Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

Abstract

We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seen in vivo, including immunostimulation of tumor growth, "sneaking through" of the tumor, and formation of a tumor "dormant state". The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL1 in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measured in vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.

Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy

“…However, these parameter values cannot just be combined. The equilibrium of the model for angiogenic signalling from Hahnfeldt et al (1999) is given by b−μ d 3 2 and is an order of magnitude larger than the carrying capacity for the model in Kuznetsov et al (1994). We therefore adjusted the values of b and d for the high equilibrium point to be in the same range as the carrying capacity for the tumor-immune system model.…”

“…Biologically validated data for the separate models are given for the model of angiogenic signalling in the paper by Hahnfeldt et al (1999) for Lewis lung carcinoma implanted in mice and for the mathematical model for tumor-immune system interactions in the paper by Kuznetsov et al (1994) based on in vivo experimental data for B-lymphoma BC L 1 in the spleen of mice. However, these parameter values cannot just be combined.…”

“…Other models are equally viable and have been considered in the literature. In Stepanova's original research an exponential model was used while Kuznetsov et al (1994) use a logistic model. A Gompertzian model has also been used in the work by de Vladar and González (2004).…”

“…The parameter values for these two mouse models clearly are not compatible, yet the underlying qualitative mathematical structures of the models separately exhibit strong robustness properties. Essentially, while the model for angiogenic signalling is fully robust with respect to parameter variations (Ledzewicz and Schättler 2007;d'Onofrio and Gandolfi 2004;Schättler et al 2011), essential features (such as multi-stability, immunosurveillance/tumor dormancy) are also preserved for the model for tumorimmune system interactions (Kuznetsov et al 1994;Ledzewicz et al 2012aLedzewicz et al , b, 2013d'Onofrio et al 2012;de Vladar and González 2004). We combine these two models and carry out a dynamical systems analysis of the equilibria, the regions of attraction of stable equilibria and bifurcation phenomena under constant, low-dose chemotherapy.…”

“…Separately both models have been biologically validated. Data for the model by Hahnfeldt et al are available for Lewis lung carcinoma implanted in mice (Hahnfeldt et al 1999) and parameters for the model by Stepanova have been estimated in the research by Kuznetsov et al (1994) based on in vivo experimental data for B-lymphoma BC L 1 in the spleen of mice. The parameter values for these two mouse models clearly are not compatible, yet the underlying qualitative mathematical structures of the models separately exhibit strong robustness properties.…”