A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion

Figure 1.  Outline of the unit cubic domain $ \Omega $ (in black) and the embedded single-branch network $ \Lambda $ (red line)

Figure 2.  Evolution of the cellular density $ U $ on a single-branch network and of the corresponding three-dimensional chemoattractant concentration $ c $ at four different times, $ t = 0.0, \, 0.08,\, 0.26 {\rm{\;and\;}} 1.2 $.

Figure 3.  Initial conditions and subsequent states of the evolution in time of the cellular density $ U $ and of the concentration of the chemoattractant $ c $ along the single-branch network, thus for $ s \in [0,0.8] $ (or, equivalently, for $ x \in [0.1,0.9] $).

Figure 4.  Evolution in time of the solution of the $ 3 $D-$ 1 $D Keller-Segel model, on a one-dimensional bifurcated network embedded in a unit three-dimensional domain, at four different times of the simulation: $ t = 0, \, 0.04, \, 0.16, \, 1 $

Figure 5.  (a)–(c) Time evolution of three initial aggregates of cells on a symmetric network and the corresponding chemical concentration at three different times, $ t = 0, \, 0.04, {\rm{\;and\;}} 0.3 $. (d) Internal visualization of the chemoattractant concentration (on a different scale of values)

Figure 6.  Evolution of the cellular density and of the chemical concentration, solutions to the multiscale Keller-Segel model, at six different times

Figure 7.  Schematic representation of the metastatic cascade, adapted from [20]. Malignant tumor cells escape from the primary tumor site by breaching the extracellular barriers (such as the basement membrane) (1), take advantage of the collagen fibrils and networks to invade the tissue (2) and enter the blood circulation (3)

Figure 8.  Time evolution of the cellular density $ U $ (blue line) and the corresponding ECM density $ m $ (red line) on the one-dimensional single-branch network, for $ x \in [0.1,0.9] $ and at four different times $ t = 0, \, 0.12, \, 0.24,\, 1.2 $

Figure 9.  Solutions of the multiscale Keller-Segel-type model (20) in the single-branch test case at three different times. The evolution of the cellular density $ U $ on the one-dimensional segment is displayed together with the density of the extracellular matrix $ m $ (left panels) and with the MDEs concentration $ c $ (right panels)

Figure 10.  The time evolution of the cellular density $ U $ and the density of the extracellular matrix $ m $ for two different networks. The invasion is slowed down ((a)–(b)) or sped up ((c)–(d)) compared to the simulation in Figure 9

Figure 11.  Initial density $ m_0 $ of the extracellular matrix for $ x\in[0,1] $

Figure 12.  Evolution in time of the cellular density $ U $ and the corresponding density of the ECM $ m $, solutions of the multiscale Keller-Segel-type model (20), at six subsequent times of the simulations

Figure 13.  Evolution in time of the cellular density $ U $ and the corresponding density of the ECM $ m $, solutions of the multiscale Keller-Segel-type model 20. The invasion is more effective without the presence of transversal branches, compared to the solutions displayed in Figures 12

Figure 14.  Comparison of the evolution in time of the cellular density $ U $ for the network $ \mathcal{N}_1 $ (left panels) and $ \mathcal{N}_2 $ (right panels) at three subsequent times of the simulations

Figure 15.  Comparison of the evolution in time of the cellular density $ U $ on the three parallel branches $ \Lambda_1 $ (red line), $ \Lambda_2 $ (green line) and $ \Lambda_3 $ (blue line) for the networks $ \mathcal{N}_1 $ (left panels) and $ \mathcal{N}_2 $ (right panels)