Controllability in a class of cancer therapy models with co-evolving resistance

TL;DR

This work uses the affine structure and low dimension of the system to explicitly construct controllable subsets of the state space enclosing sets of equilibria that entirely determine the asymptotic behavior of all trajectories that cannot lead to a cure.

Abstract

Adaptive therapy is a recent paradigm in cancer treatment aiming at indefinite, safe containment of the disease when cure is judged unattainable. In modeling this approach, inherent limitations arise due to the structure of the vector fields and the bounds imposed by toxic side-effects of the drug. In this work we analyze these limitations in a minimal class of models describing a cancer population with a slowly co-evolving drug resistance trait. Chemotherapeutic treatment is introduced as any bounded time-varying input, forcing the cells to adapt to a changing environment. We leverage the affine structure and low dimension of the system to explicitly construct controllable subsets of the state space enclosing sets of equilibria. We show that these controllable sets entirely determine the asymptotic behavior of all trajectories that cannot lead to a cure.

Figures (10)

• Figure 1: Sketch of the approach taken in this work. The landscape defined by a function $h(u,a)$ with ridges (solid red lines) and valleys (dotted red line) appearing and vanishing as the drug dosage of the chemotherapy is varied over time (cp. Waddington's classical picture ferrell2012). A treatment protocol thus yields a sequence of landscapes with respect to which the cancer population tries to optimize its resistance. To do so, the population chases a moving (local) maximum of the landscape. In this example the treatment is decreased and then increased again. The resulting patient trajectory is sketched in blue. The cell population corresponding to the trajectory is sketched at four stages with the intensity of the blue color illustrating increasing and decreasing resistance.
• Figure 2: Examples of growth landscape $h(u,a) = h_0(u) - a h_1(u)$\ref{['eq:h']} for treatment values $a \in \{0,a_M\}$. (a) Simple cost of resistance landscape. (b) An example without cost of resistance, where the growth landscape takes a unique maximum for high trait-values, remaining nearly unaffected by the treatment. (c) A more complex cost of resistance example. Under the effect of treatment, a single global maximum of the landscape is deformed into a "double well"-type potential at $a = a_M$. (d) A landscape with partial cost of resistance. It has a global maximum at high resistance for $a=0$, shielded by a local maximum near $u=0$. Parameter values and functions used can be found in Appendix \ref{['app:parameters']}.
• Figure 3: Comparison of reachable sets for the full system $\Gamma(x_0)$ and reduced system $\widetilde{\Gamma}(x_0)$ from a point $x_0 \in S$; see (\ref{['eq:S']}). The blue curve is a (absolutely continuous) sample trajectory traversing $x_j = \phi^{a_j}_{t_j}(x_{j-1})$. Because this trajectory does not intersect $E$, it is the same for full and reduced system under the rescaling of time (\ref{['eq:stime']}). The grey sample trajectory intersects $E$ for $t=\infty$ in the full system and (twice) in finite time for the reduced system.
• Figure 4: Examples for the manifold of equilibria $\mathcal{P}$ and feasible equilbria $\mathcal{P}_A$ projected on $X$ for a chosen bounded treatment range $A = [a_-,a_+] \subset [0,a_M]$. Function $h(u,a)$ chosen as in Fig. \ref{['fig:potentials']}. (a) Connected $\mathcal{P}_A = \mathcal{P}^{\textup{stab}}_A$ consisting only of stable nodes. (b) Stable node largely unaffected by the treatment range. (c) Disconnected $\mathcal{P}_A$ in two components, one comprised only of stable nodes, the other of nodes, saddles and an isolated fold bifurcation point (red dot). Dashed box shows a possible compact set $U \times N$ containing all feasible equilibria. Inset shows equilibria as $(u,a_\star(u))$ and $U \times A$. In the context of section \ref{['sec:palliative']}, the two components $I_{\mathcal{P}_A}$ are of type (1) and (3). (d) Disconnected $\mathcal{P}_A$ in three components, comprised of only nodes, or only saddles. Right most component is nearly invisible due to small effect of treatment.
• Figure 5: Example of a node-type set $\mathop{\mathrm{enc}}\nolimits \Omega_1$ surrounding a connected component with endpoints $(p_\pm,a_\pm) \in \mathcal{P}^{\textup{stab}}_A$. (a) Construction of the boundary $\Omega_1$ as a union of orbits. Due to Lemma \ref{['prop:angle_condition']}, the image $f(x,A)$ lies in the "lower" tangent halfspace to $\mathop{\mathrm{enc}}\nolimits \Omega_1$ at the point $x$. (b) The set enclosed by $\Omega_1$ with its controllability property.

Theorems & Definitions (25)

• Remark 1
• Definition 1: cf. ColoniusKliemann2000
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 2
• Lemma 3
• proof
• Definition 3: cf. hautus1977