Control by Viability in a Chemotherapy Cancer Model

Abstract

The aim of this study is to provide a feedback control, called the Chemotherapy Protocol Law, with the purpose to keep the density of tumor cells that are treated by chemotherapy below a “tolerance level” $L_c$, while retaining the density of normal cells above a “healthy level” $N_c$. The mathematical model is a controlled dynamical system involving three nonlinear differential equations, based on a Gompertzian law of cell growth. By evoking viability and set-valued theories, we derive sufficient conditions for the existence of a Chemotherapy Protocol Law. Thereafter, on a suitable viability domain, we build a multifunction whose selections are the required Chemotherapy Protocol Laws. Finally, we propose a design of selection that generates a Chemotherapy Protocol Law.

Appendix

In this section we present some concepts of viability theory involved in this paper, further developments can be found in Aubin (1990) and Aubin and Frankowska (1990).

Let $f:{\mathbb {R}}^n\times {\mathbb {R}}^p\rightarrow {\mathbb {R}}^n$ a continuous function, we consider the following constrained controlled dynamical system:

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{y}}(t)=f(y(t),u(y)),\forall \;t\in [0,+\infty [,\\ y(0)=y_0, \end{array} \right. \end{aligned}$$

(25)

where $y:[0,+\infty [\rightarrow {\mathbb {R}}^n$ is an absolutely continuous function, $u:{\mathbb {R}}^n\rightarrow {\mathcal {U}}$ is a continuous feedback control function and ${\mathcal {U}}\subset {\mathbb {R}}^p$ is a compact convex set.

Note that the system (25) can be rewritten by considering

$$\begin{aligned} G(y)=f(y,u(y)),\;\;\;\forall \;y\in {\mathbb {R}}^n. \end{aligned}$$

G is continuous since f(., .) and u(.) are continuous.

We consider hence the new system

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{y}}(t)=G(y(t)),\quad \forall \;t\in [0,+\infty [,\\ y(0)=y_0. \end{array} \right. \end{aligned}$$

(26)

The viability theory aims to study the existence of trajectories of system (26) that start in a subset $K\subset {\mathbb {R}}^n$ and remain in it thereafter. Note that the viability property is characterized throughout the contingent cone (Bouligand cone) which can be expressed as follows.

Definition 2

Let $K\subset {\mathbb {R}}^n$, and $x\in K$. The Bouligand cone $T_{K}(x)$ to K at x is defined by

$$\begin{aligned} v\in T_{K}(x)\Longleftrightarrow \lim \inf _{h\longrightarrow 0^+} \displaystyle \frac{d(x+hv,K)}{h}=0. \end{aligned}$$

We can now define rigorously the viability concept, see Aubin and Frankowska (1990).

Definition 3

Let $K\subset {\mathbb {R}}^n$, and y(.) a solution of the system (26). We say that

• y(.) is locally viable in K if there exists $T>0$ such that

  $$\begin{aligned} \forall \; t\in [0,T],\;\;\; y(t)\in K. \end{aligned}$$

• y(.) is globally viable in K (or is a viable trajectory in K) if

  $$\begin{aligned} \forall \; t\in [0,+\infty [,\;\;\; y(t)\in K. \end{aligned}$$

• K enjoys a local viability property if for any initial state $x_0\in K$, there exist $T>0$ and a solution of differential equation (26) starting at $x_0$ and locally viable in K on [0, T].

• K enjoys a global viability property (or viability property) if for any initial state $x_0\in K$, there exist a solution of differential equation (26) starting at $x_0$ and globally viable in K.

The relationship between the contingent cone and viable trajectories is illustrated in Fig. 3. The contingent cones to a set K at $x_0$, $x_1$ and $x_2$ respectively, are represented by cones with angles represented by dotted lines. We mention that if $G(x_i)\in T_K(x_i)$ for $i=0,\;1,\;2$, then the vector field at each point $x_i$ for $i=0,\;1,\;2$, is pointed inside the set K and hence the trajectory solution of the system (26) and crossing through this $x_i$ is directed inside the set K in this point. So if $G(x)\in T_K(x),\;\forall \;x\in K$ then the trajectory is entirely contained in K.

Fig. 3

Contingent cones at different points with vector fields G(x)

The following theorem derives sufficient conditions for viability property, see Aubin (1990) and Aubin and Frankowska (1990).

Theorem 3

(Nagumo) Suppose thatK is closed and G is continuous and fulfils a linear growth condition, that is, there exist$\sigma ,\;c>0$ such that

$$\begin{aligned} \parallel G(x)\parallel \le \sigma \parallel x\parallel +\;c, \;\;\; \forall \;x\in K. \end{aligned}$$

If for all$x \in K: \;\;\; G(x) \in T_{K}(x)$, thenK enjoys a viability property. In this case the setK is called a viability domain.

In the theorem the growth condition substitutes the condition that G(K) is bounded to assure the global viability as proved in Theorem 10.1.6 (set-valued map version) in Aubin and Frankowska (1990).

Since the contingent cone is important to provide conditions for viability, the computation of this tool is necessary. J. P. Aubin and H. Frankowska give a useful characterization of the contingent cone in case of a subset K defined by inequalities [see the example on page 123, Aubin and Frankowska (1990)]:

Lemma 3

Let$g=(g_1,\ldots ,g_p):{\mathbb {R}}^n\longrightarrow {\mathbb {R}}^p$ be a continuous function and consider the subset K of constraints defined by the inequalities

$$\begin{aligned} K=\{x\in {\mathbb {R}}^n| g_i(x)\ge 0,\quad i=1,\ldots ,p \}, \end{aligned}$$

and let$I(x):=\{i=1,\ldots ,p| g_i(x)=0\}$. If g is differentiable and

$$\begin{aligned} \exists v_0\in {\mathbb {R}}^n\; such\; that\; \forall i\in I(x),\;\; g'_i(x).v_0>0, \end{aligned}$$

then the contingent cone is given by

$$\begin{aligned} T_{K}(x)=\{w\in {\mathbb {R}}^n| \forall i\in I(x), g'_i(x).w\ge 0 \}. \end{aligned}$$

Now, if for all$i=1,\ldots ,p$: $g_i(x)>0$, then$x\in \mathring{K}$, and

$$\begin{aligned} T_{K}(x)={\mathbb {R}}^n. \end{aligned}$$

The last expression derives from the fact that when $x\in \mathring{K}$, the interior of K, then all tangential directions starting from x are pointed inside the set K, which implies that the contingent cone of K at x is the space ${\mathbb {R}}^n$.

Another concept involved in this paper is set-valued analysis. We define a set-valued map $\varGamma :{\mathbb {R}}^n\rightsquigarrow {\mathbb {R}}^n$ as a multifunction such that for all $x\in {\mathbb {R}}^n$,

$$\begin{aligned} \varGamma (x)\subset {\mathbb {R}}^n. \end{aligned}$$

A selection of a set-valued map $\varGamma$ is a single-valued map $w:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n$ such that

$$\begin{aligned} \forall y\in {\mathbb {R}}^n,\;\;\;\; w(y)\in \varGamma (y). \end{aligned}$$

According to assumptions on the multifunction $\varGamma$ we can provide minimal, continuous, Lipshitz etc. selections of it.

In this context, instead of a differential equation, we define differential inclusion as follows:

$$\begin{aligned} {\dot{y}}(t)\in \varGamma (y(t)),\; \text{ almost } \text{ everywhere } \;t\in [0,+\infty [. \end{aligned}$$

So the condition $G(x) \in T_{K}(x)$, for all $x\in K$, in Theorem 3 on the viability property is adapted to differential inclusion by setting [see Theorem 10.1.6, Aubin and Frankowska (1990)]

$$\begin{aligned} \forall \;x\in K,\;\;\;\varGamma (x) \cap T_{K}(x) \ne \emptyset . \end{aligned}$$

This intersection is viewed by means of selections, i.e. there exists a selection w of $\varGamma$ such that

$$\begin{aligned} \forall \;x\in K,\;\;\; w(x)\in T_{K}(x). \end{aligned}$$