Control by Viability in a Chemotherapy Cancer Model


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.

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  1. Afenya EK, Caldron CP (1996) A brief look at a normal cell decline and inhibition in acute leukemia. J Cancer Detect Prev 20(3):171–179

    Google Scholar 

  2. Alberts B, Johson A, Lewis J, Raff M, Robert K, Walter P (2002) Molecular biology of the cell, 4th edn. Garland Science, New York

    Google Scholar 

  3. Aubin JP (1990) Viability theory. Birkhäuser, Boston

    Google Scholar 

  4. Aubin JP, Frankowska H (1990) Set-valued analysis. Birkhäuser, Boston

    Google Scholar 

  5. Baudino B, D’agata F, Caroppo P, Castellano G, Cauda S, Manfredi M, Geda E, Castelli L, Mortara P, Orsi L, Cauda F, Sacco K, Ardito RB, Pinessi L, Geminiani G, Torta R, Bisi G (2012) The chemotherapy long-term effect on cognitive functions and brain metabolism in lymphoma patients. Q J Nucl Med Mol Imaging 56(6):559–568

    Google Scholar 

  6. Bratus A, Fimmel AS, Todorov Y, Semenov Y, Nuernberg YS (2012) On strategies on a mathematical model for leukemia therapy. J Nonlinear Anal Real World Appl 13:1044–1059

    Article  Google Scholar 

  7. Bratus A, Todorov Y, Yegorov I, Yurchenko D (2013) Solution of the feedback control problem in the mathematical model of leukemia therapy. J Optim Theory Appl 159:590–605

    Article  Google Scholar 

  8. Bratus A, Yegorov I, Yurchenko D (2016) Dynamic mathematical models of therapy processes against glioma and leukemia under stochastic uncertainties. Meccanica dei Materiali e delle Strutture 6(1):131–138

    Google Scholar 

  9. Dexter DL, Leith JT (1986) Tumor heterogeneity and drug resistance. J Clin Oncol 4(3):244–257

    Article  Google Scholar 

  10. Evan GI, Vousden KH (2001) Progress proliferation, cell cycle and apoptosis in cancer. Nature 411:342–348

    Article  Google Scholar 

  11. Gewirtz DA, Holt SE, Grant S (2007) Apoptosis, senescence and cancer. Humana Press, New York

    Google Scholar 

  12. Hoffbrand AV, Petit JE (1984) Essential haematology. Blackwell, Oxford

    Google Scholar 

  13. Isaeva OG, Ospove VA (2009) Different strategies for cancer treatment : mathematical modeling. J Comput Math Methods Med 10:253–272

    Article  Google Scholar 

  14. Kassara K (2006) A set-valued approach to control immunotherapy. J Math Comput Model 44:1114–1125

    Article  Google Scholar 

  15. Kassara K (2009) A unified set-valued approach to control immunotherapy. SIAM J Control Optim 48:909–924

    Article  Google Scholar 

  16. Kassara K, Moustafid A (2011) Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method. J Math Biosci 231:135–143

    Article  Google Scholar 

  17. Lasley I (2011) 21st Century cancer treatement. CreateSpace Independent Publishing Platform, Scotts Valley

    Google Scholar 

  18. Matsuda T, Takayama T, Tashiro M, Nakamura Y, Ohashi Y, Shimozuma K (2005) Mild cognitive impairment after adjuvant chemotherapy in breast cancer patients-evaluation of appropriate research design and methodology to measure symptoms. Breast Cancer 12(3):279–287

    Article  Google Scholar 

  19. Matveev AS, Savkin AV (2008) Application of optimal control theory to analysis of cancer chemotherapy regimens. Syst Control Lett 46(5):4042–4048

    Google Scholar 

  20. Ness KK, Gurney JG (2007) Adverse late effects of childhood cancer and its treatment on health and performance. Annu Rev Public Health 28:278–302

    Article  Google Scholar 

  21. Nowell PC (2002) Tumour progression: a brief historical perspective. Semin Cancer Biol 12:261–266

    Article  Google Scholar 

  22. Riah R, Fiacchini M, Alamir M (2015) Invariance-based analysis of cancer chemotherapy. IEEE conference on control application (CCA), pp 1111–1116

  23. Rubinow SI, Lebowitz JL (1976) A mathemathecal model of the acute myeloblastic leukemic state in man. Biophys J 16:897–910

    Article  Google Scholar 

  24. Schimke RT (1984) Gene amplification, drug resistance and cancer. Cancer Res 44:1735–1742

    Google Scholar 

  25. Sun Y, Campisi J, Higano C, Beer TM, Porter P, Coleman I, True L, Nelson PS (2012) Treatment-induced damage to the tumor microenvironment promotes prostate cancer therapy resistance through WNT16B. Nat Med 18:1359–1368

    Article  Google Scholar 

  26. Swift RJ, Wirkus SA (2006) A course in ordinary differential equations. Chapman and Hall/CRC, Boca Raton

    Google Scholar 

  27. Tsygvintsev A, Marino S, Kirschner DE (2013) Mathematical model of gene therapy for the treatment of cancer. In: Ledzewicz U, Schättler H, Friedman A, Kashdan E (eds) Mathematical methods and models in biomedicine. Lecture notes on mathematical modelling in the life sciences. Springer, New York, NY, pp 367–385

    Google Scholar 

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We are very grateful to the editor in chief of the journal “Acta Biotheoretica”, and the anonymous reviewers for their useful remarks and constructive suggestions that improve substantially the manuscript.

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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:

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

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

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

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

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

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Serhani, M., Essaadi, H., Kassara, K. et al. Control by Viability in a Chemotherapy Cancer Model. Acta Biotheor 67, 177–200 (2019).

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  • Cancer model
  • Feedback control
  • Viability
  • Set-valued analysis

Mathematics Subject Classification

  • 34H05
  • 34A60
  • 49 J52
  • 49K15
  • 92C50
  • 9C60