Simulating the behaviour of glioblastoma multiforme based on patient MRI during treatments

Abstract

Glioblastoma multiforme is a brain cancer that still shows poor prognosis for patients despite the active research for new treatments. In this work, the goal is to model and simulate the evolution of tumour associated angiogenesis and the therapeutic response to glioblastoma multiforme. Multiple phenomena are modelled in order to fit different biological pathways, such as the cellular cycle, apoptosis, hypoxia or angiogenesis. This leads to a nonlinear system with 4 equations and 4 unknowns: the density of tumour cells, the $\text {O}_{2}$ concentration, the density of endothelial cells and the vascular endothelial growth factor concentration. This system is solved numerically on a mesh fitting the geometry of the brain and the tumour of a patient based on a 2D slice of MRI. We show that our numerical scheme is positive, and we give the energy estimates on the discrete solution to ensure its existence. The numerical scheme uses nonlinear control volume finite elements in space and is implicit in time. Numerical simulations have been done using the different standard treatments: surgery, chemotherapy and radiotherapy, in order to conform to the behaviour of a tumour in response to treatments according to empirical clinical knowledge. We find that our theoretical model exhibits realistic behaviours.

Appendices

Proof of Proposition 2

We will show the result only for $u^{n}$ but the same steps are followed for showing that $u_{e}^{n}$ is upper-bounded. This proof works by induction on n, let’s suppose that for a $n\in \llbracket 0,N\rrbracket$ we have $u^{n}_{K}\le 1,\forall K\in \vartheta$.

Let’s $u_{K^\star }=u_{K}^{n+1}=\underset{H\in \vartheta }{\max }u_{H}^{n+1}$ and $\forall L\in \vartheta ,u_{L}=u_{L}^{n+1}$. Then by multiplying the equation of (14) associated with $K^{\star }$ by $(1-u_{K^\star })^{-}$, we get

$$\begin{aligned}&\frac{m_{K}}{\delta t}|(1-u_{K^\star })^{-}|^{2}+\frac{m_{K}}{\delta t}(1-u^{n}_{K})(1-u_{K^\star })^{-} \nonumber \\&\qquad + \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(1)}_{KL}a_{KL}^{n+1}(u_{K^\star }-u_{L})(1-u_{K^\star })^{-} \nonumber \\&\qquad -\sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(1)}_{KL}a_{KL}^{n+1}\mu _{KL}^{n+1}(c_{K}^{n+1}-c_{L}^{n+1})(1-u_{K^\star })^{-} \nonumber \\&\quad = m_{K}\rho _{1}h(c_{K}^{n+1})f_{u_{T,K}^{n+1}}(u_{K^\star })(1-u_{K^\star })^{-} \nonumber \\&\qquad -m_{K}\beta _{1}u_{K^\star }(1-u_{K^\star })^{-}-m_{K}T_{treat}(t_{n+1},u_{K^\star })(1-u_{K^\star })^{-}. \end{aligned}$$

(39)

Using proposition 1, we know that $u_{K^\star }\ge u_{L}\ge 0,\forall L\in \vartheta$. If $\varLambda ^{(1)}_{KL}<0$, we have $a_{KL}^{n+1}(1-u_{K^\star })^{-}=0$ due to the fact that $a(\cdot )$ is set to zero outside of [0, 1]. So knowing that $a_{KL}^{n+1}\ge 0$, we have

$$\begin{aligned} \varLambda ^{(1)}_{KL}a_{KL}^{n+1}(1-u_{K^\star })^{-}\ge 0, \end{aligned}$$

then observing that $(u_{K^\star }-u_{L})\ge 0$, we conclude that the third term in (39) is positive, e.g.

$$\begin{aligned} + \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(1)}_{KL}a_{KL}^{n+1}(u_{K^\star }-u_{L})(1-u_{K^\star })^{-} \ge 0. \end{aligned}$$

(40)

The function $f_{u_{T}}(\cdot )$ is extended by zero outside of [0,1], implying that $f_{u_{T,K}^{n+1}}(u_{K^\star })(1-u_{K^\star })^{-}=0$. Since $(u_{L}\le u_{K^\star })$ and

$$\begin{aligned}&\mu ^{(1)}_{\downarrow }(u_{K^\star }) + \mu ^{(1)}_{\uparrow }(u_{L}) = \mu _{1}(u_{K^\star })-\mu _{1}(0)-\int _{u_{L}}^{u_{K^\star }}(\mu _{1}'(s))^{+}ds, \\&\mu ^{(1)}_{\uparrow }(u_{K^\star }) + \mu ^{(1)}_{\downarrow }(u_{L}) = \mu _{1}(u_{K^\star })-\mu _{1}(0)+\int _{u_{L}}^{u_{K^\star }}(\mu _{1}'(s))^{-}ds, \end{aligned}$$

we have

$$\begin{aligned}&(\mu ^{(1)}_{\downarrow }(u_{K^\star }) + \mu ^{(1)}_{\uparrow }(u_{L}))(1-u_{K^\star })^{-} = -(1-u_{K^\star })^{-}\int _{u_{L}}^{u_{K^\star }}(\mu _{1}'(s))^{+}ds\le 0, \\&(\mu ^{(1)}_{\uparrow }(u_{K^\star }) + \mu ^{(1)}_{\downarrow }(u_{L}))(1-u_{K^\star })^{-} = +(1-u_{K^\star })^{-}\int _{u_{L}}^{u_{K^\star }}(\mu _{1}'(s))^{-}ds\ge 0. \end{aligned}$$

So whenever the sign of $\varLambda ^{(1)}_{KL}(c_{K}^{n+1}-c_{L}^{n+1})$

$$\begin{aligned} \varLambda ^{(1)}_{KL}(c_{K}^{n+1}-c_{L}^{n+1})\mu _{KL}^{n+1}(1-u_{K^\star })^{-}\le 0,\end{aligned}$$

and, having $a_{KL}^{n+1}\ge 0$, we conclude that the fourth term in (39) is positive, e.g.

$$\begin{aligned} -\sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(1)}_{KL}a_{KL}^{n+1}\mu _{KL}^{n+1}(c_{K}^{n+1}-c_{L}^{n+1})(1-u_{K^\star })^{-} \ge 0. \end{aligned}$$

(41)

With (40), (41) and the positivity of $T_{treat}(\cdot ,\cdot )$, we conclude also that

$$\begin{aligned}&\frac{m_{K}}{\delta t}|(1-u_{K^\star })^{-}|^{2}+\frac{m_{K}}{\delta t}(1-u^{n}_{K})(1-u_{K^\star })^{-} + \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(1)}_{KL}a_{KL}^{n+1}(u_{K^\star }-u_{L})(1-u_{K^\star })^{-} \\&\qquad -\sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(1)}_{KL}a_{KL}^{n+1}\mu _{KL}^{n+1}(c_{K}^{n+1}-c_{L}^{n+1})(1-u_{K^\star })^{-} \\&\quad = -m_{K}\beta _{1}u_{K^\star }(1-u_{K^\star })^{-}-m_{K}T_{treat}(t_{n+1},u_{K^\star })(1-u_{K^\star })^{-}\le 0. \end{aligned}$$

So all terms on left side of (39) being non-negative, they are null, in particular: $\frac{m_{K}}{\delta t}(1-u_{K^\star })^{-^2}=0$, then $(1-u_{K^\star })^{-}=0$, which gives the result of this proposition. $\square$

Proof of Proposition 3

We will show the result only for $c^{n}$ but the same steps are followed for showing the positivity of $v^{n}$. This proof works by induction on n, let’s suppose that for a $n\in \llbracket 0,N\rrbracket$ we have $c^{n}_{K}\ge 0,\forall K\in \vartheta$.

Let’s $c_{K_\star }=c_{K}^{n+1}=\underset{H\in \vartheta }{\min }c_{H}^{n+1}$ and $\forall L\in \vartheta ,c_{L}=c_{L}^{n+1}$. Then by multiplying the equation of (15) associated with $K_{\star }$ by $-c_{K_\star }^{-}$, we get

$$\begin{aligned}&\frac{m_{K}}{\delta t}c_{K_\star }^{-^2}+\frac{m_{K}}{\delta t}c_{K_\star }^{-}c_{K}^{n} - \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}D_{KL}^{(2)}\eta _{KL}^{n+1}(p(c_{K_\star })-p(c_{L}))c_{K_\star }^{-}\nonumber \\&\quad = -m_{K}\alpha _{2}u_{e}^{n+1}c_{K_\star }^{-}-m_{K}\beta _{2}c_{K_\star }^{-^2}-\gamma _{2}u_{K}^{n+1}c_{K_\star }^{-^2}. \end{aligned}$$

(42)

Using the results of Proposition 1 and 2, we have $u_{e}^{n+1},u_{K}^{n+1}\ge 0$, implying that

$$\begin{aligned} -m_{K}\alpha _{2}u_{e}^{n+1}c_{K_\star }^{-} \le 0, \quad -m_{K}\beta _{2}c_{K_\star }^{-^2} \le 0 \quad \text { and }\quad -\gamma _{2}u_{K}^{n+1}c_{K_\star }^{-^2} \le 0. \end{aligned}$$

If $D_{KL}^{(2)}<0$ then $\eta _{KL}^{n+1}c_{K_\star }^{-}=0$. Observing that $\mu _{KL}^{n+1}\ge 0$ and that the function $p(\cdot )$ is non-decreasing on ${\mathbb {R}}$, we have $(p(c_{K_\star })-p(c_{L}))\le 0$, meaning that

$$\begin{aligned}&- \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}D_{KL}^{(2)}\eta _{KL}^{n+1}(p(c_{K_\star })-p(c_{L}))c_{K_\star }^{-} \nonumber \\&\quad = -\sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}(D_{KL}^{(2)})^{+}\eta _{KL}^{n+1}(p(c_{K_\star })-p(c_{L}))c_{K_\star }^{-} \ge 0. \end{aligned}$$

(43)

Then

$$\begin{aligned}&\frac{m_{K}}{\delta t}c_{K_\star }^{-^2}+\frac{m_{K}}{\delta t}c_{K_\star }^{-}c_{K}^{n} - \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}D_{KL}^{(2)}\eta _{KL}^{n+1}(p(c_{K_\star })-p(c_{L}))c_{K_\star }^{-}\\&\quad = -m_{K}\alpha _{2}u_{e}^{n+1}c_{K_\star }^{-}-m_{K}\beta _{2}c_{K_\star }^{-^2}-\gamma _{2}u_{K}^{n+1}c_{K_\star }^{-^2} \le 0. \end{aligned}$$

The terms on the left side of (42) being non-negative, we conclude that they are all null. In particular $c_{K_\star }^{-^2}=0$, which completes the proof of the proposition.$\square$

Proof of Proposition 4

This works by induction on m.

Let’s consider the application: ${\mathscr {W}}=({\mathscr {W}}_{u},{\mathscr {W}}_{c},{\mathscr {W}}_{u_{e}},{\mathscr {W}}_{v}):({\mathbb {R}}^{\#\vartheta })^{4}\rightarrow ({\mathbb {R}}^{\#\vartheta })^{4}$ where $\forall K\in \vartheta$, for $n\in \llbracket 0,N\rrbracket$ and $\forall m\ge 0$

$$\begin{aligned}&\circ ({\mathscr {W}}_{u}(y))_{K} = m_{K}\frac{y_{K}-u_K^{n}}{\delta t} +\sum _{\sigma _{KL}\in \epsilon _{K}}\varLambda _{KL}^{(1)}a_{KL}^{m+1}(y_{K}-y_{L})\\&\quad -\sum _{\sigma _{KL}\in \epsilon _{K}}\varLambda ^{(1)}_{KL}a_{KL}^{m+1}\mu _{KL}^{m+1}(c_{K}^{m}-c_{L}^{m}) - m_{K}\rho _{1}h(c_{K}^{m})f_{u_{T,K}^{m}}(y_{K})\\&\quad +m_{K}\beta _{1}y_{K}+m_{K}T_{treat}(t_{m+1},y_{K}), \\&\circ ({\mathscr {W}}_{c}(y))_{K} =\frac{m_{K}}{\delta t}(y_{K}-c_{K}^{n}) + \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}D_{KL}^{(2)}\eta _{KL}^{m+1}(p(y_{K})-p(y_{L})) \\&\quad - m_{K}\alpha _{2}u_{e,K}^{m}+m_{K}\beta _{2}y_{K}+\gamma _{2}u_{K}^{m}y_{K}, \\&\circ ({\mathscr {W}}_{u_{e}}(y))_{K} =\frac{m_{K}}{\delta t}(y_{K}-u_{e,K}^{n}) + \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}(y_{K}-y_{L}) \\&\quad -\sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}{\tilde{\mu }}_{KL}^{m+1}(v_{K}^{m}-v_{L}^{m}) \\&\quad - m_{K}\rho _{3}f_{u_{T,K}^{m}}(y_{K})+m_{K}\beta _{3}y_{K} , \\&\circ ({\mathscr {W}}_{v}(y))_{K} = \frac{m_{K}}{\delta t}(y_{K}-v_{K}^{n}) + \sum _{\sigma _{KL}\in {\mathscr {E}}_{K}}D_{KL}^{(4)}{\bar{\eta }}_{KL}^{m+1}(p(y_{K})-p(y_{L})) \\&\quad - m_{K}\alpha _{4}g(c_{K}^{m})u_{K}^{m}+m_{K}\beta _{4}y_{K}+\gamma _{4}u_{e,K}^{m}y_{K}. \end{aligned}$$

Observe that if ${\mathscr {W}}(u^{m+1},c^{m+1},u_{e}^{m+1},v^{m+1})=0$ then $(u^{m+1},c^{m+1},u_{e}^{m+1},v^{m+1})$ is a solution of (26)–(29).

The aim in the first step is to prove that $\exists k\ge 0,\forall \Vert y\Vert _{2}>k : ({\mathscr {W}}(y),y)\ge 0$. So we develop the expression of $({\mathscr {W}}(y),y)=({\mathscr {W}}_{u}(y_{u}),y_{u})+({\mathscr {W}}_{c}(y_{c}),y_{c})+({\mathscr {W}}_{u_{e}}(y_{u_{e}}),y_{u_{e}})+({\mathscr {W}}_{v}(y_{v}),y_{v}),y=(y_{u},y_{c},y_{u_{e}},y_{v})$ to build inequalities. We have

$$\begin{aligned}&\circ ({\mathscr {W}}_{u}(y),y) = \sum _{K\in \vartheta }\frac{m_{K}}{\delta t}y_{K}^{2}-\sum _{K\in \vartheta }\frac{m_{K}}{\delta t}u_K^{n}y_{K}+\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda _{KL}^{(1)}a_{KL}^{m+1}(y_{K}-y_{L})^{2}\\&\quad -\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda ^{(1)}_{KL}a_{KL}^{m+1}\mu _{KL}^{m+1}(c_{K}^{m}-c_{L}^{m})(y_{K}-y_{L}) - \sum _{K\in \vartheta }m_{K}\rho _{1}h(c_{K}^{m})f_{u_{T,K}^{m}}(y_{K})y_{K}\\&\quad +\sum _{K\in \vartheta }m_{K}\beta _{1}y_{K}^{2}+\sum _{K\in \vartheta }m_{K}T_{treat}(t_{m+1},y_{K})y_{K}, \\&\circ ({\mathscr {W}}_{c}(y),y) =\sum _{K\in \vartheta } \frac{m_{K}}{\delta t}y_{K}^{2}-\sum _{K\in \vartheta } \frac{m_{K}}{\delta t}c_{K}^{n}y_{K} + \sum _{\sigma _{KL}\in {\mathscr {E}}}D_{KL}^{(2)}\eta _{KL}^{m+1}(p(y_{K}) \\&\quad -p(y_{L}))(y_{K}-y_{L}) -\sum _{K\in \vartheta } m_{K}\alpha _{2}u_{e,K}^{m}y_{K}+\sum _{K\in \vartheta }m_{K}\beta _{2}y_{K}^{2}+\sum _{K\in \vartheta }\gamma _{2}u_{K}^{m}y_{K}^{2}, \\&\circ ({\mathscr {W}}_{u_{e}}(y),y) =\sum _{K\in \vartheta }\frac{m_{K}}{\delta t}y_{K}^{2}-\sum _{K\in \vartheta }\frac{m_{K}}{\delta t}u_{e,K}^{n}y_{K} + \sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}(y_{K}-y_{L})^{2} \\&\quad -\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}{\tilde{\mu }}_{KL}^{n+1}(v_{K}^{m}-v_{L}^{m})(y_{K}-y_{L}) \\&\quad -\sum _{K\in \vartheta } m_{K}\rho _{3}f_{u_{T,K}^{m}}(y_{K})y_{K}+\sum _{K\in \vartheta }m_{K}\beta _{3}y_{K}^{2}, \\&\circ ({\mathscr {W}}_{v}(y),y)= \sum _{K\in \vartheta }\frac{m_{K}}{\delta t}y_{K}^{2}-\sum _{K\in \vartheta }\frac{m_{K}}{\delta t}v_{K}^{n}y_{K} + \sum _{\sigma _{KL}\in {\mathscr {E}}}D_{KL}^{(4)}{\bar{\eta }}_{KL}^{m+1}(p(y_{K})\\&\quad -p(y_{L}))(y_{K}-y_{L}) - \sum _{K\in \vartheta }m_{K}\alpha _{4}g(c_{K}^{m})u_{K}^{m}y_{K}+\sum _{K\in \vartheta }m_{K}\beta _{4}y_{K}^{2}+\sum _{K\in \vartheta }\gamma _{4}u_{e,K}^{m}y_{K}^{2}. \end{aligned}$$

With Lemma 1, we know that $\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda _{KL}^{(1)}a_{KL}^{m+1}(y_{K}-y_{L})^{2}\ge 0$, $\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}(y_{K}-y_{L})^{2}\ge 0$ and with Lemma 2 we have $\sum _{\sigma _{KL}\in {\mathscr {E}}}D_{KL}^{(2)}\eta _{KL}^{m+1}(p(y_{K})-p(y_{L}))(y_{K}-y_{L})\ge 0$, $\sum _{\sigma _{KL}\in {\mathscr {E}}}D_{KL}^{(4)}{\bar{\eta }}_{KL}^{m+1}(p(y_{K})-p(y_{L}))(y_{K}-y_{L})\ge 0$ because $p(\cdot )$ is non decreasing on ${\mathbb {R}}$.

Then we have $\forall \lambda >0$

$$\begin{aligned}&-\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda ^{(1)}_{KL}a_{KL}^{m+1}\mu _{KL}^{m+1}(c_{K}^{m}-c_{L}^{m})(y_{K}-y_{L}) \\&\quad \ge -\frac{\lambda ^{2}}{2}\sum _{\sigma _{KL}\in {\mathscr {E}}}(\varLambda _{KL}a_{KL}^{m+1}\mu _{KL}^{m+1}(c_{K}^{m}-c_{L}^{m}))^{2} - \frac{1}{2\lambda ^{2}}\sum _{\sigma _{KL}\in {\mathscr {E}}}(y_{K}-y_{L})^{2} \\&\qquad -\frac{\lambda ^{2}}{2}\sum _{\sigma _{KL}\in {\mathscr {E}}}(\varLambda _{KL}a_{L^{\infty }({\mathbb {R}})}\mu _{L^{\infty }({\mathbb {R}})}(c_{K}^{m}-c_{L}^{m}))^{2} - \frac{2}{\lambda ^{2}}(\#{\mathscr {E}})\Vert y\Vert _{2}^{2}, \end{aligned}$$

and

$$\begin{aligned}&-\sum _{\sigma _{KL}\in {\mathscr {E}}}\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}{\tilde{\mu }}_{KL}^{m+1}(v_{K}^{m}-v_{L}^{m})(y_{K}-y_{L}) \\&\quad \ge -\frac{\lambda ^{2}}{2}\sum _{\sigma _{KL}\in {\mathscr {E}}}(\varLambda ^{(3)}_{KL}{\tilde{a}}_{KL}^{m+1}{\tilde{\mu }}_{KL}^{m+1}(v_{K}^{m}-v_{L}^{m}))^{2} -\frac{1}{2\lambda ^{2}}\sum _{\sigma _{KL}\in {\mathscr {E}}}(y_{K}-y_{L})^{2}\\&\quad \ge -\frac{\lambda ^{2}}{2}\sum _{\sigma _{KL}\in {\mathscr {E}}}(\varLambda ^{(3)}_{KL}{\tilde{a}}_{L^{\infty }({\mathbb {R}})}{\tilde{\mu }}_{L^{\infty }({\mathbb {R}})}(v_{K}^{m}-v_{L}^{m}))^{2} - \frac{2}{\lambda ^{2}}(\#{\mathscr {E}})\Vert y\Vert _{2}^{2}. \end{aligned}$$

So, we have those inequalities

$$\begin{aligned}&({\mathscr {W}}_{u}(y_{u}),y_{u})\ge C^{(u)}_{1}\Vert y_{u}\Vert _{2}^{2}-C^{(u)}_{2}\Vert y_{u}\Vert _{2}-C^{(u)}_{3}, \\&({\mathscr {W}}_{c}(y_{c}),y_{c}) \ge C^{(c)}_{1}\Vert y_{c}\Vert _{2}^{2}-C^{(c)}_{2}\Vert y_{c}\Vert _{2}, \\&({\mathscr {W}}_{u_{e}}(y_{u_{e}}),y_{u_{e}}) \ge C^{(u_{e})}_{1}\Vert y_{u_{e}}\Vert _{2}^{2}-C^{(u_{e})}_{2}\Vert y_{u_{e}}\Vert _{2}-C^{(u_{e})}_{3}, \\&({\mathscr {W}}_{v}(y_{v}),y_{v}) \ge C^{(v)}_{1}\Vert y_{v}\Vert _{2}^{2}-C^{(v)}_{2}\Vert y_{v}\Vert _{2}, \end{aligned}$$

with the following expression for the constants, choosing $\lambda$ such that $\lambda ^{2}>\frac{2(\#{\mathscr {E}})\delta t}{m^{down}_{{\mathscr {M}}}}$ and denoting $m^{down}_{{\mathscr {M}}}=\underset{K\in \vartheta }{\min }\text { }m_{K}\le m_{K}\le \underset{K\in \vartheta }{\max }\text { }m_{K} = m^{up}_{{\mathscr {M}}}$

$$\begin{aligned}&C^{(u)}_{1} =\frac{m^{down}_{{\mathscr {M}}}}{\delta t}-\frac{2(\#{\mathscr {E}})}{\lambda ^{2}}>0,\qquad C^{(u)}_{2} =\frac{m^{up}_{{\mathscr {M}}}}{\delta _{t}}(\#\vartheta )+m^{up}_{{\mathscr {M}}}\rho _{1}f_{L^{\infty }({\mathbb {R}})}(\#\vartheta )>0,\\&C^{(u)}_{3} = \frac{\lambda ^{2}}{2}\sum _{\sigma _{KL}\in {\mathscr {E}}}(\varLambda _{KL}^{(1)}a_{L^{\infty }({\mathbb {R}})}\mu _{L^{\infty }({\mathbb {R}})}(c_{K}^{m}-c_{L}^{m}))^{2}>0, \\&C^{(c)}_{1} = \frac{m^{down}_{{\mathscr {M}}}}{\delta t}>0,\qquad C^{(c)}_{2} = m^{up}_{{\mathscr {M}}}(\#\vartheta )(\frac{\underset{K\in \vartheta }{\max }(c_{K}^{m})}{\delta t}+\alpha _{2})>0, \\&C^{(u_{e})}_{1} =\frac{m^{down}_{{\mathscr {M}}}}{\delta t}-\frac{2(\#{\mathscr {E}})}{\lambda ^{2}}>0, \qquad C^{(u_{e})}_{2} =\frac{m^{up}_{{\mathscr {M}}}}{\delta _{t}}(\#\vartheta )+m^{up}_{{\mathscr {M}}}\rho _{3}f_{L^{\infty }({\mathbb {R}})}(\#\vartheta )>0,\\&C^{(u_{e})}_{3} = \frac{\lambda ^{2}}{2}\sum _{\sigma _{KL}\in {\mathscr {E}}}(\varLambda _{KL}^{(3)}{\tilde{a}}_{L^{\infty }({\mathbb {R}})}{\tilde{\mu }}_{L^{\infty }({\mathbb {R}})}(v_{K}^{m}-v_{L}^{m}))^{2}>0, \\&C^{(v)}_{1} = \frac{m^{down}_{{\mathscr {M}}}}{\delta t}>0, \qquad C^{(v)}_{2} = m^{up}_{{\mathscr {M}}}(\#\vartheta )(\frac{\underset{K\in \vartheta }{\max }(v_{K}^{m})}{\delta t}+\alpha _{4})>0. \end{aligned}$$

It induces that

$$\begin{aligned}&({\mathscr {W}}(y),y) \ge \min (C^{(u)}_{1},C^{(c)}_{1},C^{(u_{e})}_{1},C^{(v)}_{1})\Vert y\Vert _{2}^{2} \\&\quad -\max (C^{(u)}_{2},C^{(c)}_{2},C^{(u_{e})}_{2},C^{(v)}_{2})\Vert y\Vert _{2}-C^{(u)}_{3}-C^{(u_{e})}_{3}. \end{aligned}$$

So there exists $k>0$ from which $\forall \Vert y\Vert _{2}^{2}\ge k:({\mathscr {W}}(y),y)>0$.

Suppose that there is no $z\in {\mathbb {R}}^{4\times \#\vartheta }:{\mathscr {W}}(z)=0$, in that case we can define the application ${\mathscr {S}}:y\in {\mathscr {B}}(0,k)\mapsto -k\frac{{\mathscr {W}}(y)}{\Vert {\mathscr {W}}(y)\Vert }\in {\mathscr {B}}(0,k)$. ${\mathscr {S}}$ is continuous due to ${\mathscr {W}}$, so according to the Brouwer fixed-point theorem there exists a fixed point ${\tilde{y}}$ of ${\mathscr {S}}$ on ${\mathscr {B}}(0,k)$:

$$\begin{aligned} {\tilde{y}}=-k\frac{{\mathscr {W}}({\tilde{y}})}{\Vert {\mathscr {W}}({\tilde{y}})\Vert } \end{aligned}$$

(44)

Taking the norm of ${\tilde{y}}$ from (44), we get $\Vert {\tilde{y}}\Vert =k>0$ but taking the inner product of (44) with ${\tilde{y}}$ we get : $\Vert {\tilde{y}}\Vert ^{2} = -k\frac{({\mathscr {W}}({\tilde{y}}),{\tilde{y}})}{\Vert {\mathscr {W}}({\tilde{y}})\Vert }\le 0$. Thus there exists z with ${\mathscr {W}}(z)=0$. So there exists a solution $(u^{m+1},c^{m+1},u_{e}^{m+1},v^{m+1})$ to (26)–(29). $\square$