# A New Chemotactic Mechanism Governs Long-Range Angiogenesis Induced by Patching an Arterial Graft into a Vein

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. VEGF Distribution Model

#### 2.2. VEGFA Quantification in In Vivo Angiogenesis Model

#### 2.3. VEGFA Distribution Equation Based on In Vivo Quantification Data

#### 2.4. New Blood Vessel Length Measurement in In Vivo Angiogenesis Model

#### 2.5. Sprout Tip Distribution Equation with New Chemotactic Velocity

## 3. Discussion

## 4. Material and Methods

#### 4.1. Numerical Simulation Method and Results

#### 4.1.1. Derivation of Numerical Scheme

#### 4.1.2. Numerical Simulation Results

#### 4.2. Material and Methods: In Vivo

#### 4.2.1. Quantification of Vegfa in In Vivo Angiogenesis Model

#### 4.2.2. Measurement of New Blood Vessel Length

#### 4.2.3. Numerical Simulation

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

VEGF | Vascular endothelial growth factor |

AVF | Arteriovenous fistula |

AVM | Arteriovenous malformation |

VEGFR | Vascular endothelial growth factor receptor |

mm | Millimeter |

SD | Standard deviation |

MN | Minnesota |

## Appendix A

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**Figure 1.**VEGF distribution at the tissue area and approximation curve of VEGF concentration from day 0 to 14 in in vivo experimental. (

**a**) An illustration of the eight tissue areas (Fp, F1, F2, F3, F4, F5, F6, and F7) that have been taken to measure the VEGFA concentration. The arterial patch graft is placed at Fp and the pre-existing artery is assumed to exist at around F4/F5 areas. (

**b**) VEGF concentration from experimental data showing that the VEGF concentration was higher in the pre-existing area (F4/5) than in the patch area (Fp). (

**c**) The measured VEGF concentrations (pg/mL) at the pre-existing vessel change with time. This indicates that there were three rabbit cases for each time point. We plotted each case on the graph, and its fitted graph was derived. The curve was obtained by numerically fitting the data to Equation (4): $a=46.2525$, $b=0.2131$, $\sigma =0.1548$, $\gamma =0.3485$.

**Figure 2.**(

**a**) An illustration of VEGFA concentration changes over time. The level concentration of VEGFA changes according to $\{b+sin(2\pi \sigma t)exp(-\gamma t\left)\right\}$ term in Equation (4). The VEGFA concentration is always higher near the pre-existing blood vessels other than in the other area. This illustration is based on the experimental data in Section 2.2. (

**b**) VEGFA concentration profile using Equations (4) and (5) at F4/F5 by setting parameters as follows: $x,y\in (0,1)$, $a=1$, $b=0.39$, $\sigma =0.23/2$, $\gamma =0.15$, $\u03f5=2.9$, ${x}_{p}=0$, ${y}_{p}=0.5$, $x=1$, and $y=0.5$. This VEGFA concentration changes graph has a qualitatively similar profile as that in vivo. The VEGFA concentration changes graph has a similar profile as in vivo experiments (see Figure 1b). (

**c**) VEGF distribution of Equations (4) and (5) on domain $x,y\in (0,1)$ at $t=0$.

**Figure 3.**(

**a**) New blood vessels (yellow arrowheads) are sprouting and growing from arterioles (red arrowheads). The measurement distance is between the outer circumference of the arteriole and the farthest point of the new blood vessel. The specimen was stained with Elastica-Masson staining. Scale bar: 250 $\mathsf{\mu}$m. (

**b**) Graph of the average length of the new blood vessel from day 0 to day 14. At each time point, every fifteen lengths were measured in three cases, and the average was calculated.

**Figure 4.**(

**a**) The numerical lattices were defined on domain $\mathrm{\Omega}=(0,1)\times (0,1))$. We defined two different lattices: the main lattice (blue) and sub-lattice (red). (

**b**) Lattices for the calculation of scheme ${\left({v}_{m}\right)}_{i,j}^{k}$. (

**c**) Lattices for the calculation of scheme ${G}_{i,j}^{k}$. (

**d**) We may shift the weight values ${P}_{i+1,j}^{k,1}$, ${P}_{i-1}^{k,2}$, ${P}_{i,j+1}^{k,3}$, and ${P}_{i,j-1}^{k,4}$ to ${P}_{i,j}^{k,1}$, ${P}_{i,j}^{k,2}$, ${P}_{i,j}^{k,3}$, and ${P}_{i,j}^{k,4}$, so that each of these P on $(i,j)$ can be assumed as the probability of a sprout tip to migrate to the left, right, up, and bottom, respectively. Additionally, ${P}_{i,j}^{k,0}$ is regarded as the probability of a sprout tip to stay on $(i,j)$.

**Figure 5.**The simulation results of the new blood vessel growth with the background of VEGF concentration level, captured on days $1,2,7,10,11$, and 14. Until $t=2$, the new blood vessels grew faster before the sprout tips experience the rest period. After $t=7$, the new blood vessels regrew until reaching the periphery of the arterial patch at $t=10$. Only one blood vessel can reach the arterial patch due to anastomosis.

**Figure 6.**Average changes of new vessels’ length growth speed until it reaches the periphery of the arterial patch between in vivo and simulations from day 0 until day 14. (

**a**) The average speed of vessel length growth in mm based on in vivo experiments. (

**b**) The average speed of vessel length growth in mm based on simulations.

**Figure 7.**Error bars show the mean and standard deviation of the in vivo and simulations of the sprouting new blood vessels based on day 0 until day 14 until it reached the periphery of the arterial patch. (

**a**) Mean and standard deviation of vessel length growth in mm based on in vivo experiments. (

**b**) Mean and standard deviation of vessel length growth in mm based on simulations.

**Figure 8.**VEGF distribution at $t=0$ with the pre-existing vessel on the right and arterial patch on the left boundary.

**Figure 9.**Comparison of the new blood vessel growth and the VEGFA gradient changes. When the VEGFA concentration is decreasing over time, the growth of the new vessels become slower. We consider this growth of the new blood vessels at the decreasing rate of VEGFA to be called the rest period, ${R}_{p}$, and at the increasing rate of VEGFA is called the growth period, ${G}_{p}$. (

**a**) VEGFA concentration changing over time (days) and attenuates gradually. (

**b**) Measurement of the new blood vessel in days.

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**MDPI and ACS Style**

Minerva, D.; Othman, N.L.; Nakazawa, T.; Ito, Y.; Yoshida, M.; Goto, A.; Suzuki, T.
A New Chemotactic Mechanism Governs Long-Range Angiogenesis Induced by Patching an Arterial Graft into a Vein. *Int. J. Mol. Sci.* **2022**, *23*, 11208.
https://doi.org/10.3390/ijms231911208

**AMA Style**

Minerva D, Othman NL, Nakazawa T, Ito Y, Yoshida M, Goto A, Suzuki T.
A New Chemotactic Mechanism Governs Long-Range Angiogenesis Induced by Patching an Arterial Graft into a Vein. *International Journal of Molecular Sciences*. 2022; 23(19):11208.
https://doi.org/10.3390/ijms231911208

**Chicago/Turabian Style**

Minerva, Dhisa, Nuha Loling Othman, Takashi Nakazawa, Yukinobu Ito, Makoto Yoshida, Akiteru Goto, and Takashi Suzuki.
2022. "A New Chemotactic Mechanism Governs Long-Range Angiogenesis Induced by Patching an Arterial Graft into a Vein" *International Journal of Molecular Sciences* 23, no. 19: 11208.
https://doi.org/10.3390/ijms231911208