Optimal Control for Mathematical Models

for Anti-Angiogenic Treatments

(joint research with Urszula Ledzewicz of Southern Illinois University in Edwardsville)

 

 

 

 

 

The reason for the failure of most cancer chemotherapy treatments lies in both intrinsic and acquired drug resistance. Malignant cancer cell populations are highly heterogeneous and fast duplications combined with genetic instabilities provide just one of several mechanisms which allow for quickly developing acquired resistance to anti-cancer drugs. In addition, intrinsic resistance makes some cancer cells not susceptible to many cytotoxic agents. Healthy cells on the other hand are genetically very stable and do not develop similar features. So, while the cancer population becomes increasingly more resistant to the anti-cancer drugs, they keep on killing the healthy cells eventually leading to a failure of the therapy.

 

One approach to cancer treatments that tries to circumvent the problem of drug resistance is tumor anti-angiogenesis. A growing tumor, after it reaches just a few millimeters in diameter, no longer can rely on blood vessels of the host for its supply of nutrients, but it needs to develop its own system for blood supply. In this process, called angiogenesis, there is a bi-directional reciprocal signaling between endothelial cells, which provide the lining for the newly forming blood vessels of the tumor, and tumor cell growth. Vascular endothelial growth factor (VEGF) and to a lesser extent basic fibroblast growth factor (bFGF) are produced by the tumor and stimulate endothelial cell growth. Endothelial cells in return sustain tumor growth by forming vessels used for the supply of nutrients to the tumor. Overall, angiogenesis can be viewed as a complex balance of tightly regulated stimulatory and inhibitory mechanisms balanced by micro-environmental factors.


In the model developed by Hahnfeldt, Panigrahy, Folkman and Hlatky (Cancer Research, 1999)  these effects are summarized in a two-dimensional dynamical system with the numbers of primary tumor cells, p, and the carrying capacity of the vasculature, q, as variables. The latter is the maximum tumor volume possible by the vessel network. A growth function describes the size of the tumor dependent on the volume of endothelial cells and is chosen as Gompertzian with a variable carrying capacity defined by q in the original model (other models are equally realistic and may be considered).

 

(1)

dp/ dt = -ξpln(p/q)


Endothelial cells have receptors which make them sensitive to inhibitors of inducers of angiogenesis like, for example, endostatin. The overall dynamics is a balance between stimulation and inhibition and its basic structure is of the form

 

(2)

dq/ dt = -μq + S(p,q) - I(p,q) – Gqu

 

where μ describes the loss of endothelial cells due to natural causes (death etc.), I and S denote inhibition and stimulation terms, respectively, and the term Gqu represents a loss of endothelial cells due to additional outside inhibition. The variable u represents the control in the system and corresponds to the angiogenic dose rate while G is a constant that represents the anti-angiogenic killing parameter.

 

The three models considered below differ in the form of the inhibition and stimulation terms I and S.

 

 

 

Model

 

 

I(p,q)

 

S(p,q)

 

Hahnfeldt, Panigrahy, Folkman and Hlatky

 

 

dp2/3q

 

bp

 

Ergun, Camphausen and Wein

 

 

dp4/3

 

bp2/3

 

d’Onofrio and Gandolfi

 

 

dp2/3q

 

bq

 

 

 

In our research we analyze the following optimal control problem:

 

 

For a free terminal time T minimize the value p(T) subject to the dynamics given by equations (1) and (2) over all Lebesgue measurable functions u: [0,T] → [0,a] which satisfy an isoperimetric constraint of the form ∫0T u(t)dt≤A .

 

 

 

Model by Hahnfeldt, Panigrahy, Folkman and Hlatky, [Cancer Research, 59, (1999),  pp.  4770-4775]:

* Application of optimal control to a system describing tumor anti-angiogenesis, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, Japan, July 2006, pp. 478-484

 

* Anti-angiogenic therapy in cancer treatment as an optimal control problem, submitted for publication to SIAM J. on Control and Optimization, currently under revision

 

 

Modification considered by A. Ergun, K. Camphausen and L.M. Wein [Bulletin of Mathematical  Biology, 65, (2003), pp. 407-424]:

* A synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors, Proceedings of the 44th IEEE Conference on Decision and Control (CDC), Sevilla, Spain, December 2005, pp. 934-939

 

* Optimal control for a system modelling tumor anti-angiogenesis, ICGST - International J. on Automatic Control and Systems Engineering (ACSE), 6, 2006, pp. 33-39

 

 


Modification considered by A. d'Onofrio and A. Gandolfi, [Mathematical Biosciences, 191, (2004), pp. 159-184]:

* Analysis of a mathematical model for tumor anti-angiogenesis, submitted for publication to Optimal Control, Applications and Methods, currently under revision

 

 

An analysis of singular arcs for the general form of inhibition and stimulation is given in the paper

·       On a class of systems describing tumor anti-angiogenesis under Gompertzian growth, WSEAS Transactions on Systems, to appear, 2007

      

 

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