Tumor Anti Angiogenesis

Tumor Anti Angiogenesis
State dimension:	1
Differential states:	3
Discrete control functions:	1

The tumor anti-angiogenesis problem is a classical benchmark in biomedical optimal control. This description is taken from [1].

It is based on the mathematical model of Hahnfeldt et al. [2] for tumor growth under angiogenic signalling, as studied from an optimal control perspective by Ledzewicz and Schättler [3]. Angiogenesis is the process by which tumors recruit a network of blood vessels to sustain their growth; anti-angiogenic therapy targets this process by administering inhibitor agents. The two biological states are the tumor volume and the carrying capacity of the vascular endothelium (which sets the maximum supportable tumor size). A third state tracks the cumulative inhibitor dose, enforcing a total-dose constraint. The control is the instantaneous inhibitor dose rate. The objective is to minimise the tumor volume at the final time. The problem is known to exhibit a rich optimal control structure, including bang–bang arcs, singular arcs, and boundary arcs corresponding to the dose constraint.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & p(t_{\mathrm{f}})\\ \text{subject to}\\ \dot{p}(t)& =& -\xi p(t)\ln \left(\frac{p(t)}{q(t)}\right),\\ \dot{q}(t)& =& bp(t)-\mu q(t)-(dp(t{)}^{2/3}+eu(t))q(t),\\ \dot{y}(t)& =& u(t),\\ (p(0),q(0),y(0))& =& (p_0,q_0,0),\\ y(t_{\mathrm{f}})& \le & A,\\ u(t)& \in & [0,a]\mathrm{\forall }t\in [0,t_{\mathrm{f}}],\end{array}$

where $p(t)$ is the tumor volume (mm³), $q(t)$ is the endothelial carrying capacity (mm³), and $y(t)$ is the cumulative inhibitor dose (mg/kg) administered up to time $t$. The tumor dynamics follow the Gompertz law $\dot{p}=-\xi p\ln (p/q)$, which models logistic-like growth bounded by the carrying capacity $q$. The carrying capacity decreases due to natural endothelial cell loss ($\mu q$), inhibitor-induced endothelial cell loss ($euq$), and stimulated shedding proportional to tumor surface area ($d{p}^{2/3}q$), while increasing due to endothelial recruitment by the tumor ($bp$). The state $y(t)$ enforces the total-dose constraint $y(t_{\mathrm{f}})\le A$ as an isoperimetric inequality.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$\xi$	0.084	Tumor growth rate (day−1)
$b$	5.85	Endothelial birth rate coefficient (day−1)
$\mu$	0.02	Endothelial natural loss rate (day−1)
$d$	0.00873	Inhibition coefficient for endothelial shedding (mm−2 day−1)
$e$	0.66	Inhibitor effectiveness on endothelium (kg mg−1 day−1)
$a$	75	Maximum inhibitor dose rate (mg kg−1 day−1)
$A$	15	Maximum cumulative inhibitor dose (mg kg−1)
$p_0$	12000	Initial tumor volume (mm3)
$q_0$	15000	Initial endothelial carrying capacity (mm3)
$t_{\mathrm{f}}$	1.2	Horizon of the control problem (days)

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  Tumor volume, carrying capacity, cumulative dose, and optimal inhibitor schedule for a reference solution.

Miscellaneous and Further Reading

The equilibrium $p^{*}=q^{*}$ corresponds to a tumor at maximum sustainable size. Without treatment, $p(t)$ converges to this steady state. Ledzewicz and Schättler [3] characterise the optimal control using Pontryagin's minimum principle and show that the optimal policy consists of a sequence of full-dose ($u=a$), no-dose ($u=0$), and singular arcs, depending on the initial conditions and remaining dose budget. This formulation and detailed analysis can be found in [1], [2], and [3].

References

[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl
[2] Hahnfeldt, P., Panigrahy, D., Folkman, J., & Hlatky, L. (1999). Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775.
[3] Ledzewicz, U., & Schättler, H. (2007). Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM Journal on Control and Optimization, 46(3), 1052–1079.