# Application of Population Dynamics to Study Heterotypic Cell Aggregations in the Near-Wall Region of a Shear Flow

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## Abstract

Our research focused on the polymorphonuclear neutrophils (PMNs) tethering to the vascular endothelial cells (EC) and the subsequent melanoma cell emboli formation in a shear flow, an important process of tumor cell extravasation from the circulation during metastasis. We applied population balance model based on Smoluchowski coagulation equation to study the heterotypic aggregation between PMNs and melanoma cells in the near-wall region of an *in vitro* parallel-plate flow chamber, which simulates *in vivo* cell-substrate adhesion from the vasculatures by combining mathematical modeling and numerical simulations with experimental observations. To the best of our knowledge, a multiscale near-wall aggregation model was developed, for the first time, which incorporated the effects of both cell deformation and general ratios of heterotypic cells on the cell aggregation process. Quantitative agreement was found between numerical predictions and *in vitro* experiments. The effects of factors, including: intrinsic binding molecule properties, near-wall heterotypic cell concentrations, and cell deformations on the coagulation process, are discussed. Several parameter identification approaches are proposed and validated which, in turn, demonstrate the importance of the reaction coefficient and the critical bond number on the aggregation process.

## Keywords

Melanoma cells Leukocytes Endothelium Collisions Aggregates Probabilities Cell adhesion Shear conditions## Notations

*A*The reaction constant as a combination of

*A*_{c},*N*_{r}*, N*_{l}*, K*_{on}and*K*_{off}(s^{−1})*A*_{c}The contact area between a PMN and a tumor cell (

*μ*m^{2})*C*_{a}*, We, Re*The Capillary number, the Weber number and the Reynolds number

- Coll
_{n} The collision number between two kinds of cells (events)

*C*_{P},*C*_{T},*C*The concentration of PMNs, tumor cells and bulk cell concentration, respectively, in the near-wall region (cells mL

^{−1})*d*The diameter of an undeformed PMN cell (

*μ*m)*g*Acceleration due to gravity (m s

^{−2})*H*,*H*_{dep},*h*The height of a deformed PMN cell attached to the substrate, the height defined by the depth of field of the microscope objective used in the experimental observations and the distance from the substrate to the center of the cell, which represented the position of the cell (

*μ*m)*I*_{ini}*, I*_{tf}The increasing multiple of initial condition or tethering frequency

*J*The incoming flux to collision region near PMN per concentration (kg

^{−1}m^{3})*K*_{on},*K*_{on}^{(n)}The forward reaction rate per unit density for bond formation (s

^{−1}*μ*m^{−2})*K*_{off},*K*_{off}^{(n)}The backward reaction rate per cell for bond breakage (s

^{−1})*n*Outward unit normal (vector) of collision region

*n**The approximation of the smallest number of bonds required for firm adhesion (bonds)

*N*_{r},*N*_{l}The concentration of receptor and ligand on cells (

*μ*m^{−2})*N*_{PT}*, N*_{P}The number of tethered PMN–tumor cell doublets and tethered PMN monomers on the substrate (cells)

*P*_{n}(*t*),*P*_{n}The probability of having

*n*bonds.*r*_{p}*, r*_{t}The radius of an undeformed PMN cell and a tumor cell (

*μ*m)*T*_{f}The tethering frequency of cells: including firmly adhered cell and rolling cells (events s

^{−1}per view)*v*_{avg},*v*_{rel}The average settling velocity for cells above height

*H*_{dep}and the relative velocity (difference between velocities) of two kinds of cells (*μ*m s^{−1})*v*_{s}^{0},*v*_{s},*v*_{c}The free settling velocity, settling velocity and convection velocity of cells in the parabolic flow profile (

*μ*m s^{−1})*α*The aggregation percentage

*β*_{PT},*β*(*i*,*j*;*i*′,*j*′)The coagulation kernel of tumor cells in the near-wall region to the tethered PMNs (

*μ*m^{3}s^{−1})- \( \hat{\beta }_{\text{PT}} ,\hat{\beta } \)
The collision rate between tethered PMNs and tumor cell monomers (

*μ*m^{3}s^{−1})- \( \dot{\gamma } \)
The shear rate in the close-wall region (s

^{−1})*є*_{PT}The adhesion efficiency between tethered PMNs and tumor cell monomers

*μ*The viscosity of the fluid flow (Poise)

- ρ
_{m}, ρ_{t} The fluid density, the tumor cell density (kg m

^{−3})*σ*Membrane tension (N m

^{−1})- Ψ
The concentration of cells within the region of height

*H*_{dep}- Ω
The collision region

## Notes

### Acknowledgments

The authors thank Dr. Meghan Hoskins for providing simulation data on hydrodynamic force. This work was supported by the National Institutes of Health grant CA-125707 and National Science Foundation grant CBET-0729091.

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