Evolutionary Design of Cancer Chemotherapies

Minaya Villasana, and

* Gabriela Ochoa

http://www.ldc.usb.ve/~gabro/

References

M. Villasana and G. Ochoa (2004) Heuristic design of cancer chemotherapies. IEEE Transactions on Evolutionary Computation (In Press)

M. Villasana y A. Radunskaya (2003) A Delay Differential Equation Model for Tumor-Growth, Journal of Matheatical Biology, vol 47, pp 270-294

Houck, C. and Joines, J. and Kay, M. (1995) A Genetic Algorithm for Function Optimization: A Matlab Implementation. North Carolina State University

Nikolaus Hansen and Andreas Ostermeier (2001) Completely Derandomized Self-Adaptation in Evolution Strategies, Evolutionary Computation, 9:2, pp 159-195

Content

Cancer and the cell cycle Chemotherapy Optimal control Aim of the study The model of tumour growth Optimal control problem The Algorithms (GA, ES, SA) Experiments and Results

Cancer Cancer is the uncontrolled growth of

cells due to damage to DNA (mutations) In adult life, normal cells grow and divide

to form new cells only when the body needs them (to replace worn-out or dying cells and to repair injuries)

Mutations can sometimes disrupt this orderly process. New cells form when the body does not need them, and old cells do not die when they should

These extra cells produce a tumour that may be cancerous

The cell cycle Cycle of events from one cell

division to the next. The phases of the cell cycle are:

G0 is a period where cells exist in a quiescent state.

G1 is the first growth phase. S, during which the DNA is

replicated, where S stands for the Synthesis of DNA.

G2 is the second growth phase, also the preparation phase for

M or mitosis is, the actual division of the cell into two daughter cells

Chemotherapy

Four major types of treatment for cancer: surgery, radiation, chemotherapy, and biologic therapies

Chemotherapy: treatment with powerful drugs that are most often given by mouth or by injection

Most chemotherapeutic drugs work by impairing mitosis (cell division), effectively targeting fast-dividing cells. Some drugs cause cells to commit apoptosis (effectively "cell suicide")

Other fast dividing cells such those responsible for hair growth and replacement of epithelium in the intestine are also affected (scientists have yet to be able to locate specific features of

malignant cells that would make them uniquely targetable)

Chemotherapy (Cycle-phase-specific drugs) Chemotherapy is given in cycles, each followed by a

recovery period Drug schedule: duration and number of the cycles.

(application and recovery periods) Taxol (paclitaxel) drug used today for treating breast,

ovarian, head and neck cancers Action of Taxol, 3 different mechanisms: (1) inhibits

mitosis,(2) induces apoptosis, and (3) enhances tumour radio sensitivity

The optimal scheduling for Taxol is not yet known

Control and Optimal Control

Control systems: dynamical systems whose laws are not entirely fixed, but depend on parameters called controls

A suitable choice of the controls can force the system to achieve a desired goal (Ex.: in a driving vehicle, the controls are the accelerator, the brakes and the steering wheel)

Control theory can be applied to other areas such as the growth process in organisms and populations

In general, several choices of controls to steer a system from an initial state to a goal state. A selection among those succesful controls can be made to minimize some quantity (cost)

Examples of cost functions: time to a desired goal, energy, costs Optimal Control Problem (OCP): Mimimize the cost function over all

admisible controls

Aim of the study

To design drug schedules with Taxol as the only chemotherapeutic agent

Patient dynamic: mathematical model of tumour growth, interactions with immune cells, and application of a cycle-phase specific drug

An optimal control problem is formulated, and evolutionary algorithms are used to solve it

Drive the system towards the basin of attraction of the tumour-free fixed point

The model of tumour growth

Cycling toumour populations divided into phases (G0 is not considered). Two compartments cells in mitosis ,M, and in interphase,I, (G1+ S+G2)

Interactions of tumour cells and drug with the Immune system, I, (represented by cytotoxic T cells, CTL)

The size of the Immune population represents the patient`s health. Its ability to fight cancer is included in the model

On major difference with previuos work: use of Delay Differential Equations (DDE) They appear naturally when considering the

cell cycle Previous work supports use of DDE in

modeling cell proliferation

I

S M

uτ

Tumour cells reside in interphase τ units of time, before continuing in the cylce to mitosis

Equations of the model' derivatives with respect to time

S: tumour cells in interphase (G1+S+G2)

M: tumour cells in mitosis

I: immune sytem cells (CTL)

u: concentration of the drug. Linear convex combination of u1, u2

a1, a4: rates at which cells reproduce, together with τ regulate the pace of cell division

d1I, d2S, d3M: proportions of natural cell death or apoptosis.

MI, SI: competition terms, losses due to encounters among the different cell types (ci)

1-e-k2u : effect of the drug. Removal of cells

u1, u2: drug decay modeled with two elimination rates (bi-exponencial curve): 1) Fast rate: blood stream, 2) Slower decay: tissue.

c(t): control function, drug injected at time t

Basin of attraction

The system has up to 5 fixed points (depending on parameter values)

One fixed point is always present: S = 0,M = 0, I =k/d1 (k = bone marrow production of

immune cells, d1 natural cell death)

Represents a tumour-free environment with positive immune population (desirable scenario)

Basin of attraction of the tumour free fixed point (calculated numerically)

Basin of attraction: set of initial conditions, for wich the orbits go towards an equilibrium.

Optimal control problem

What is the best course of treatment with the single agent taxol on de model described, so that the tumour is eradicated while the immune system remains above a threshold?

c(t): (control) amount of drug introduced at time t Goal: minimize the average and final tumuor size

Min TI(tf) + TM(tf) + 1/tf ∫TI(tf)+TM(tf)dt Subject to Equations in (1), Added restriction: I ≥ Imin

Characteristics of the solutions It is shown analytically that the

OCP problem admits bang-bang solutions.

Bang-bang solutions: the control (i.e. the drug injected at time t) attains its maximum or minimum value (for bounded controls)

The problem reduces to determine the best switching times of the solution. i.e., the times in which the drug begins or ceases to be administered

0

1

Problem encoding (first approach)

Bounds of the control variables According to the medical literature, Taxol

maximun tolerated dose is 5 days of infusion

Problem encoding

Two types of control variables are distinguished (both encoded as float numbers) Administration-time lengths ([0.2, 5] , [0.2, 10] days) Resting-time lengths

A parameter, P, indicates the number of switching times. We found empirically that nine (application/resting) cycles were enough to drive the tumor into the basin of attraction (i.e. P = 18)

Algorithms

Two evolutionary algorithms (freeware implementations in Matlab) Genetic Algorithm: GAOT Genetic Algorithms for Optimization

Toolbox (Houck, C. et al., 1995) Evolution Strategies: CMA-ES derandomized ES with covariance

matrix adaptation (Hansen N. Ostermeier A., 2001) A simulated annealing algorithm was also implemented For comparison purposes, we set a maximum of 3,000

evaluations for each algorithm Each function evaluation required the integration of a

DDE system for large periods of time. Excesively slow runs! Parameter tuning was not feasible

Genetic Algorithm

GAOT, tested in a series of problems against SA. GA with real valued encoding superior to both binary GA and SA

Several genetic operators suited for float encoding (Michalewicz, 92) (frequency of application as suggested by GAOT) Mutation: uniform (4), non-uniform (4), multi-non-uniform(6) Recombination: simple (4), arithmetic (4), heuristic (2)

Selection: Normalized geometric ranking Generational GA, Population size: 30 Fixed termination criterium: 100 generations

CMA Evolution Strategies

Mutation strength adaptation, distinctive component of ESs. The mutation strength σ (a single number in basic ES) is replaced by

an N x N matrix (Covarience matrix) Several covariance matrix adaptation methods have been proposed CMA: Cumulative Mutation Strength Adaptation,

attempts to derandomize the process of mutation adjusting. deterministic rather than based on variation and selection acumulates and analyzes information over a number of time steps

Was shown to have convergence velocity improvements over other ESs on a large test suite

CMA ES provides default parameter values: λ = 4 + |3 lnN|, μ = |λ/2|, weights for recombination

We set the number of iterations = 250

Simulated Annealing

Extended Bounds

Summary and Conclusions

The design of efficient drug schedules is formulated as an optimal control problem admitting bang-bang solutions

The problem is stated as finding the optimal switching times

Modern heuristics methods are a good choice for optimising complex systems

All methods produced efficient drug schedules ES has the best speed of convergence and

quality of solutions