# American Institute of Mathematical Sciences

January  2021, 26(1): 443-481. doi: 10.3934/dcdsb.2020284

## Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence

 1 Kanazawa University, Faculty of Mathematics & Physics, Kakuma, Kanazawa 920-1192, Japan 2 University of St. Andrews, School of Mathematics & Statistics, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland, UK 3 Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Darmstadt, Germany 4 Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany 5 Johannes Gutenberg-Universität Mainz, Institut für Mathematik, Staudingerweg 9, 55128 Mainz, Germany

Received  May 2020 Revised  August 2020 Published  January 2021 Early access  September 2020

We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character.

We prove global existence of weak solutions to a simplified model version and perform numerical simulations for the full setting under several phenotype switching and motility scenarios. We also compare (via simulations) this model with the corresponding haptotaxis-chemotaxis one featuring indirect chemorepellent production and provide a discussion about possible model extensions and mathematical challenges.

Citation: Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 443-481. doi: 10.3934/dcdsb.2020284
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Simulation results of Experiment 5 — constant phenotypic switch rates with the ECM-with-stripes initial conditions (41). In this and the rest of the simulation results, the densities of MCCs and PCCs are represented with isolines colored according to the displayed colorbars. The density of the ECM is visualized by a variable-intensity color that follows the corresponding colorbar. The MCCs, in their taxis-biased random motion, follow the gradients of the ECM and accordingly their density increases over the stripes of the ECM. The ECM is depleted by both cell subpopulations, which also limit its reconstruction. The PCCs obey a logistic-type growth and fill the free space left by the ECM and MCCs; they moreover undergo phenotypic transitions back-and-forth to MCCs according to the PMT and MPT rates $\lambda$ and $\gamma$
; the ECM is depleted by the action of both MCCs and PCCs while the PCCs fill the space left by the ECM and MCCs">Figure 2.  Simulation results of Experiment 1 — constant phenotypic switch rates with the randomly-structured ECM initial conditions (42). Through their haptotaxis-biased random migration, the MCCs identify the higher ECM density regions and accordingly invade the surrounding environment. The ECM and PCCs exhibit similar behavior as in the ECM-with-stripes case shown in Figures 1; the ECM is depleted by the action of both MCCs and PCCs while the PCCs fill the space left by the ECM and MCCs
The PMT rate $\lambda$ and its relation to the MPT rate $\gamma$ with respect to the amount of occupied cell-cell receptors ($\zeta$) and cell-tissue receptors ($y$) in case $\gamma_0 = 1$
, the MCCs can infer higher densities at sites with larger ECM gradients, and lower ones where cell-tissue interfaces are less sharp (e.g., at the center of fiber strands crossing), thus allowing for less cells to move beyond the main fiber tracts">Figure 4.  Simulation results of Experiment 2 — dynamic phenotypic switch rates using the ECM-with-stripes initial conditions (41). When comparing with the corresponding simulation in Experiment 1 (constant phenotypic transition rates) shown in Figure 1, the MCCs can infer higher densities at sites with larger ECM gradients, and lower ones where cell-tissue interfaces are less sharp (e.g., at the center of fiber strands crossing), thus allowing for less cells to move beyond the main fiber tracts
, the MCCs' invasion is here slightly more cohesive, allowing for fewer, but larger local maxima. The density of the PCCs is thereby slightly lower than in Figure 2">Figure 5.  Simulation results of Experiment 2 — dynamic phenotypic switch rates with the randomly-structured ECM initial conditions (42). Comparing with the simulation of Experiment 1 (constant phenotypic transition), shown in Figure 2, the MCCs' invasion is here slightly more cohesive, allowing for fewer, but larger local maxima. The density of the PCCs is thereby slightly lower than in Figure 2
, the effect of the acidity can be seen in the more extensive spread of MCCs, due to chemorepellence by a self-diffusing signal, along with reduced proliferation due to hypoxia, and enhanced ECM degradation throughout the domain">Figure 6.  Simulation results of Experiment 3 – acidity driven migration with the ECM-with-stripes initial conditions (41). In addition to the MCCs, PCCs, and ECM, we also visualize here the pH levels. When comparing with Experiment 2 (dynamic phenotypic switch without acidity), Figure 4, the effect of the acidity can be seen in the more extensive spread of MCCs, due to chemorepellence by a self-diffusing signal, along with reduced proliferation due to hypoxia, and enhanced ECM degradation throughout the domain
apply here as well, when correspondingly comparing with Experiment 2 (dynamic phenotypic switch without acidity), Figure 5">Figure 7.  Simulation results of Experiment 3 – acidity driven migration with the random-structured ECM initial conditions (42). Remarks analogous to those made in Figures 6 apply here as well, when correspondingly comparing with Experiment 2 (dynamic phenotypic switch without acidity), Figure 5
, we note that there is a similar extent of tumor spread, however with MCCs forming very localized, relatively large aggregates (see also e.g., the closeup in Figure 9), while the PCC density remains almost the same">Figure 8.  Simulation results of Experiment 4 — degenerate diffusion, with the ECM-with-stripes initial conditions (41). Compared to the non-degenerate diffusion in Experiment 2, shown in Figure 4, we note that there is a similar extent of tumor spread, however with MCCs forming very localized, relatively large aggregates (see also e.g., the closeup in Figure 9), while the PCC density remains almost the same
. MCCs form localized aggregates while the PCC density remains similar as in Experiment 2 shown in Figure 4">Figure 9.  Closeup of the densities at $t = 10$ in Experiment 4 — degenerate diffusion, with the ECM-with-stripes initial conditions (41). Compare to Figure 8. MCCs form localized aggregates while the PCC density remains similar as in Experiment 2 shown in Figure 4
, we note that the degenerate case leads to higher, more localized MCC densities, mainly near the invasion front">Figure 10.  Simulation results of Experiment 4 — degenerate diffusion with the randomly structured ECM (42). When comparing with the non-degenerate diffusion in Experiment 2, shown in Figure 5, we note that the degenerate case leads to higher, more localized MCC densities, mainly near the invasion front
we see only a slight impact of the cell reconstruction of tissue; the results are almost identical, maybe with a slightly higher concentration of the MCCs towards the invasion front and higher ECM degradation in the inner part of the tumor">Figure 11.  Simulation results of Experiment 5 — ECM remodeling by cancer cells with the ECM-with-stripes initial conditions (41). Compared to Experiment 2 (dynamic phenotypic transition with self-remodeling of the matrix) shown in Figure 4 we see only a slight impact of the cell reconstruction of tissue; the results are almost identical, maybe with a slightly higher concentration of the MCCs towards the invasion front and higher ECM degradation in the inner part of the tumor
, it is clear that the cell reconstruction of the ECM leads to a more fragmented invasion of the MCCs invasion and to higher concentrations along the propagating fronts. We moreover see that the PCCs exhibit a non-smooth boundary/periphery in their support, and that the reconstruction of the ECM is localized where the MCCs are located">Figure 12.  Simulation results of Experiment 5 — ECM remodeling by cancer cells on a randomly-structured ECM (42). When compared with Experiment 2 (dynamic phenotypic transition with self-remodeling of the matrix), shown in Figure 5, it is clear that the cell reconstruction of the ECM leads to a more fragmented invasion of the MCCs invasion and to higher concentrations along the propagating fronts. We moreover see that the PCCs exhibit a non-smooth boundary/periphery in their support, and that the reconstruction of the ECM is localized where the MCCs are located
, the results are almost identical; no particular anoikis effect is visible">Figure 13.  Simulation results of Experiment 6 — anoikis effect on an ECM with initial condition (41). Compared to Experiment 2, shown in Figure 4, the results are almost identical; no particular anoikis effect is visible
, the effect of anoikis becomes visible: the tumor pattern is more heterogeneous (mainly due to the evolution of MCCs) with correspondingly lower PCC density in regions with stronger degraded ECM">Figure 14.  Simulation results of Experiment 6 — anoikis effect on the randomly structured ECM initial conditions (42). Compared to Experiment 2, shown in Figure 5, the effect of anoikis becomes visible: the tumor pattern is more heterogeneous (mainly due to the evolution of MCCs) with correspondingly lower PCC density in regions with stronger degraded ECM
Construction of the randomly-structured ECM with a sequence of grid refinements steps. The first stage of this process is the construction of a random $8\times8$ grid (top left panel) with values normally distributed in $[0,1]$. This grid is progressively refined to the final (for this case) resolution of $256\times256$ (bottom right panel). At every refinement step the number of computational cells is doubled along each dimension and the new values are obtained by a) averaging the values of the neighboring cells of the coarser grid, and b) adding some random and normally distributed noise. Periodic interpolations are employed at the "boundary" the discretization domain. It can be clearly seen that the coarse structure of the ECM that was randomly chosen in the $8\times8$ matrix is still visible in the refined $256\times 256$ grid
Butcher tableau for the explicit (upper) and the implicit (lower) parts of the third order IMEX scheme ARK3(2)4L[2]SA we use in (B.5), see also [45]
 $0$ $\frac{1767732205903}{2027836641118}$ $\frac{1767732205903}{2027836641118}$ $\frac{3}{5}$ $\frac{5535828885825}{10492691773637}$ $\frac{788022342437}{10882634858940}$ $1$ $\frac{6485989280629}{16251701735622}$ $-\frac{4246266847089}{9704473918619}$ $\frac{10755448449292}{10357097424841}$ $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$ 0 0 $\frac{1767732205903}{2027836641118}$ $\frac{1767732205903}{4055673282236}$ $\frac{1767732205903}{4055673282236}$ $\frac{3}{5}$ $\frac{2746238789719}{10658868560708}$ $-\frac{640167445237}{6845629431997}$ $\frac{1767732205903}{4055673282236}$ 1 $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$ $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$
 $0$ $\frac{1767732205903}{2027836641118}$ $\frac{1767732205903}{2027836641118}$ $\frac{3}{5}$ $\frac{5535828885825}{10492691773637}$ $\frac{788022342437}{10882634858940}$ $1$ $\frac{6485989280629}{16251701735622}$ $-\frac{4246266847089}{9704473918619}$ $\frac{10755448449292}{10357097424841}$ $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$ 0 0 $\frac{1767732205903}{2027836641118}$ $\frac{1767732205903}{4055673282236}$ $\frac{1767732205903}{4055673282236}$ $\frac{3}{5}$ $\frac{2746238789719}{10658868560708}$ $-\frac{640167445237}{6845629431997}$ $\frac{1767732205903}{4055673282236}$ 1 $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$ $\frac{1471266399579}{7840856788654}$ $-\frac{4482444167858}{7529755066697}$ $\frac{11266239266428}{11593286722821}$ $\frac{1767732205903}{4055673282236}$
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