Mathematical modelling of tumour growth and interaction with host tissue and the immune system

In this thesis, three mathematical models describing the growth of solid tumour incorporating the host tissue and the immune system response are developed and investigated. The initial model describes the dynamics of the growing tumour and immune response before being extended in the second model by introducing a time-varying dendritic cell-based treatment strategy. Finally, in the third model, we present a mathematical model of a growing tumour using a hybrid cellular automata. These models can provide information to pre-experimental work to assist in designing more effective and efficient laboratory experiments related to tumour growth and interactions with the immune system and immunotherapy.

A linear-elastic model of anisotropic tumour growth

This paper examines the effect of anisotropic growth on the evolution of mechanical stresses in a linear-elastic model of a growing, avascular tumour. This represents an important improvement on previous linear-elastic models of tissue growth since it has been shown recently that spatially-varying isotropic growth of linear-elastic tissues does not afford the necessary stress-relaxation for a steady-state stress distribution upon reaching a nutrient-regulated equilibrium size. Time-dependent numerical solutions are developed using a Lax-Wendroff scheme, which show the evolution of the tissue stress distributions over a period of growth until a steady-state is reached. These results are compared with the steady-state solutions predicted by the model equations, and key parameters influencing these steady-state distributions are identified. Recommendations for further extensions and applications of this model are proposed.

Oxygen transport in a three-dimensional microvascular network incorporated with early tumour growth and preexisting vessel cooption: Numerical simulation study

We propose a dynamic mathematical model of tissue oxygen transport by a preexisting three-dimensional microvascular network which provides nutrients for an in situ cancer at the very early stage of primary microtumour growth. The expanding tumour consumes oxygen during its invasion to the surrounding tissues and cooption of host vessels. The preexisting vessel cooption, remodelling and collapse are modelled by the changes of haemodynamic conditions due to the growing tumour. A detailed computational model of oxygen transport in tumour tissue is developed by considering (a) the time-varying oxygen advection diffusion equation within the microvessel segments, (b) the oxygen flux across the vessel walls, and (c) the oxygen diffusion and consumption with in the tumour and surrounding healthy tissue. The results show the oxygen concentration distribution at different time points of early tumour growth. In addition, the influence of preexisting vessel density on the oxygen transport has been discussed. The proposed model not only provides a quantitative approach for investigating the interactions between tumour growth and oxygen delivery, but also is extendable to model other molecules or chemotherapeutic drug transport in the future study.

3D numerical simulation of avascular tumour growth: effect of hypoxic micro-environment in host tissue

A three-dimensional (3D) mathematical model of tumour growth at the avascular phase and vessel remodelling in host tissues is proposed with emphasis on the study of the interactions of tumour growth and hypoxic micro-environment in host tissues. The hybrid based model includes the continuum part, such as the distributions of oxygen and vascular endothelial growth factors (VEGFs), and the discrete part of tumour cells (TCs) and blood vessel networks. The simulation shows the dynamic process of avascular tumour growth from a few initial cells to an equilibrium state with varied vessel networks. After a phase of rapidly increasing numbers of the TCs, more and more host vessels collapse due to the stress caused by the growing tumour. In addition, the consumption of oxygen expands with the enlarged tumour region. The study also discusses the effects of certain factors on tumour growth, including the density and configuration of preexisting vessel networks and the blood oxygen content. The model enables us to examine the relationship between early tumour growth and hypoxic micro-environment in host tissues, which can be useful for further applications, such as tumour metastasis and the initialization of tumour angiogenesis.

Kallikrein-related peptidase 4 induces tumour microenvironment alterations that support tumour growth

Kallikrein-related peptidase 4 (KLK4) is a protease with elevated production in prostate cancer versus benign tissue. KLK4 expression is associated with prostate cancer risk, and its activity favours tumour progression through increasing cell motility and growth. Importantly, over-production of KLK4 in prostate glandular cells precedes tumour formation, positioning the enzyme to play a role in early remodelling of the tumour microenvironment, a process essential for tumour growth. We sought to identify the proteins and downstream signalling pathways targeted by KLK4 activity, to define its role in tumour microenvironment remodelling and evaluate the efficacy of KLK4 inhibition as a cancer therapy.

Investigation of various growth mechanisms of solid tumour growth within the linear-quadratic model for radiotherapy

The standard linear-quadratic survival model for radiotherapy is used to investigate different schedules of radiation treatment planning to study how these may be affected by different tumour repopulation kinetics between treatments.

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise oil the Gompertzian growth of tumours. Our results are obtained by Solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In Our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under Study. We perform simulations to analyze various aspects, of the probability distribution. of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (tau) to the steady-state distribution as a function of (i) of the correlation strength (lambda) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (alpha). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.

The pyrrolo-1,5-benzoxazepine, PBOX-6, inhibits the growth of breast cancer cells in vitro independent of estrogen receptor status and inhibits breast tumour growth in vivo

Members of a novel series of pyrrolo-1,5-benzoxazepine (PBOX) compounds have been shown to induce apoptosis in a number of human leukemia cell lines of different haematological lineage, suggesting their potential as anti-cancer agents. In this study, we sought to determine if PBOX-6, a well characterised member of the PBOX series of compounds, is also an effective inhibitor of breast cancer growth. Two estrogen receptor (ER)-positive (MCF-7 and T-47-D) and two ER-negative (MDA-MB-231 and SK-BR-3) cell lines were examined. The 3,4,5-dimethylthiazol-2-yl-2,5-diphenyl-tetrazolium bromide (MTT) assay was used to determine reduction in cell viability. PBOX-6 reduced the cell viability of all four cell lines tested, regardless of ER status, with IC(50) values ranging from 1.0 to 2.3 microM. PBOX-6 was most effective in the SK-BR-3 cells, which express high endogenous levels of the HER-2 oncogene. Overexpression of the HER-2 oncogene has been associated with aggressive disease and resistance to chemotherapy. The mechanism of PBOX-6-induced cell death was due to apoptosis, as indicated by the increased proportion of cells in the pre-G1 peak and poly(ADP-ribose) polymerase (PARP) cleavage. Moreover, intratumoural administration of PBOX-6 (7.5 mg/kg) significantly inhibited tumour growth in vivo in a mouse mammary carcinoma model (p=0.04, n=5, Student's t-test). Thus, PBOX-6 could be a promising anti-cancer agent for both hormone-dependent and -independent breast cancers.

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) J. Math. Biol. 45(2), 125-152. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.

Walker 256 tumour growth causes marked changes of glutamine metabolism in rat small intestine

The effect of Walker 256 tumour growth on the metabolism of glucose and glutamine in the small intestine of rats was examined. Walker 256 tumour has been extensively used as an experimental model to induce cancer cachexia in rats. Walker 256 tumour growth decreased body weight and small intestine weight and length. The activities of glucose-6-phosphate dehydrogenase and phosphate-dependent glutaminase were reduced in the proximal, median and distal portions of the intestine. Glutamine oxidation was reduced in the proximal portion only. The decrease in glutaminase activity was not due to a low synthesis of the protein as indicated by Western blotting analysis. Hexokinase and citrate synthase activities were not changed by the tumour. These findings led us to postulate that tumour growth impairs glutamine metabolism of small intestine but the mechanism involved remains to be elucidated. Copyright (C) 2001 John Wiley Sons, Ltd.