A Mathematical Study of the Role of tBregs in Breast Cancer

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.

Appendices

Appendix

Appendix A Preliminary Results

In this section, we prove that IVP (we remind that this acronym is used for the initial value problem $\left\{ (1),(2)\right\}$) has a unique solution, which is non-negative for non-negative initial conditions and for positive parameter values. We also prove that our solution is global, i.e. it does not explode for some finite positive value of time t. The above conditions are necessary in order to assure that IVP yields biologically realistic results.

Proposition 14

(Uniqueness and non-negativity) For every $\left( T_0,N_0,R_0,B_0\right) \in \mathbb {R}_{\ge 0}^4$, IVP has a unique (local) solution $\left( T, N, R, B\right) :\left[ 0,\tau \right) \rightarrow \mathbb {R}_{\ge 0}^4$ for some $\tau >0$.

Proof

It is easy to see that the conditions of the Picard-Lindelöf theorem are fulfilled, since every function of the right-hand side of system (1) is continuous, just like its partial derivative with respect to every variable. Thus, we have that there exists a unique solution to IVP. In fact, we can extended the solution and consider it in the maximal non-negative interval of existence.

Next, we prove the non-negativity of the solution. Rewriting (1a) in the following form

$$\begin{aligned} \frac{{\mathrm{d}}T}{{\mathrm{d}}t}(t) +(cN(t)-a)T(t)=-abT^2(t) \,, \end{aligned}$$

we notice that we have a Bernoulli equation for the variable T, and thus its solution is

$$\begin{aligned} T(t) = \frac{T_0 e^{\int _0^t \left( a-cN(s) \right) \, \text {d}s}}{1+T_0ab \int _0^t \! e^{{\int _0^s \left( a-cN(\xi ) \right) \, \text {d}\xi }} \, \text {d}s} \,, \end{aligned}$$

which is non-negative, if $T_0$ is non-negative.

Using the fact that $\sigma > 0$, we turn our attention to (1b). We have that

$$\begin{aligned} \frac{{\mathrm{d}}N}{{\mathrm{d}}t}(t) > -(\gamma R(t) + \theta _N)N(t) \,, \end{aligned}$$

and using Grönwall’s inequality, we have that

$$\begin{aligned} N(t) \ge N_0e^{- \int _0^t \left( \gamma R(s) + \theta _N \right) \, \text {d}s} \,, \end{aligned}$$

which means that $N(t)\ge 0$, when $N_0\ge 0$.

Using a similar method as above, from (1c) we get

$$\begin{aligned} R(t) \ge R_0e^{\int _0^t \left( m_BB(s)-\theta _R \right) \, \text {d}s} \,, \end{aligned}$$

which means that $R(t)\ge 0$, when $R_0\ge 0$.

We then use the separation of variables method to solve (1d) for the variable B. Its solution is

$$\begin{aligned} B(t) = B_0e^{\int _0^t \left( m_TT(s) - \theta _B \right) \, \text {d}s} \,. \end{aligned}$$

(A1)

Clearly, if $B_0\ge 0$, then $B(t)\ge 0$. $\square$

Proposition 15

(Boundedness of T) The set $\left[ 0,1/b\right]$ is positively invariant for the component T of the solution of IVP.

Proof

Since the solution of IVP is non-negative, we have from (1a) that

$$\begin{aligned} \frac{{\mathrm{d}}T}{{\mathrm{d}}t}(t) = \underbrace{aT(1-bT) - cNT}_{q_1(T)} \le \underbrace{aT(1-bT)}_{q_2(T)} \; . \end{aligned}$$

We assume the following two initial value problems:

$$\begin{aligned}&\frac{{\mathrm{d}}T}{{\mathrm{d}}t} = q_1(T), \quad T(0)=T_0 \ge 0 \quad \text { and} \end{aligned}$$

(A2)

$$\begin{aligned}&\frac{{\mathrm{d}}z}{{\mathrm{d}}t}= q_2(z), \quad z(0) = \frac{1}{b} \; . \end{aligned}$$

(A3)

Assuming that $T_0 \le z(0) = 1/b$, and since $q_1, q_2$ are Lipschitz functions on $\mathbb {R}$ that satisfy the inequality $q_1(T) \le q_2(T)$, from the comparison theorem we have that $T(t) \le z(t)$ for t in the maximal non-negative interval of existence of the solution of IVP. Solving initial value problem (A3), yields $z = 1/b$. Hence, $T(t) \le 1/b$, with the assumption that the initial value of T is smaller or equal to 1/b. $\square$

Proposition 16

(Globality) If $T_0 \le 1/b$, then the solution of IVP is global.

Proof

From (1d) and using the fact that $T(t) \le 1/b$, we have that

$$\begin{aligned} \frac{{\mathrm{d}}B}{{\mathrm{d}}t}(t) \le -\theta _BB(t)+\frac{m_T}{b}B(t) \,, \end{aligned}$$

and by using Grönwall’s inequality, we have that

$$\begin{aligned} B(t) \le B_0e^{(\frac{m_T}{b}-\theta _B)t} \,. \end{aligned}$$

Moreover, from (1c) and using the fact that $\theta _RR \ge 0$ we get

$$\begin{aligned} \frac{{\mathrm{d}}R}{{\mathrm{d}}t}(t) \le \kappa + m_BB(t)R(t) \,, \end{aligned}$$

and by using Grönwall’s inequality, we have that

$$\begin{aligned} R(t) \le e^{m_B \int _0^t \! B(s) \, \text {d}s} \left( R_0 + \kappa \int _0^t \! e^{ - m_B \int _0^\xi \! B(s) \, \text {d}s} \, \text {d}\xi \right) \,. \end{aligned}$$

Finally, from (1b) and using the fact that $\gamma RN \ge 0$ we get

$$\begin{aligned} \frac{{\mathrm{d}}N}{{\mathrm{d}}t}(t)\le \sigma - \theta _N N(t) \,, \end{aligned}$$

and by using Grönwall’s inequality, we have that

$$\begin{aligned} N(t) \le \frac{\sigma }{\theta _N} - \frac{\sigma }{\theta _N}e^{-\theta _Nt} + e^{-\theta _Nt}N_0 \,. \end{aligned}$$

Since the solution is bounded on any compact non-negative interval, we deduce its (positive) globality. $\square$

Appendix B Parameter Estimation

Here we explain our reasoning behind our choice of parameters. Most of the parameters in our model have been chosen based on methods and data that can also be found in Bitsouni and Tsilidis (2022).

1.1 Appendix B.1 The Tumour

Based on the data fitting experiments conducted in Bitsouni and Tsilidis (2022), we chose the logistic function to model the breast cancer growth, with the tumour growth rate being $a = 0.15$ day$^{-1}$ and the inverse of the tumour carrying capacity being $b = 1 \cdot 10^{-9}$ cell$^{-1}$. The cell lines used for the estimation of these parameters are CN34BrM, MDA-231 and SUM1315.

1.2 Appendix B.2 The NK Cells

Healthy young adults have a total NK production rate of (15 ± 7.6) $\cdot 10^6$ $\text {cell} \cdot \text {litre}^{-1}$ $\cdot \, \mathrm {day}^{-1}$, while healthy older adults have one of (7.3 ± 3.7) $\cdot 10^6$ $\text {cells} \cdot \text {litre}^{-1}$ $\cdot \,\mathrm {day}^{-1}$ (Zhang et al. 2007). Since the average amount of blood in the human body is about 5 litre (Starr et al. 2012), the constant source of NK cells is in the range

$$\begin{aligned} \sigma \in \left[ 1.8 \cdot 10^7 \text { cell} \cdot \text {day}^{-1},\, 1.13 \cdot 10^8 \text { cell} \cdot \text {day}^{-1} \right] \,. \end{aligned}$$

The half-life of NK cells in humans is 1 to 2 weeks (Zhang et al. 2007) which, assuming exponential decay of NK cells, yields a range for $\theta _N$ of $({\frac{\ln 2}{14}}, {\frac{\ln 2}{7}})=(0.049, 0.099)$. Here, we choose an NK cells half-life of 11 days with a corresponding programmable NK death rate of

$$\begin{aligned} \theta _N = \frac{\ln 2}{11 \text { day}} \approx 6.301 \cdot 10^{-2} \text { day}^{-1}\,. \end{aligned}$$

Approximately 4 to 29% of circulating lymphocytes are NK cells (Keohane et al. 2015). The average number of lymphocytes per microlitre is 1000 to 4800 cells (Abbas et al. 2014), and since the average human has an average of 5 litres of blood, we have that the total population of lymphocytes in a human is 5$\cdot$10$^9$ to 24$\cdot$10$^9$ cells. Therefore, the total population of NK cells in blood is $N_{min}=$ 2$\cdot$10$^8$ to $N_{max}=$ 6.96$\cdot$10$^9$ cells. At the healthy equilibrium, our model suggests the population of NK cells to be $\frac{\sigma \theta _R}{\gamma \kappa +\theta _N \theta _R}$, which means that

$$\begin{aligned} N_{min} \le \frac{\sigma \theta _R}{\gamma \kappa +\theta _N \theta _R} \le N_{max} \Leftrightarrow \frac{\theta _R ( \sigma - \theta _N )}{N_{max} \kappa } \le \gamma \le \frac{\theta _R ( \sigma - \theta _N )}{N_{min} \kappa }\,. \end{aligned}$$

Replacing the minimum value of $\sigma$, and the maximum value of $\theta _N$ and $\kappa$ in the above inequality yields the minimum value of $\gamma$, while replacing the maximum value of $\sigma$, and the minimum value of $\theta _N$ and $\kappa$ (we derive a range for $\kappa$ in Appendix B.3), yields the maximum value of $\gamma$. Finally, after calculating the above two quantities, the resulting range for the parameter $\gamma$ is

$$\begin{aligned} \gamma \in \left[ 1.796 \cdot 10^{-12} \text { cell}^{-1} \cdot \text {day}^{-1} , 4.52 \cdot 10^{-9} \text { cell}^{-1} \cdot \text {day}^{-1} \right] \,. \end{aligned}$$

1.3 Appendix B.3 The Tregs

For the constant source of Tregs, our model suggests that a healthy organism has an average number of $\kappa /\theta _R$ Tregs, since this is the coordinate which corresponds to Tregs in the healthy equilibrium. From Pang et al. (2013) we get that Tregs are 5 to 10% of the total CD4$^+$ T cells population circulating in blood, while from Abbas et al. (2014) we get that the percentage of CD4$^+$ T cells among the total population of circulating lymphocytes ranges from 50 to 60%. Hence, the percentage of Tregs among the total population of circulating lymphocytes is 2.5 to 6%. The average number of lymphocytes per microlitre is 1000 to 4800 cells (Abbas et al. 2014), and since the average human has an average of 5 litres of blood, we have that the total population of lymphocytes in a human is 5$\cdot$10$^9$ to 24$\cdot$10$^9$ cells. Therefore, the total population of Tregs in blood is 1.25$\cdot$10$^8$ to 1.44$\cdot$10$^9$ cells. Solving the equation

$$\begin{aligned} \textit{total population of Tregs} = \frac{\kappa }{\theta _R}, \end{aligned}$$

for $\kappa$ and replacing the range of values for the total population of Tregs as found above and the value of $\theta _R$ as found in the following paragraph, we finally get the constant source of Tregs to be in the interval

$$\begin{aligned} \kappa \in [4.8137 \cdot 10^6 \text { cell} \cdot day^{-1},5.5454 \cdot 10^7 \text { cell} \cdot day^{-1}]\,. \end{aligned}$$

The half-life of Tregs is found to be about 18 days (Mabarrack et al. 2008). Thus, assuming Tregs follow exponential decay we have that

$$\begin{aligned} \theta _R = \frac{\ln 2}{18 \text { day}} \approx 3.851 \cdot 10^{-2} \text { day}^{-1}. \end{aligned}$$

The rate of NK cell death due to Tregs, $\gamma$, is assumed to be

$$\begin{aligned} \gamma = 1 \cdot 10^{-10} \text { cell}^{-1} \cdot \text {day}^{-1}. \end{aligned}$$

1.4 Appendix B.4 The tBregs

As B cells are less studied than T cells and NK cells, we are not able to find the half-life of Bregs, let alone tBregs, as they are recently discovered. So, we estimate the rate of programmable tBreg cell death to be around $\theta _B = 0.4 \text { day}^{-1}$.

The same holds for the rate of breast-cancer-induced tBreg activation, $m_T$, which we assume it to be either $m_T = 5.2 \cdot 10^{-15}$ cell$^{-1}$ $\cdot$ day$^{-1}$ or $m_T = 5 \cdot 10^{-10}$ cell$^{-1}$ $\cdot$ day$^{-1}$.