# A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis

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## Abstract

**:**

## 1. Introduction

## 2. Modeling

- variables: time $t\ge 0$, position $\mathbf{x}\in {\mathbb{R}}^{N}$, velocity of glioma cells $\mathbf{v}\in V=s{\mathbb{S}}^{N-1}\subset {\mathbb{R}}^{N}$, velocity of ECs $\mathbf{\vartheta}\in \Theta =\sigma {\mathbb{S}}^{N-1}\subset {\mathbb{R}}^{N}$; we assume the cell speeds $s,\sigma >0$ to be constant; ${\mathbb{S}}^{N-1}$ denotes the unit sphere in ${\mathbb{R}}^{N}$;
- $\widehat{\mathbf{v}}=\frac{\mathbf{v}}{\left|\mathbf{v}\right|}$, $\widehat{\mathbf{\vartheta}}=\frac{\mathbf{\vartheta}}{\left|\mathbf{\vartheta}\right|}$ are unit vectors denoting the directions of vectors $\mathbf{v}\in V$ and $\mathbf{\vartheta}\in \Theta $, respectively;
- $p(t,\mathbf{x},\mathbf{v})$: (mesoscopic) density function of glioma cells and $M(t,\mathbf{x})={\int}_{V}p(t,\mathbf{x},\mathbf{v})d\mathbf{v}$: macroscopic density of glioma;
- $w(t,\mathbf{x},\mathbf{\vartheta})$: (mesoscopic) density function of ECs and $W(t,\mathbf{x})={\int}_{\Theta}w(t,\mathbf{x},\mathbf{\vartheta})d\mathbf{\vartheta}$: macroscopic density of ECs.
- $q(\mathbf{x},\mathbf{\theta})$: (known) directional distribution function of tissue fibers with orientation $\theta \in {\mathbb{S}}^{N-1}$. It holds that ${\int}_{{\mathbb{S}}^{N-1}}q(\mathbf{x},\mathbf{\theta})d\mathbf{\theta}=1$; for $\omega ={\int}_{V}q(\mathbf{x},\widehat{\mathbf{v}})d\mathbf{v}={s}^{N-1}$, it holds then that ${\int}_{V}\frac{q(\mathbf{x},\widehat{\mathbf{v}})}{\omega}d\mathbf{v}=1$;
- ${\mathbb{E}}_{q}\left(\mathbf{x}\right):={\int}_{{\mathbb{S}}^{N-1}}\mathbf{\theta}q(\mathbf{x},\mathbf{\theta})d\mathbf{\theta}$: mean fiber orientation. We also denote ${\tilde{\mathbb{E}}}_{q}\left(\mathbf{x}\right):={\int}_{V}\mathbf{v}\frac{q(\mathbf{x},\widehat{\mathbf{v}})}{\omega}d\mathbf{v}$ and call it mean fiber direction;
- ${\mathbb{V}}_{q}\left(\mathbf{x}\right):={\int}_{{\mathbb{S}}^{N-1}}(\mathbf{\theta}-{\mathbb{E}}_{q}\left(\mathbf{x}\right))\otimes (\mathbf{\theta}-{\mathbb{E}}_{q}\left(\mathbf{x}\right))q(\mathbf{x},\mathbf{\theta})d\mathbf{\theta}$: variance–covariance matrix for orientation distribution of tissue fibers, and, correspondingly, ${\tilde{\mathbb{V}}}_{q}\left(\mathbf{x}\right)={\int}_{V}(\mathbf{v}-{\tilde{\mathbb{E}}}_{q}\left(\mathbf{x}\right))\otimes (\mathbf{v}-{\tilde{\mathbb{E}}}_{q}\left(\mathbf{x}\right))\frac{q(\mathbf{x},\widehat{\mathbf{v}})}{\omega}d\mathbf{v}$;
- $h(t,\mathbf{x})$: concentration of protons (acidity), a macroscopic quantity;
- $g(t,\mathbf{x})$: concentration of VEGF, also a macroscopic quantity;
- $\lambda >0$: constant glioma turning rate;
- $\eta (\mathbf{\vartheta},g(t,\mathbf{x}\left)\right)>0$: EC turning rate.

#### 2.1. Glioma Cells

#### 2.2. Endothelial Cells

#### 2.3. Full Macroscopic System

## 3. Numerical Simulations

#### 3.1. Nondimensionalization and Choice of Parameters

#### 3.2. Description of Tissue

#### 3.3. Initial Conditions

#### 3.4. Numerical Method

#### 3.5. Numerical Experiments

#### Experiment 1

#### Experiment 2

**Experiment 1**, but with ${\gamma}_{1}={\gamma}_{2}$, i.e., the flux-limited self-diffusion and pH-taxis being equally weighted. The results are illustrated in Figure 3. The behavior of solution components is qualitatively similar to the previous scenario, except the pseudopalisade-like patterns persist for a longer time; they are succeeded by a more uniform tumor, EC spread occupies the whole domain (also migrating into the formerly most hypoxic areas), and higher cell densities are exhibited.

#### Experiment 3

#### Experiment 4: Effects of flux limitations

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Evolution of the solution components at several times in the framework of Experiment 1 (full model: Model (25)).

**Figure 3.**Evolution of the solution components at several times in the framework of Experiment 2 (equally weighted influence of flux-limited repellent pH-taxis and self-diffusion).

**Figure 4.**Experiment 3: effect of angiogenesis. Left columns: Difference between tumor density and acidity concentration, as obtained with full System (25), and those computed from the variant Model (29) with proton uptake rate ${\zeta}_{h}=3.6\times {10}^{-8}$/h. Right columns: Same differences, but between computations done with Equation (25) and with Equation (30).

**Figure 5.**Experiment 4: effect of flux-limited motility terms. Differences between tumor density and proton concentration computed with Model (25) (in the parameter setting of Experiment 2) and those obtained for a model with glioma motility terms only involving myopic diffusion and pH-taxis without flux saturation (i.e., with glioma dynamics as in Model (31)).

**Figure 6.**1D patterns for two different choices of the glioma diffusion coefficient ${D}_{T}$. 2nd row: tumor patterns for Model (25). 3rd row: tumor patterns when Equation (25a) is replaced by Model (31). 4th row: differences between respective patterns in rows 2 and 3. 5th and 6th rows: differences between acidity and EC densities computed with Model (25) and with Model (31) replacing Equation (25a).

**Figure 7.**Differences between tumor and acidity patterns (1D) for the model comparison done in the left column of Figure 4 (framework of Experiment 3). Different choices are made for the glioma diffusion coefficient ${D}_{T}\left(x\right)$, as shown in the rightmost column.

Parameter | Meaning | Value | Reference |
---|---|---|---|

${K}_{M}$ | glioma carrying capacity | 0.3–0.8 $\phantom{\rule{4.pt}{0ex}}\mathrm{cells}/\mathsf{\mu}{\mathrm{m}}^{2}$ | this work, [35] |

${K}_{h}$ | acidity threshold for cancer cell death | ${10}^{-6.4}\phantom{\rule{4.pt}{0ex}}\mathrm{mol}/\mathrm{L}$ | [36] |

s | speed of glioma cells | 15–20 $\mathsf{\mu}\mathrm{m}/\mathrm{h}$ | [37,38] |

$\lambda $ | turning frequency coefficient | 360/h | [26,39] |

${\gamma}_{h}$ | proton production rate | $3.6\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ mol/(L·h) | this work, [40] |

${\zeta}_{h}$ | proton removal rate | $3.6\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$–3.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$/h | this work |

${D}_{h}$ | acidity diffusion coefficient | $1.8\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$–3.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$$\mathsf{\mu}$m${}^{2}$/h | this work, [17] |

${\mathsf{\mu}}_{1}$ | glioma growth/decay rate, influenced by acidity | (4.16–8.3) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$/h | [41,42] |

${\mathsf{\mu}}_{2}$ | glioma growth rate, influenced by ECs | (3–3.95) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$/h | this work |

$\alpha $ | advection constant | ${10}^{3}$ | [23] |

${\gamma}_{1}$ | weight coefficient related to acidic stress | $3\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$–3 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$/$\mathsf{\mu}$m | this work |

${\gamma}_{2}$ | weight coefficient related to cell population pressure | ${10}^{-4}$–${10}^{-3}$/$\mathsf{\mu}$m | this work |

$\sigma $ | speed of ECs | 15–20 $\mathsf{\mu}$m/h | [22,43] |

${\eta}_{0}$ | turning rate of ECs | 36/h | this work |

${\mathsf{\mu}}_{w}$ | EC proliferation rate | (1.25–2.08) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$/h | this work, [22] |

${\chi}_{a}$ | coefficient of chemotactic sensitivity of ECs | 0.72–21.6 h | this work |

${K}_{W}$ | carrying capacity of ECs | 0.1–0.3 cells/$\mathsf{\mu}$m${}^{2}$ | this work, [44] |

${D}_{g}$ | diffusion coefficient of VEGF | $3.6\phantom{\rule{3.33333pt}{0ex}}\times $${10}^{2}$–$1.04\times {10}^{5}$$\mathsf{\mu}$m${}^{2}$/h | [45,46] |

${K}_{g}$ | VEGF threshold value | $8\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$–${10}^{-6}$ mol/L | this work, [47] |

${\gamma}_{g}$ | VEGF production rate | $3.6\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ mol/(L·h) | [45] |

${\zeta}_{g}$ | VEGF uptake rate | $3.6\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$–3.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$/h | this work |

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**MDPI and ACS Style**

Kumar, P.; Surulescu, C.
A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis. *Symmetry* **2020**, *12*, 1870.
https://doi.org/10.3390/sym12111870

**AMA Style**

Kumar P, Surulescu C.
A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis. *Symmetry*. 2020; 12(11):1870.
https://doi.org/10.3390/sym12111870

**Chicago/Turabian Style**

Kumar, Pawan, and Christina Surulescu.
2020. "A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis" *Symmetry* 12, no. 11: 1870.
https://doi.org/10.3390/sym12111870