Multi-scale biological and physical modelling of the tumour micro-environment

Abstract

Paced by advances in high performance computing, and algorithms for multi-physics and multi-scale simulation, a number of groups have recently established numerical models of flowing blood systems, where cell-scale interactions are explicitly resolved. To be biologically representative, these models account for some or all of: (1) fluid dynamics of the carrier flow, (2) structural dynamics of the cells and vessel walls, (3) interaction and transport biochemistry, and, (4) methods for scaling to physiologically representative numbers of cells. In this article, our interest is the modelling of the tumour micro-environment. We review the broader area of cell-scale resolving blood flow modelling, while focusing on the particular interactions of tumour cells and white blood cells, known to play an important role in metastasis.

Figure 1.

(a,b) Side-view microscopy and μ-PIV visualizations of an endothelium adherent PMN in a flow chamber for low and high shear conditions (50, 258 s⁻¹ respectively) [10] (with permission); (c,d) CFD model results for shear rates in (a,b) (prescribed cell shapes) [11]; (e,f) Comparison of measured and predicted velocities for shear rates in (a,b) [11]; (g,h) side- and top-view microscopy and μ-PIV images of a PMN–TC aggregate [27] (with permission).

Figure 2.

Two timesteps in an adaptive Cartesian-mesh model of a rigid TC interacting with a rigidly modelled deformed adherent PMN [11].

Figure 3.

Successive timesteps in the simulation of a rigid TC interacting with a deforming adherent PMN [11].

Figure 4.

Representation of the relevant adhesion molecule expressions.

Figure 5.

Elements of PMN–TC tethering simulations. (a) Predicted separation distance vs. time between adherent PMN and a TC [11], (b) number of bonds formed vs. time [11], (c) generalized bond search/formation [44], (d) predicted path of the TC for a PMN–TC tethering event, (e) final tethered configuration.

Figure 6.

(a) Predicted TC trajectories for three of the 45 DNS runs used to train the emulator. (b) Prediction of the adhesion outcome in a preliminary study with 45 combinations of TC height and shear rate shows a good correspondence between simulation and emulator. Dots represent the full simulation, colored according to adhesion outcomes: no collision between TC and PMN, collision between TC and PMN but no adhesion formed, and adhesion. Contours show the adhesion probability predicted by the emulator, colored according to the value of the probability (right bar). As expected, the probability of adhesion increases with the decrease of height and shear rates. The contour accurately separates adhesion from collision without adhesion.