# Autologous Gradient Formation under Differential Interstitial Fluid Flow Environments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Assumptions of the Model

_{rel}term were guessed. These parameters have not been well-studied and should be experimentally determined for exact values in the future, especially because a sensitivity analysis reveals that k

_{rel}accounts for large fluctuations in model outcome (Supplemental Figure S3). In addition, we only model one protease, MMP9, and one chemokine, CXCL12. There are many different proteases that may be at work, however, to impact the gradient formation based on ECM degradation and CXCL12 inactivation [31] and there exist multiple isoforms of CXCL12 with different binding affinities for GAGs which are not taken into account in this model. Lastly, the control volume that we have chosen was assigned uniform parameters for porosity and permeability which do not change in relation to the protease in contrast to what would be seen with ECM remodeling in vitro or in vivo. We do not include a term for the protease degradation and we do not take into account any interaction between the cell and the chemokine such as recycling effects or receptor binding. Assumptions governing the fluid flow are in line with other models, namely that the fluid flow follows Darcy’s law (Brinkman form). Assumptions regarding the chemokine and protease are generally similar to other papers looking at autologous chemotaxis [34], but some of the aforementioned CXCL12 assumptions differ [37]. More detailed information specific to each manipulation can be found in the Supplemental Methods.

#### 2.2. Equations and Implementation of the Model

_{on}and k

_{off}values, experimentally determined by Munson et al. [18]. In addition the cell secretes a protease, which we are modeling as MMP9, that acts to cleave the bound CXCL12 [32] and thus inactivate it. This inactivation in the model is handled by assuming a k

_{rel}term that functions to decrease the bound CXCL12 concentration, such that the inactivated CXCL12 will not appear in the simulated results of the chemokine gradient. The protease is an enzyme in the model, meaning it does not get altered by its proteolysis function and so its reaction is zero. The equations are shown below:

_{m}is the mass transfer coefficient, D is the diffusion coefficient, D

_{sphere}is the diameter of the sphere, ρ is the fluid density, µ is the fluid viscosity, and V is the average fluid velocity.

_{i}is the concentration taken at the respective point on the surface of the sphere, either upstream or downstream of flow. All parameters and their values for the model can be found in Table 1.

#### 2.3. Conditions and Quantitative Values Used in the Model

^{−5}m/s) and the outlet being a uniform pressure (p = 0 Pa). From the mass transport side, a Neumann boundary condition is applied around the outside of the cell representing secretion of the chemokine CXCL12 and the protease MMP9. An open boundary condition is specified for all of the outer walls of the control volume where convective inflow and outflow of the chemical species can occur. This represents an open space where the chemokine can disperse as if the surrounding space is a large volume, which is applicable for our system. Reactions (Equations (5)–(7)) are prescribed to take place within the control volume. Initial values for the chemokine and protease were set to 100 nM and 1 nM, respectively. Geometry of the figure is set up to include a rectangular prism 100 µm × 50 µm × 50 µm (L × W × H) corresponding to a control volume encompassing a cell, which is modeled as a sphere with diameter of 10 µm. The sphere is placed 25 µm from the inlet boundary. Meshing was done through the COMSOL software to apply a free tetrahedral mesh to the geometry. A predefined mesh calibrated for fluid dynamics was used with a ‘finer’ element size applied for each domain based on a mesh refinement analysis (Supplemental Figure S2). For specific model conditions applicable to individual figures, refer to the supplementary materials.

## 3. Results

#### 3.1. Effects of Transport Parameter Changes on Pericellular Gradients

_{rel}term is increased we see a lesser concentration of bound CXCL12 and eventually a disruption of the gradient as the bound CXCL12 is mostly cleaved and inactivated. Changing the k

_{on}or k

_{off}values impacts the magnitude of the bound CXCL12–increasing k

_{on}will increase the bound CXCL12 and increasing k

_{off}will decrease the bound CXCL12 as expected. Lastly, for velocity we saw that higher velocities led to steeper differences of concentration upstream and downstream of the cell (Figure 1F). As velocity is increased, so too is the % concentration as the chemokine is affected by the convection of fluid flow. The difference across the cell can be used to calculate a concentration gradient (Figure 1G), which can be used to simply describe the steepness of the gradient across the cell and thus the gradient that a cell may feel. This value is denoted as % concentration and will be used throughout the manuscript. As expected, we see a significant correlation (r = 0.991, p < 0.05) between the velocity and the % concentration (Figure 1H).

#### 3.2. Directionality of Flow Alters Gradient

#### 3.3. Temporal Fluctuations in Velocity Yield Variable Gradients

#### 3.4. Background Concentration Can Negate Bound CXCL12 Gradient

^{3}(Figure 4C). The position of the cell is also important where background concentration is concerned. As the cell is modeled farther from the source of the background concentration, the pericellular gradient becomes less disrupted to the point that at 70 µm from the source the autologous gradient is still observed for the length of time that the background concentration is applied (Figure 4D). If the cell is closer than 70 μm the background concentration completely abrogates pericellular gradient formation and actually causes a higher concentration of bound CXCL12 to develop upstream of the cell. If the background concentration is removed, the cell reaches a new set point of gradient formation which is notably lower than the starting % concentration the closer the cell is to the source.

#### 3.5. The Invading Cell Needs to Be a Certain Distance from Tumor Border for Gradient to Develop

#### 3.6. Cell Type and Morphology Affects Gradient Formation

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

_{rel}. Figure S4: Parametric sweeps of input variables and their corresponding effects on % concentration. Figure S5: Additional multidirectional flow modeling. Figure S6: Time effect on gradient formation for baseline condition. Figure S7: Cell size and orientation impact gradient formation. Supplemental Methods.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Modulation of transport parameters yields expected responses in gradient formation. (

**A**) Representative images of three different diffusion coefficient combinations which create varying bound CXCL12 gradients around the cell. (

**B**) Quantification of % concentration (bound CXCL12) around the cell. (

**C**) Representative images of three combinations of reaction coefficients showing the differential response of gradient formation. (

**D**) Quantification of bound CXCL12 around the cell with varying reaction coefficients. (

**E**) Representative images of gradient formation with increasing velocity. (

**F**) Quantification of % concentration at varying velocities. (

**G**) Schematic of model and % concentration calculation. (

**H**) Regression of relationship between % concentration and velocity, showing positive correlation between the two.

**Figure 2.**Multidirectional flow and its impact on gradient formation around a single cell. (

**A**) MRI analysis of interstitial fluid flow magnitude and direction overlayed on histological sample showing flow heterogeneity and two specific regions of interest (1) and (2). Scale bar 0.5 mm. (

**B**) Schematic of flow directed evenly on the top and bottom of the cell (

**left**) and resulting flow pathways overlayed on top of bound CXCL12 gradient (

**right**) around a single cell (10µm diameter). (

**C**) Schematic of flow directed above and to the left of the cell (

**left**) and resulting flow pathway (magenta arrows) with corresponding resultant of flow direction (green arrow) overlayed on bound CXCL12 gradient (

**right**).

**Figure 3.**Transient solutions of gradient formation around single cells based on physiological time-dependent changes to superficial flow rate. (

**A**) Schematic of CED increasing fluid flow (

**top**) followed by the resulting concentration gradient over time (

**middle**), and the ramp function input into COMSOL (

**bottom**). (

**B**) Schematic of surgical resection impact of fluid flow (

**top**), the resulting concentration gradient over time (

**middle**), and the pulse function input into COMSOL (

**bottom**). (

**C**) Schematic of oscillating CSF flow coming from the Virchow-Robin space into the parenchyma and interacting with tumor cells (

**top**), the resulting concentration gradient over time (

**middle**), and the oscillating function input into COMSOL (

**bottom**).

**Figure 4.**Background concentration reduces the concentration gradient around single cell. (

**A**) Schematic of background concentration around a secreting cell (

**left**) and timelapse video of background concentration added at 150 cs (

**right**). Note that timelapse images are not to the same scale in order to better show background concentration. (

**B**) Modulating cellular secretion rate of CXCL12 while keeping a constant background CXCL12 concentration of 100 nM. (

**C**) Amount of background bound CXCL12 needed to decrease the cellular gradient below 5%. (

**D**) Impact of distance between cell and source of CXCL12 background on gradient formation.

**Figure 5.**Concentration gradients felt by an invading cell beyond the tumor border at varying distance (

**A**) Immunohistochemistry of tumor border and invading cell (green) with CXCR4 (red) and CXCL12 (blue) (

**left**) and schematic representing model parameters of interest (

**right**). Scale bar = 50 µm (

**B**) Representative image of gradient formation of cell with tumor border CXCL12 secretion. (

**C**) Quantification of gradient formation around invading cell depending on distance from tumor border and velocity of IFF.

**Figure 6.**Concentration gradients depend on location along a morphologically accurate tumor cell. (

**A**) Confocal image of GL261. (

**B**) 3D rendering and mesh used in the model. (

**C**) Representative images of cell imports into model and resulting concentration profiles. (

**D**) Quantification of % concentration at different points on the cell.

Parameter | Variable | Value | Unit | Source |
---|---|---|---|---|

Density of fluid | ρ | 1 | g/mL | Anne Lui, Can J Anaesth, 1998 [39] |

Dynamic viscosity of fluid | µ | 0.7–1 | mPa·s | Bloomfield, Pediatr Neurosurg, 1998 [40] |

Temperature | T | 310.15 | K | Physiological temp |

Inlet velocity | v | 0.1–100 | µm/s | Munson JM, Can Man and Res, 2014 [7] |

Porosity | ε | 0.3 | Linninger A, IEEE Trans. on Biomed. Eng. 2007 [41] | |

Permeability | κ | 1.00 × 10^{−11} | cm^{2} | Munson JM, Cancer Research 2013 [18] |

Diffusion coefficient, chemokine | D_cxcl12 | 120 | µm^{2}/s | Fleury M, Biophysics Journal, 2006 [34] |

Diffusion coefficient, protease | D_protease | 80 | µm^{2}/s | Fleury M, Biophysics Journal, 2006 [34] |

Mass transfer coefficient, chemokine | k_cxcl12 | 2.80 × 10^{−5} | m/s | Calculated Value |

Mass transfer coefficient, protease | k_protease | 1.60 × 10^{−5} | m/s | Calculated Value |

Bulk concentration, chemokine | bulk_cxcl12 | 100 | nM | Estimated Value |

Bulk concentration, protease | bulk_protease | 1 | nM | Estimated Value |

Heparan sulfate concentration | HS | 2.60 × 10^{−3} | mM | Estimated Value |

Radius of sphere | r | 5.00 × 10^{−6} | M | Approximate Cell Diameter |

Chemokine binding rate | k_on | 9.30 × 10^{4} | 1/(M·s) | Munson JM, Cancer Research 2013 [18] |

Chemokine unbinding rate | k_off | 1.16 × 10^{−5} | 1/s | Munson JM, Cancer Research 2013 [18] |

Chemokine release rate from protease | k_rel | 1.00 × 10^{4} | 1/(M·s) | Estimated Value |

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Stine, C.A.; Munson, J.M.
Autologous Gradient Formation under Differential Interstitial Fluid Flow Environments. *Biophysica* **2022**, *2*, 16-33.
https://doi.org/10.3390/biophysica2010003

**AMA Style**

Stine CA, Munson JM.
Autologous Gradient Formation under Differential Interstitial Fluid Flow Environments. *Biophysica*. 2022; 2(1):16-33.
https://doi.org/10.3390/biophysica2010003

**Chicago/Turabian Style**

Stine, Caleb A., and Jennifer M. Munson.
2022. "Autologous Gradient Formation under Differential Interstitial Fluid Flow Environments" *Biophysica* 2, no. 1: 16-33.
https://doi.org/10.3390/biophysica2010003