# An Image-Based Framework for the Analysis of the Murine Microvasculature: From Tissue Clarification to Computational Hemodynamics

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- 1.
- Fixation: Samples were fixed using paraformaldehyde (PFA) after transcardiacally mice perfusion and dissection before post-fixation with PFA.
- 2.
- Sectioning: Here, $500\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m thick brain slices were sectioned using a vibratome.
- 3.
- Clearing: Sections were cleared using the CUBIC protocol.
- 4.
- Staining: Delipidated sections were stained with FITC-Lectin and an arteriole-specific dye Alexa Fluor 633 hydrazide.
- 5.
- Imaging: Here, $500\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m slices were analyzed using an advanced two-photon microscopy.

#### 2.1. Fixation, Sectioning and Tissue Optical Clearing

#### 2.2. Tissue Staining

#### 2.3. Two-Photon Excitation Microscopy

#### 2.4. Image Analysis

#### Vessel Measurements

#### 2.5. One-Dimensional (1D) Modeling

#### 2.5.1. Governing Equations

#### 2.5.2. Boundary Conditions

- (a)
- Pressures were imposed at the inlet and outlets. With that, there was no need to know the flow direction in all in- and outflows respectively, as the flow direction in the segments adjusted to fulfill the pressure boundary conditions.
- (b)
- Boundary nodes (1 segment nodes) inside the geometries limits were assigned with a zero flow condition and with zero hematocrit. These nodes show the presence of broken vessels inside the geometry that could be produced during the segmentation. It is important to notice that these vessels have no physiological meaning but need to be treated.
- (c)
- Three different sets of pressure boundary conditions were assigned depending on the segment to which the boundary node was attached to: venule, arteriole or capillary:
- 1.
- At the arterial inflow, a pressure of 50 mmHg was given. The arterial pressure outflow was set to 40 or 45 mmHg depending on its nearness to the inflow. With that, the risk of a short circuit was eliminated.
- 2.
- At the venular outflows, a pressure of 10 mmHg was given.
- 3.
- In the capillary in/outflows, two cases were studied, following Lorthois and coworkers [5]:
- Case 1:
- Zero flow condition: Flow is set to zero in all the capillary outflows. In this case, the flow goes from the arterial inlet passing through the whole geometry until it reaches a venular outlet. As reported [5], this condition would underestimate the flow in the geometry as it isolates it from its virtual neighbors.
- Case 2:
- Constant pressure condition: A constant capillary boundary pressure was calculated so that the net capillary flow (the sum of the flow in all the inlets and outlets) was zero; thus, everything that enters through the arterioles exits through the venules. In other words, this pressure was adjusted such that the total flow entering the arteriolar network was the same as the total flow entering the venular network. In this way, the net flux to all the boundary capillary segments was zero. As a consequence, the net flux leaving the studied brain region through capillaries to supply neighboring areas was exactly compensated by the net flux arriving from neighboring areas through capillaries. As shown in the literature, this condition forces the flux lines to be perpendicular to the ends of the computational domain, maximizing the exchanges of fluid with the neighboring region. For this reason, this condition overestimates the flow in the geometry as it maximizes the flow exchange between the region itself and its virtual neighbors. [64].

- (d)
- The boundary hematocrits values were set $0.45$ at the arterial inlets, $0.4$ at the arterial outlets, $0.2$ at the venular outlets and a random value between $0.2$ and $0.6$ for the rest of the capillary boundary nodes, mimicking the chaotic and haeterogenic nature of this variable in the capillary beds [5,64].

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Tissue staining: The tissue was divided in cubic samples of $500\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m: (

**a**) endothelial labeling with lectin; (

**b**) cubic samples; (

**c**) close-up view of the single cubic sample box in red in (

**b**).

**Figure 3.**Image treatment: (

**a**) Original images (

**left**) and result of filtering (

**right**). (

**b**) Zoomed versions extracted from (

**a**). Original slice (

**up**), filtered slice (

**down**). (

**c**) Segmented volumen. (

**d**) Result after artifacts erasing.

**Figure 5.**Morphology of the 5 cerebrovascular regions of the murine cortex considered in this study. The heat map represents by color the distribution of the value of the diameters within the microcirculatory synthetic network.

**Figure 6.**Statistical analysis of the considered regions: cumulative distribution functions of diameters (

**a**), lengths (

**b**), surface areas (

**c**), volumes (

**d**) and tortuosity (

**e**).

**Figure 7.**Computed blood flow (in [nL/min]) within the 5 cerebrovascular regions of the murine cortex. The heat map (in logarithmic scale) represents by colors the average blood flow distribution within the microcirculatory synthetic network.

**Figure 8.**Computed hematocrit (in [%]) within the 5 cerebrovascular regions of the murine cortex. The heat map represents by colors the hematocrit percentage per segment within the microcirculatory synthetic network.

**Figure 9.**Computed endothelial shear stress of the 5 cerebrovascular regions of the murine cortex. The heat map represents by colors the average value of the shear stress per segment within the microcirculatory synthetic network.

Region | Nodes | Boundary Nodes | Segments | Dimensions ($\mathsf{\mu}$m) |
---|---|---|---|---|

$\#1$ | 1250 | 210 | 1561 | $698\times 459\times 310$ |

$\#2$ | 751 | 136 | 932 | $466\times 432\times 207$ |

$\#3$ | 968 | 134 | 1265 | $559\times 365\times 318$ |

$\#4$ | 948 | 198 | 1164 | $517\times 532\times 220$ |

$\#5$ | 999 | 72 | 1292 | $508\times 383\times 345$ |

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## Share and Cite

**MDPI and ACS Style**

Mañosas, S.; Sanz, A.; Ederra, C.; Urbiola, A.; Rojas-de-Miguel, E.; Ostiz, A.; Cortés-Domínguez, I.; Ramírez, N.; Ortíz-de-Solórzano, C.; Villanueva, A.;
et al. An Image-Based Framework for the Analysis of the Murine Microvasculature: From Tissue Clarification to Computational Hemodynamics. *Mathematics* **2022**, *10*, 4593.
https://doi.org/10.3390/math10234593

**AMA Style**

Mañosas S, Sanz A, Ederra C, Urbiola A, Rojas-de-Miguel E, Ostiz A, Cortés-Domínguez I, Ramírez N, Ortíz-de-Solórzano C, Villanueva A,
et al. An Image-Based Framework for the Analysis of the Murine Microvasculature: From Tissue Clarification to Computational Hemodynamics. *Mathematics*. 2022; 10(23):4593.
https://doi.org/10.3390/math10234593

**Chicago/Turabian Style**

Mañosas, Santiago, Aritz Sanz, Cristina Ederra, Ainhoa Urbiola, Elvira Rojas-de-Miguel, Ainhoa Ostiz, Iván Cortés-Domínguez, Natalia Ramírez, Carlos Ortíz-de-Solórzano, Arantxa Villanueva,
and et al. 2022. "An Image-Based Framework for the Analysis of the Murine Microvasculature: From Tissue Clarification to Computational Hemodynamics" *Mathematics* 10, no. 23: 4593.
https://doi.org/10.3390/math10234593