The Reduced Cancitis model

Abstract

A novel mechanism based model - the Cancitis model - describing the interaction of blood cancer and the inflammatory system is proposed. The immune response is divided into two components, one where the elimination rate of malignant stem cells is independent of the level of the blood cancer and one where the elimination rate depends on the level of the blood cancer. A dimensional analysis shows that the full 6-dimensional system of nonlinear ordinary differential equation may be reduced to a 2-dimensional system - the reduced cancitis model - using Fenichel's theory. The original 18 parameters appear in the reduced model in 8 groups of parameters. The reduced model is analyzed and steady states and their dependence especially on the exogenous inflammatory stimuli are analyzed. A semi analytic investigation reveals the stability properties of the steady states. Finally, we prove positivity of the system and the existence of an attracting trapping region in the positive octahedron guaranteeing global existence and uniqueness of solutions. The possible topologies of the dynamical system are completely determined as having a Janus structure, where two qualitatively different topologies appear for different sets of parameters. To classify this Janus structure we propose a novel concept in blood cancer - a reproduction ratio R. It determines the topological structure depending on whether it is larger or smaller than a threshold value. Furthermore, it follows that inflammation, acted by the exogenous inflammatory stimulation, may determine the onset and development of blood cancer. The body may manage initial blood cancer as long as the self-renewal rate is not too high, but fails to manage it if an inflammation appears. Thus, inflammation may trigger and drive blood cancers. Finally, the mathematical analysis suggests novel treatment strategies and it is used to discuss the in silico ect of existing treatment, e.g. interferon or T-cell therapy, and the impact of malignant cells becoming resistant.

Animations

Here we show some animations of the development of the reduced cancitis model for various choices, and changes, of parameters. The right-hand side of the animations show the development of the two compartments over time, while the left-hand side shows the corresponding phase-space behaviour. The phase-space animation also displays the nullclines and the directional field, as well as numerically calculated equilibria, for the given choice of parameters.

Demonstration of changing the reproduction ratio, R

Demonstration of changing the secondary reproduction number, S

Animation of typical disease development, with treatment. Note the right-hand side displays allele burden instead of reduced numbers.

Parameterspace animation

The following two animations shows which admissible equilibria are present for a range of values for the reproduction ratio, R, the secondary reproduction number, S, and the reduced inflammation parameter, J.

All parameters-choices has only one stable steady state. The legend displays which it is for the given color. In the parenthesis shorthands for all admisible steady states are displayed. T: Trivial. H: Hematopoietic. L: Leukemic. C: Co-existence.

The following animation is similar to the one above, but for a wider range of J, in particular, including J < 0.5