Optimal Control for Mathematical Models

for Cancer Chemotherapy

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

Analysis of Cell-Cycle Specific Mathematical Models for Cancer Chemotherapy:

In this research we considered the analysis of existing cell-cycle specific models for cancer chemotherapy (originally formulated by A. Swierniak and M. Kimmel) as an optimal control problem. The state variables describe the numbers of cancer cells in various phases of the cell-cycle that are clustered into compartments in order to properly model the actions of the drugs. For example, mitosis is typically combined with the second growth phase since it is difficult to distinguish between cells in these phases and this is were most cytotoxic agents (killing agents) are active. Other drugs act as blocking agents which delay progression through the cell-cycle and are mostly effective in the synthesis stage. For some cancers (like leucemia) so-called recruiting agents are used to make cells leave the dormant stage (cells simply are not vulnerable there) and reenter the cell-cycle where they can be attacked.

We considered various models as optimal control problems minimizing an objective that was linear in the control, the drugs used. The use of a linear objective departs from the more commonly employed quadratic forms in the literature and leads to mathematically more difficult problems. In our papers we showed that optimal protocols are bang-bang (i.e. consist of full dose sessions with rest periods in between) and that so-called singular controls (which would administer the drugs are varying lower doses) are not optimal. In fact, they are maximizing rather than minimizing. This agrees with standard medical practice in this country. However, the structure of optimal controls is vastly different from what is obtained with the use of a quadratic objective.

Some References:    unless noted otherwise all papers are co-authored with U. Ledzewicz

* Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications (JOTA), 114, (3), 2002, pp. 609-637

* Analysis of a Cell-Cycle Specific Model for Cancer Chemotherapy, J. of Biological Systems, 10, (3), 2003, pp. 183-206,

* Optimal control for a bilinear model with recruiting agent in cancer chemotherapy, Proceedings of the 42^nd IEEE Conference on Decision and Control (CDC), Maui, Hawaii, December 2003, pp. 2762-2767

* Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. of Applied Mathematics and Computer Science, 13, (3), 2003, pp. 357-368 (with U. Ledzewicz and A. Swierniak)

Models including Pharmacokinetics (PK) and/or Pharmacodynamics (PD):

In view of the tremendous complexity of the medical problem that is cancer treatment, for the analysis of mathematical models it makes sense to start with simplified models and then incorporate increasingly more complex and medically more realistic features into the model. In this sense a commonly made simplification in the literature (that also is made in the models underlying the publications listed above) is to identify the drug dosage with its concentration and even more, with its effects. In reality these clearly are different phenomena and their relations are studied under the names of pharmacokinetics (PK) and pharmacodynamics (PD). Pharmacokinetic equations model the drug’s concentration in the body plasma and pharmacodynamics models the effectiveness of the drugs. In short, PK/PD stands for the description of the full process, also known as drug delivery in the medical literature.

In the papers below we investigated whether the addition of PK/PD does change the qualitative structure of solutions and when it would suffice to consider the simplified models. We have shown that the addition of linear models for PK and/or PD to chemotherapy models in fact does not change the qualitative structure and only cause small perturbations quantitatively. While linear models are commonly used for PK, realistic relations for PD (sigmoidal or saturation models) introduce strong nonlinearities into the model. In this case convexity properties of the functions defining PD are strongly tied in with whether the Legendre-Clebsch condition for optimality of singular arcs is satisfied or not. For a sigmoidal model of drug delivery our initial results suggest a structure of optimal controls which provide a quick initial boost (bang-bang) and then regulate the concentration through slowly varying infusions (singular).

Some References:  

* The influence of PK/PD on the structure of optimal controls in cancer chemotherapy models, Mathematical Biosciences and Engineering, 2 (3), 2005, pp. 561-578

* Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, to appear

Papers on Drug Resistance:

The tendency towards optimal singular controls, i.e. varying partial doses, shown when PK/PD is included, becomes even more prominent as the model is expanded to include drug resistance, probably the single most important and what has been called “certainly the most frustrating” one of all the limiting factors of conventional drug therapies. In several papers we formulated and analyzed models for the scheduling of cancer drugs in the presence of developing drug resistance both for a single drug and multi-drug treatments. The mathematical question is not how drug resistance can be avoided (this simply is not possible), but how through the scheduling of the drugs the onset of drug resistance could be delayed as long as possible. We have shown that after a period of full dose therapy, as resistance builds up, singular controls become optimal. This reflects the changing balance between the damage done to cancer and healthy cells. Once resistance passes a certain threshold, it becomes “optimal” to no longer give drugs. Similar conclusions are valid if more than one drug is considered. These conclusions fully agree with medical experience when resistance to anti-cancer drugs makes many treatments fail and can easily be understood in a continuous-time Markov chain model.

Some References:  

* Finite dimensional models of drug resistant and phase specific cancer chemotherapy, J. of Medical Informatics and Technologies, 8, 2004, pp. 5-13

* Drug resistance in cancer chemotherapy as an optimal control problem, Discrete and Continuous Dynamical Systems, Series B, 6, (1), 2006, pp. 129-150