Chaos in a Tumor Growth Model with Delayed Responses of the Immune System

A simple prey-predator-type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El-Gohary (2008) in that it allows delay effects in the growth process of the hunting cells. Qualitative and numerical analyses for the stability of equilibriums of the model are presented. Length of the time delay that preserves stability is given. It is found that small delays guarantee stability at the equilibrium level (stable focus) but the delays greater than a critical value may produce periodic solutions through Hopf bifurcation and larger delays may even lead to chaotic attractors. Implications of these results are discussed

Cancer therapeutic potential of combinatorial immuno- and vaso-modulatory interventions

Currently, most of the basic mechanisms governing tumor-immune system interactions, in combination with modulations of tumor-associated vasculature, are far from being completely understood. Here, we propose a mathematical model of vascularized tumor growth, where the main novelty is the modeling of the interplay between functional tumor vasculature and effector cell recruitment dynamics. Parameters are calibrated on the basis of different in vivo immunocompromised Rag1-/- and wild-type (WT) BALB/c murine tumor growth experiments. The model analysis supports that tumor vasculature normalization can be a plausible and effective strategy to treat cancer when combined with appropriate immuno-stimulations. We find that improved levels of functional tumor vasculature, potentially mediated by normalization or stress alleviation strategies, can provide beneficial outcomes in terms of tumor burden reduction and growth control. Normalization of tumor blood vessels opens a therapeutic window of opportunity to augment the antitumor immune responses, as well as to reduce the intratumoral immunosuppression and induced-hypoxia due to vascular abnormalities. The potential success of normalizing tumor-associated vasculature closely depends on the effector cell recruitment dynamics and tumor sizes. Furthermore, an arbitrary increase of initial effector cell concentration does not necessarily imply a better tumor control. We evidence the existence of an optimal concentration range of effector cells for tumor shrinkage. Based on these findings, we suggest a theory-driven therapeutic proposal that optimally combines immuno- and vaso-modulatory interventions

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