Spatial evolution of tumors with successive driver mutations

Tibor Antal, P. L. Krapivsky, and M. A. Nowak

Phys. Rev. E 92, 022705 – Published 12 August 2015

Abstract

We study the influence of driver mutations on the spatial evolutionary dynamics of solid tumors. We start with a cancer clone that expands uniformly in three dimensions giving rise to a spherical shape. We assume that cell division occurs on the surface of the growing tumor. Each cell division has a chance to give rise to a mutation that activates an additional driver gene. The resulting clone has an enhanced growth rate, which generates a local ensemble of faster growing cells, thereby distorting the spherical shape of the tumor. We derive formulas for the abundance and diversity of additional driver mutations as function of time. Our model is semi-deterministic: the spatial growth of cancer clones is deterministic, while mutants arise stochastically.

Received 15 April 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022705

Authors & Affiliations

Tibor Antal¹, P. L. Krapivsky², and M. A. Nowak³

¹School of Mathematics, Edinburgh University, Edinburgh EH9 3FD, United Kingdom
²Department of Physics, Boston University, Boston, Massachusetts 02215, USA
³Program for Evolutionary Dynamics, Department of Mathematics, and Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 92, Iss. 2 — August 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Slices of a tumor with a single mutant clone at times $t = 2, 4, 16.6, 20$ . The original tumor is initiated at the origin, and the mutant is initiated at $t = 1$ at $(1, 0, 0)$ and has fitness $v = 1.5$ . The tumor at the initiation time is drawn as a thin line at each stage to show the length scale. The boundaries of the original tumor are depicted by thick lines emanating from $(1, 0)$ , other thick lines represent the outer boundary of the mutant clone. The central part and the side parts of the outer boundary of the mutant clone are described by different functional forms. On the lower left picture the original tumor is just captured by the mutant clone, this happens at $t_{c} = e^{π / β} = 16.6087 \dots$ ; on the lower right one the mutant overgrows the enclosed original tumor.
Reuse & Permissions
Figure 2
The final barnacle shape of the original tumor after it has been captured by the mutant clone. Shown results corresponds to the fitness values $v = 1.3, 1.5, 2$ . The mutant is always initiated at $t = 1$ , and the capture takes place at $t_{c} = e^{π / β} = 43.9052, 16.6087, 6.13371$ , respectively, for the illustrated fitness values. The circle represents the original tumor at the initiation time $t = 1$ of the mutant clone.
Reuse & Permissions
Figure 3
Illustration for the spreading of the mutant clone (a two-dimensional cut is drawn). On the left panel the initiation of a mutant clone is shown. A tiny segment of the surface of the original clone (a shaded region) is almost flat. After a small time interval $d t$ , the surface is composed by a circular arc and straight segments. On the right panel the evolution of the surface of the tumor is shown for later times. The angle $ϕ$ stays constant during the process.
Reuse & Permissions
Figure 4
The shape of the mutant clone (only a two-dimensional cut is drawn). The mutant clone is initiated at $(1, 0)$ . One part of the mutant position is a finite wedge originated from $(1, 0)$ with half-angle $ϕ$ and hence the outer boundary being a circle of radius $v (t - 1)$ . The thick line indicates the boundary of the original tumor, and the curve between the original tumor and the mutant clone is given by the parametric curve $r (θ)$ . The top black dot represents a general point on the outer boundary of the mutant clone. It is reached by the mutant clone originating from the middle black dot in time $t - r (θ)$ and propagating at speed $v$ .
Reuse & Permissions
Figure 5
Shape of the tumor with a single mutant clone at times $t = 2, 3$ , and 5.5. The mutant is initiated at $t = 1$ and has fitness $v = 1.5$ . The boundary of the mutant clone is red.
Reuse & Permissions
Figure 6
Shape of the tumor with many mutant clones arrived at different times with different fitness values. Mutant clones are initiated stochastically at constant rate on the surface of the original tumor, and then they grow deterministically at constant rate. The growth rates of the mutant clones were chosen randomly between 1 and 2 for this illustration.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics