agent-based mathematical modelling, perturbations of treatment characteristics are investigated and optimal treatment protocols are suggested. An integro differential equation with distributed parameters is developed to characterise the function of the E1B genes in an oncolytic adenovirus. Subsequently, by using a bifurcation analysis of a coupled-system of ordinary differential equations for oncolytic virotherapy, regions of bistability are discovered, where increased injections can result in either tumour eradication or tumour stabilisation. Through an extensive hierarchical optimisation to multiple data sets, drawn from in vitro and in vivo modelling, gel-release of a combined oncolytic virotherapy and immunotherapy treatment is optimised. Additionally, using an agent-based modelling approach, delayed-infection of an intratumourally administered virus is shown to be able to reduce tumour burden. This thesis develops new mathematical models that can be applied to a range of cancer therapies and suggests engineered treatment designs that can significantly advance current therapies and improve treatments

Stochastic Hybrid Automata with delayed transitions to model biochemical systems with delays

To study the effects of a delayed immune-response on the growth of an immuno- genic neoplasm we introduce Stochastic Hybrid Automata with delayed transi- tions as a representation of hybrid biochemical systems with delays. These tran- sitions abstractly model unknown dynamics for which a constant duration can be estimated, i.e. a delay. These automata are inspired by standard Stochastic Hybrid Automata, and their semantics is given in terms of Piecewise Determin- istic Markov Processes. The approach is general and can be applied to systems where (i) components at low concentrations are modeled discretely (so to retain their intrinsic stochastic fluctuations), (ii) abundant component, e.g., chemical signals, are well approximated by mean-field equations (so to simulate them efficiently) and (iii) missing components are abstracted with delays. Via sim- ulations we show in our application that interesting delay-induced phenomena arise, whose quantification is possible in this new quantitative framewor

A multiscale model for collagen alignment in wound healing

It is thought that collagen alignment plays a significant part in scar tissue formation during dermal wound healing. We present a multiscale model for collagen deposition and alignment during this process. We consider fibroblasts as discrete units moving within an extracellular matrix of collagen and fibrin modelled as continua. Our model includes flux induced alignment of collagen by fibroblasts, and contact guidance of fibroblasts by collagen fibres. We can use the model to predict the effects of certain manipulations, such as varying fibroblast speed, or placing an aligned piece of tissue in the wound. We also simulate experiments which alter the TGF-β concentrations in a healing dermal wound and use the model to offer an explanation of the observed influence of this growth factor on scarring

Delay model of tumor-immune system interactions with hyperthermia treatment

The interaction of the tumor-immune system was initially based on the immunosurveillance hypothesis that immune cells can identify and kill tumor cells, leading to the use of a prey-predatory model for the description of tumor-immune cell interactions. However, the current biomedical findings reveal a pathway to immunoediting, which hypothesizes the ability of tumors to inhibit, seal, and counteract effector cells. Contrary to the discovery of non-oscillating dynamic biomedicine in solid tumors, existing models show oscillating solutions. Thus, the formulation of an immunoediting model that corresponds to the interaction of the tumor-immune system is sacrosanct in the search for effective malignant tumor treatment. The research suggests an immunoediting delay model of tumor-immune system interactions that combine tumor-immune cytokines derived from tumors to counteract effector cells. Qualitative analysis of this model gives an idea of the conditions for the stability of non-aggressive (benign) tumors and the instability of aggressive (malignant) tumors. The numerical results for these two conditions do not indicate an oscillating solution. Although the elimination of tumors is seen in the case of non-aggressive tumors, the suppression of effector cells and uncontrolled growth of tumors characterize the results for aggressive tumors. To find the best treatment, a sensitivity analysis is performed to ensure the role of the model parameters in the development of the tumor. The analysis reveals the best treatment options to kill tumor cells and strengthen the performance of immune cells. The sensitivity analysis results inform the merger of hyperthermia treatments in the proposed model to investigate the effects of thermal induction on immune cell performance and tumor regression. Discrete-time delays were used to investigate whether hyperthermia treatment was safe for patients who had received other treatments, but no cure occurred. The global stability of hyperthermia treatment is obtained using the Lyapunov function. Furthermore, an optimal heat control strategy for treating malignant tumor hyperthermia is obtained to minimize the effect of heat on normal cells while ensuring the elimination of malignant tumors. This research establishes a unique thermal optimal solution that improves the performance of the effector cell without difficulty