Effect of circulating tumor cell aggregate configuration on hemodynamic transport and wall contact

Abstract

Selectin-mediated adhesion of circulating tumor cells (CTCs) to the endothelium is a critical step in cancer metastasis, a major factor contributing to the mortality of cancer. The formation of tethers between tumor cells and endothelial selectins initiates cell rolling, which can lead to firm adhesion, extravasation and the formation of secondary metastases. Tumor cells travel through the bloodstream as single cells, or as aggregates known as circulating tumor microemboli (CTM). CTM have increased survivability and metastatic potential relative to CTCs, and the presence of CTM is associated with worse patient prognosis. The motion of cells and cellular aggregates in flow is a function of their size and shape, and these differences influence the frequency and strength of their contact with the endothelium. In this study, a computational model consisting of the hydrodynamic component of the Multiparticle Adhesive Dynamics simulation analyzed the effects of model aggregate conformation and orientation on adhesive binding potential. Model aggregates of the Colo205 colorectal cancer cell line were created, consisting of two, three, and four cells in simple geometrical conformations. Contact time, contact area, and time integral of contact area were measured as a function of fluid shear rate, initial centroid height, and initial orientation for model aggregates that experienced hydrodynamic collisions with the plane wall. It was found that larger CTM conformations with intermediate nonsphericities had the highest adhesion potential. The results of this study shed light on the correlation between environmental conditions and extravasation efficiency, which could inform the development of new anti-metastatic drugs.

Keywords: Cancer metastasis; Circulating tumor cell; Circulating tumor microemboli; Hydrodynamic collision.

Fig. 1.

Schematic diagram of Multiparticle Adhesive Dynamics simulation. Linear shear flow brings the aggregate within a reactive distance of the plane wall, at which point a hydrodynamic collision occurs. The major axis, minor axis, and centroid height of the aggregate are highlighted, and the particle surface roughness (ε_w) and plane surface roughness (ε_w) layers are shown. Reactive distance is defined here as any distance between the roughness layers ≤ 20 nm, the equilibrium length of a sLe^x/E-selectin bond.

Fig. 2.

Generation of model circulating tumor microemboli. Model aggregates used in the MAD simulation were created by modifying a 96-element QUAD9 sphere mesh. The four central elements of a given side of the sphere were removed (A), and that sphere was then duplicated. The duplicated sphere was then reoriented so that the open faces of each sphere were aligned (B). The nodes along the open face of each sphere were then merged so that the resulting aggregate contained one continuous set of nodes (C). This process could be repeated for any face of any sphere, in order to create the desired aggregates. The nodes could also be manually adjusted in order to alter the geometry of the aggregate.

Fig. 3.

QUAD9 element models of Colo205 aggregates. QUAD9 element meshes were generated for doublet, triplet, and 4-mer aggregates using a MATLAB algorithm developed to create particle geometries using a 96 element sphere mesh as a template. Aggregates were created as clusters of spheres of equal size, and then the nodal coordinates were adjusted so that the model particles (A–C) matched the appearance of cultured Colo205 aggregates (D–F) imaged using an inverted microscope.

Fig. 4.

QUAD9 element sphere aggregates. Model aggregates of equal sized spheres were created for simulations of CTM behavior in Stokes flow. Aggregates consisted of clusters of 2, 3, or 4 cells in common geometrical configurations. Particles in (A–F) contained 184, 272, 272, 360, 352, and 360 QUAD9 elements, respectively.

Fig. 5.

Optical and scanning electron microscopy images of Colo205 cells. Representative results of optical and scanning electron microscope (SEM) imaging used to quantify Colo205 morphology. For optical images (A), the line tool in ImageJ was used to measure the lengths of the major and minor axes of individual Colo205 cell in order to determine mean cell diameter. For SEM images (B), the oval selection tool in ImageJ was used to outline the edge of the imaged cells, and the multi-point selection tool was then used to mark the base and peak of surface roughness elements. The distance between the paired points was measured to determine the mean height of the roughness features extending from the surface of the cell.

Fig. 6.

Rotation rate of model aggregates matches analytical solutions. The normalized angular velocity vs. orientation angle of rod-like aggregates placed far from a bounding wall was found to be in excellent agreement with solutions given by Eqs. (2.2) and (2.5). The maximum rotation occurred when the aggregates were aligned perpendicular to flow (θ = 0, π, 2π, 3π, …), and the minimum occurred when aggregates were aligned parallel to flow ($\theta =\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},\frac{7\pi }{2},\dots$).

Fig. 7.

Stokes drag force of aggregates matches analytical solutions. The Stokes drag force of aggregates placed far from a bounding wall was found to be in excellent agreement with analytical solutions given by Eq. (16). The relative error for all aggregates was less than 1%.

Fig. 8.

Model circulating tumor microemboli flow behavior is unaffected by shear rate. The trajectory of height of the lowest node point of tetrahedron 4-mer particles, normalized by volume-equivalent sphere radius, was plotted as a function of dimensionless time. Model CTM move in a repeating trajectory particular to its conformation and initial orientation, with the low point decreasing and increasing as aggregate monomers move towards and away from the surface. This pattern was independent of the fluid shear rate; only minor differences were observed in the local maxima and minima of low point heights. A collision event, outlined by the dashed box, corresponds with a blunting of the low point trajectory. During a collision the particle makes contact with the surface, and while the particle still translates and rotates, the low point bottoms out at a minimum, until the particle moves away from the surface and resumes an oscillating pattern. For nondimensionalization, the low point was normalized by the volume-equivalent sphere radius and time was normalized by shear rate.

Fig. 9.

Hydrodynamic simulations are stable over an extended period of time. Dimensionless centroid height trajectory of a model CTM doublet as a function of dimensionless time. Shear rate = 1000 s⁻¹. The model particle oscillated about its initial centroid height of 1 R_e with a steady pattern for an extended period of time. For nondimensionalization, the centroid height was normalized by the volume-equivalent sphere radius and time was normalized with shear rate.

Fig. 10.

Model circulating tumor microemboli undergo collisions then exhibit pole vaulting behavior. (A) Time-lapse trajectory of a triangular triplet with an initial centroid height 0.25 R_e from the plane surface in flow, shear rate = 2000 s⁻¹. Fluid flow induced particle translation and rotation, which brought aggregate monomers within a reactive distance of the plane surface. (B) Contact with the surface corresponded with the dimensionless centroid height reaching a minimum. Upon forming contact with the surface, the particle pushed off of the surface, and the trajectory stabilized at a centroid height higher than its initial position. This particle motion is referred to as a “pole vault”. (C) The dimensionless velocity was proportional to centroid height, and decreased during the collision and then increased as the particle pole vaulted. For nondimensionalization, the height was normalized by the volume-equivalent sphere radius, time was normalized by shear rate, and particle velocity was normalized by the fluid flow velocity at the initial centroid height.

Fig. 11.

Collision maps of model circulating tumor microemboli particles. Collision maps comprising the cumulative collision events for doublet (A–B), rodlike triplet (C–D), and square 4-mer (E–F) aggregate geometries for simulations with shear rates = 500, 1000, and 2000 s⁻¹. Increased color intensity was proportional to increased total contact time. (A), (C), and (F) display contact events that occurred on the leading edge of the aggregate, and (B) and (D) display contact events that occur on the trailing edge of the aggregate. The view in (F) highlights the fact that collisions did not occur on any other edge of the particle.

Fig. 12.

Collision efficiency is a function of shear rate and particle geometry. Comparison of total contact duration (A), maximum contact area (B), and time integral of contact area (TICA) as a function of shear rate. Increased shear rate decreases the duration of a collision, although the surface area within reactive distance of a given collision (B) remains constant. TICA (C) represents a relative measure of adhesion potential, with a higher number correlating to an increased potential to undergo a reactive event. It combines the contribution of contact area and surface area into one metric.

Fig. 13.

Collision occurrence for randomly oriented particles. Comparison of collision occurrence for model particles, with 3 random particle orientations per initial centroid height. For initial heights marked with an “X,” at least one particle in the group experienced a collision with the surface. For initial heights marked with a circle, no collisions were observed during the entirety of the simulation.

Fig. 14.

Model circulating tumor microemboli adopt stable trajectories after forming collisions. Trajectories of randomly oriented rod-like triplet with initial centroid height of 0.25, 0.5, 1 R_e freely rotating and translating in fluid with shear rate = 1000 s⁻¹. The particles with the two lowest initial heights formed collisions, which corresponded to the centroid trajectories bottoming out at a minimum. Despite the trajectories reaching different minima, upon pole vaulting their trajectories stabilized at similar heights. For all particles with centroid heights above 0.5 R_e, no collisions were observed, and centroid heights oscillated with repeating patterns at consistent heights.

Fig. 15.

Comparison of collision metrics for randomly oriented particles. For model CTM conformations where at least one arbitrarily oriented particle formed collisions with the plane surface, total contact time (A), maximum contact area (B), and mean time integral of contact area (TICA) (C) collected for each initial orientation were averaged. Average values were plotted as mean ± standard deviation. Collisions were observed only for triplet and 4-mer aggregates at initial centroid height of 0.25 or 0.5 R_e, with fluid shear rate = 1000 s⁻¹ for all initial heights. The conformations that formed collisions had similar total contact times, max contact area, and adhesion probability, as indicated by similar TICA values. Averaging the collision metrics of arbitrarily oriented aggregates minimized the differences in adhesion probability seen when the triplet-triangle, triplet-rod. 4-mer-square, and 4-mer-tetrahedron were initially oriented horizontally. The doublet and rod-like 4-mer, in contrast, did not form collisions with any of the initial orientations simulated.