Tumor growth instability and the onset of invasion

Motivated by experimental observations, we develop a mathematical model of chemotactically directed tumor growth. We present an analytical study of the model as well as a numerical one. The mathematical analysis shows that: (i) tumor cell proliferation by itself cannot generate the invasive branching behavior observed experimentally, (ii) heterotype chemotaxis provides an instability mechanism that leads to the onset of tumor invasion, and (iii) homotype chemotaxis does not provide such an instability mechanism but enhances the mean speed of the tumor surface. The numerical results not only support the assumptions needed to perform the mathematical analysis but they also provide evidence of (i), (ii), and (iii). Finally, both the analytical study and the numerical work agree with the experimental phenomena.

Figure 1

(Color online) Depicted is an overlaid image of human U87 brain tumor cells which were stably transfected with a green fluorescent protein (EGFP) Histone 2B marker for nuclei. The image is taken from a central cross section of the MTS cultured in a three-dimensional extracellular matrix (Matrigel, Becton Dickinson, USA) environment in vitro. Note the chainlike invasive patterns. The image was taken one day post transferring the MTS from liquid medium to Matrigel (scale bar $=100\mu \mathrm{m}$).

Figure 2

$M$ density field for the two limiting cases: ${\mu }_M⪡{\mu }_U$ (dotted line) and ${\mu }_M⪢{\mu }_U$ (dashed line). The solid line represents the density field of tumor cells $U$.

Figure 3

Sketch of the heterotype chemoattractant effect on tumor cell invasive drift. Dashed lines represent level sets of $Q$ and arrows represent the local normal velocity of the tumor cells due to chemo- taxis (the length of the arrows is proportional to the normal velocity).

Figure 4

(a) Qualitative behavior of the trajectory of the density field $C$ in the phase plane. The arrows indicate the direction of time evolution. Curves $N1\text{–}N3$ are the three types of nullclines that can be expected for our system. (b) Same trajectory of the density field $C$ with respect to the spatial coordinate $x$. Again, arrows indicate time evolution.

Figure 5

One-dimensional tumor proliferation. Solid lines represent the $U$ density field for the model given by Eqs. (1, 6). Dashed lines represent $U$ for the solution of Fisher’s equation, i.e., Eq. (6) with $M=K-{\lambda }_M∕{\lambda }_{U}U$.

Figure 6

(a) Numerical simulation of an initially circular tumor embedded in a matrigel medium $M$ with slow dynamics $({\mu }_M⪡{\overline{\mu }}_U)$. Different curves represent different times with the initial time $t_0$ corresponding to the inner perfect circle. (b) Time evolution of the tumor spheroid average radius, $R_{TS}$, for ${\mu }_U=0.01,{\mu }_Q=10,{\mu }_M=0.00001,{\lambda }_M=0.05,a_Q=0.75,b_Q=0.5$ and ${\chi }_Q=0\left(\text{squares}\right)$ and ${\chi }_Q=2$ (solid circles).

Figure 7

(a) Numerical simulation of tumor branching induced by a heterotype chemotactic source located at $x=L_x$. The plot represents the tumor density field $U(x,y,t=20000{ϵ}_t)$. (b) Cross section of the tumor in panel (a) for different times between $2000{ϵ}_t$ and $20000{ϵ}_t$.

Figure 8

Cross section of a tumor for the case of coupled dynamics between matrigel $M$, tumor cells $U$ and homotype chemoattractant $C$ for different times between $t_0={ϵ}_t$ and $t_f=30000{ϵ}_t$. Solid lines: $U$ subject to homotype chemotaxis; dashed lines: $U$ subject to no homotype chemotaxis. Dotted line: $C$ (times a factor of four) for $t=10000{ϵ}_t$.