A mathematical study of the role of tBregs in breast cancer

TL;DR

A model for the mathematical study of immune response to breast cancer is proposed and studied for a compact study of the dynamical role in cancer promotion of a relatively recently described subgroup of regulatory B cells, which are evoked by the tumour.

Abstract

A model for the mathematical study of immune response to breast cancer is proposed and studied, both analytically and numerically. It is a simplification of a complex one, recently introduced by two of the present authors. It serves for a compact study of the dynamical role in cancer promotion of a relatively recently described subgroup of regulatory B cells, which are evoked by the tumour.

Figures (13)

• Figure 1: Interactions between the cells in the model described in bitsouni2021mathematical. Solid line (---): Stimulating effect. Dashed line (- -): Inhibiting effect. Dotted line ($\cdot \cdot \cdot$): Steady systemic supply. Figure adapted from bitsouni2021mathematical, with the inclusion of the present, yet previously non-depicted, steady systemic supply.
• Figure 2: Interactions between the cells in the simplified model. Solid line (---): Stimulating effect. Dashed line (- -): Inhibiting effect.
• Figure 3: Projection of the bifurcation diagram of system \ref{['reducedModel']} onto the $\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$--$c$ plane. Solid line (---): Stable equilibrium. Dashed line (- -): Unstable equilibrium. $\square$: Transcritical bifurcation 1. $\bigcirc$: Transcritical bifurcation 2. $\mathcal{VL}$: Region in which $c$ is very low, i.e. $c<c_2$ when $c_2>0$. $\mathcal{L}$: Region in which $c$ is low, i.e. $c<c_1$ when $c_2<0$ and $c_2<c<c_1$ when $c_2>0$. $\mathcal{H}$: Region in which $c$ is high, i.e. $c>c_1$.
• Figure 4: Phase portrait of system \ref{['reducedModel']} with $B_0 = 0$, for different values of $c$. The equilibrium point $(\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu,\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu,\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu)$ pictured by the red sphere moves along the black line, which is given by $\{(T,N,R) \in \mathbb{R}_{\ge 0 }^3: N=\frac{\sigma \theta _R}{\gamma \kappa +\theta _N \theta _R}, R= \frac{\kappa }{\theta _R}\}$. The value of $\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ tends to 0, as $c$ increases, whereas tends to $\frac{1}{b}$, as $c$ tends to 0.
• Figure 5: Ensemble simulations of IVP, with $B_0$ taking values in the 30-point discretisation of the interval $\left[ 0, 5\cdot 10^{8} \right] \, \cdot \, \text{cells}$, $T_0 = 5$ cells, and $c=2 \cdot 10^{-10}\; \text{cell}^{-1} \, \cdot \, \text{day}^{-1}$ (region $\mathcal{L}$). When $B\to 0$, $T$ eventually tends to $\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu>0$ of $E_2$.

Theorems & Definitions (33)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof