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

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.

Appendix

Computing the Collision Rate $\hat{\beta }_{\text{PT}}$

We use the spherical coordinates to express all the related parameters so as to evaluate the integral in Eq. (8) with a given velocity profile. In our rectangular coordinates, we suppose the chamber cross section (Fig. 3) is in yz-plane and the x-axis points to the reader, so that

$$ v = \left( {v_{x} ,v_{y} ,v_{z} } \right)\quad {\text{and}}\quad v_{x} = 0,\;v_{y} = v_{c} ,\;v_{z} = - v_{s} . $$

We take the standard spherical coordinate system with the origin being the center of the sphere where the arc shaped PMN lies on and the two bottom points of PMN having spherical coordinates $\left( {r,\varphi_{0} ,{\frac{\pi }{2}}} \right)$ and $\left( {r,\varphi_{0} ,{\frac{3\pi }{2}}} \right).$ Assume that the deformable PMN preserves its volume $V = {\frac{4}{3}}\pi r_{\text{p}}^{3} ,$ where r _p is the radius of an undeformed PMN, we thus have that

$$ V = {\frac{2}{3}}\pi \left( {1 - { \cos }\,\varphi_{0} } \right)r^{3} $$

which gives a closed form for H, r and $\varphi_{0}$ via the following explicit formulae:

$$ r = {\frac{V}{{\pi H^{3} }}} + {\frac{H}{3}}\quad {\text{and}}\quad \varphi_{0} = { \arccos }\left( {1 - {\frac{H}{r}}} \right). $$

Now consider a tumor cell which is colliding with this PMN. Assume that the contact point has coordinate (r + r _t, Θ, φ), where r _t is the radius of a tumor cell. Thus, h, the distance between the center of tumor cell and the substrate, is

$$ h = \left( {r + r_{\rm t} } \right){ \cos }\,\varphi - r\,{ \cos }\,\varphi_{0} . $$

Our task is to compute the coagulation kernel under different flow conditions. For any given parameters $\dot{\gamma }$ and μ, we compute H by the fitted function f in Eq. (16) first, and compute φ₀ and r next. We may then plug in all these parameters and the flow velocity profile into the Eq. (8) to compute the kernel by estimating the integral over the collision region Ω. That is, we estimate

$$ \hat{\beta }_{\text{PT}} = - \int\limits_{0}^{\pi } {\int\limits_{0}^{2\pi } {F(\varphi ,\theta )|_{{F \le 0,h \ge r_{\text{t}} }} d\varphi d\theta } } $$

where the integrand is given by

$$ F\left( {\varphi ,\theta } \right) = { \sin }\,\varphi (r + r_{\text{t}} )^{2} \left( {{ \sin }\,\varphi \,{ \sin }\,\theta v_{y} + { \cos }\,\varphi v_{z} } \right). $$

This integral is evaluated numerically via quadrature rules. The parameters that we need in the calculation are listed in Table 6.

Table 6 Values for specific parameters

Sensitivity to the Maximum Number of Bonds N

The algebraic system

$$ \begin{aligned} - AP_{0} + P_{1} = 0 \hfill \\ AP_{n - 1} - (A + n)P_{n} + (n + 1)P_{n + 1} = 0 \hfill \\ - AP_{N - 1} - NP_{N} = 0 \hfill \\ \end{aligned} $$

has a general solution $P_{n} = {\frac{{A^{n} P_{0} }}{n!}},$ for n = 1, 2, 3,…, with A and P ₀ satisfying

$$ \sum\limits_{n = 0}^{N} {P_{n} } = P_{0} \sum\limits_{n = 0}^{N} {{\frac{{A^{n} }}{n!}} = 1} . $$

Under a given flow condition, the reaction constant A is fixed, by taking $N \to \infty ,$

$$ \sum\limits_{n = 0}^{\infty } {P_{n} = P_{0} \sum\limits_{n = 0}^{\infty } {{\frac{{A^{n} }}{n!}}} = P_{0} e^{A} = 1} . $$

Therefore, the solution $P_{n}$ converges as $N \to \infty$ to ${\frac{{e^{ - A} A^{n} }}{n!}}$. By numerical simulation, we could actually find that when N ≥ 300, the solution and the limit are almost identical (Fig. 13). So we can truncate the original system to N = 300.

Figure 13

The steady state solution distribution of adhesion efficiency with the system size taken to be 300, 500, and 1000