Abstract
In the case of some specific cancers, immunotherapy is one of the possible treatments that can be considered. Our study is based on a mathematical model of patient-specific immunotherapy proposed in Kronik et al. (PLoS One 5(12):e15,482, 2010). This model was validated for clinical trials presented in Michael et al. (Clin Cancer Res 11(12):4469–4478, 2005). It consists of seven ordinary differential equations and its asymptotic dynamics can be described by some t-periodic one-dimensional dynamical system. In this paper we propose a generalised version of this t-periodic system and study the dynamics of the proposed model. We show that there are three possible types of the model behaviour: the solution either converges to zero, or diverges to infinity, or it is periodic. Moreover, the periodic solution is unique, and it divides the phase space into two sub-regions. The general results are applied to the PC specific case, which allow to derive conditions guaranteeing successful as well as unsuccessful treatment. The results indicate that a single vaccination is not sufficient to cure the cancer.
Highlights
Over the last few decades, cancer immunotherapy has become a growing area of active research, with a wide variety of approaches developed and clinically implemented in different cancer indications
The suggested treatments encounter significant problems when translated into clinic, eventually showing limited efficacy and low response rate, even though sporadic cases of remarkable effectiveness are observed in individual patients. The latter observation raises the possibility that effective design of immune intervention is undermined by our limited understanding of the complicated dynamics of interactions between the two major players, the immune system and the cancer, and how it varies from patient to patient (Agur and Vuk Pavlovic’ 2012)
The model proposed in Kronik et al (2010) and analysed in the present work, describes the interaction between advanced prostate cancer (PCa) and immune system under vaccination treatment
Summary
Over the last few decades, cancer immunotherapy has become a growing area of active research, with a wide variety of approaches developed and clinically implemented in different cancer indications. If the model’s parameters are retrieved from experiments, or by matching model output versus real clinical data, one can obtain a general insight into the systems’ dynamics, but rather predict the quantitative response of the disease to the treatment This allows designing and testing various alternative treatment strategies that would be most beneficial, on a population or individual level. It was demonstrated that individual parameters can be identified at early stages of the treatment, and the model can be used to predict further dynamics of response and eventually optimise the individual treatment schedule, given a welldefined end-point; e.g., stabilisation of PSA levels (Kogan et al 2012) Such a model may be relevant to a clinical application of immunotherapy, with a potential to dramatically change the approach to treatment design. In that case giving the treatment periodically, we asymptotically obtain a t-periodic right-hand side of the equation, and the general theorem from Sect. 2 applies
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