Evolutionary Design of Cancer Chemotherapies

Gabriela Ochoahttp://www.ldc.usb.ve/~gabro/

M. Villasana and G. Ochoa (2004) IEEE Transactions on Evolutionary Computation (In Press)

Content

Cancer and the cell cycle Chemotherapy Optimal control theory Aim of the study The model of tumour growth Problem formulation 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

G0

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. These cycles can be very different, depend on each patient and clinician

Drug protocol: duration and number of the cycles. (application and recovery periods)

Taxol (paclitaxel) cycle-phase-specific 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 protocol for Taxol is not yet known

Optimal Control Theory (1)

A suitable choice of the controls can force a control 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, where the control are the added nutrients or pesticides

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

Optimal Control Theory (2)

Examples of cost functions: time taken to reach a desired goal, energy utilized, manufacturing costs, etc.

Optimal Control Theory: Problem of minimising the cost function over all admissible controls

A certain type of continuous-time OCPs, with bounded control, is known a priori to have optimal control which is always at the bounds

it is called Bang-bang control and is fully characterised by the switch times, which are the only variables that need to be optimised

Aim of the study

To design drug protocols 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 (OCP) is formulated, where the control is the added drug (Taxol)

Evolutionary algorithms are used to solve the OCP

The model of tumour growth

Cycling tumour 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

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

the cell cycle Previous work supports use of DDE in

modelling cell proliferation

I

S M

uτ

Tumour cells reside in interphase τ units of time, before continuing in the cycle to mitosis State variables: number of cancer cells, and immune cells. Controls: drug dosages

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)

Fixed point always present: (S,M,I) = (0, 0, k/d1) (k = bone

marrow production of immune cells, d1 natural cell death)

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 which 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 given threshold?

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

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

Min I(tf) + M(tf) + 1/tf ∫I(tf)+M(tf)dt Subject to Equations in (1), Added restriction: I ≥ Imin ( I should maintain its initial

state)

Characteristics of the solutions It is shown analytically

(Ponttryagin’s Maximum Principle) that optimal solutions are bang-bang.

Treatment protocols that alternate maximum doses of chemotherapy with rest periods where no drug is administereds

The problem reduces to determine the best switching times (application/resting)

0

1

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

tolerated dose is 5 days of infusion at 30mg/m2/day every 3 weeks.

A lower limit of 3 hour infusions, is a common practice when Taxol is used

Standard protocols for Taxol: a week of doses followed by a resting periods of about 28 days

Administration times: [0.2, 5] days Resting times: [0, 30] days (again following current

practice)

Model parameters

Parameters were estimated from data available in the literature, specifically breast cancer data (cell cycle time,

proportion of cells proliferating and extent of cell loss) Drug parameters, using references about Taxol (effects of the

drug, decay rates, etc) The immune system is comprised of many types of cells.

Cytotoxic T cells (CTL) were modelled since they play an important role in combating cancer

The parameters chosen represent a patient with a rapidly growing tumour and an immune system not able to control the tumour progression

Initial function outside the basin of attraction (S(0), M(0), I(0)) = (1.3, 1.2, 0.9)

Problem encoding (first approach) Binary strings: where 0

represents no drug administered during the respective time-stage and 1 drug administered

Time is discretised in adequate time-stages, in this case 3 hours (the lower limit for drug infusions)

Algorithm: GA with binary encoding

Drug protocols not-suitable for practice: very short cycles (factors such as side effects, and emotional stress to the patient should be considered)

Problem encoding

Float encoding: Two types of control variables are distinguished Administration-time lengths: [0.2, 5] days Resting-time lengths: [0, 30] days

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. Excessively slow runs! Parameter tuning was not feasible

Genetic Algorithm

GAOT, tested in a series of problems against SA. GA with real valued encoding (enhanced with local search -SQP) superior to both binary GA and SA

Several genetic operators suited for float encoding (Michalewicz, 92) (freq. of application as suggested by GAOT)

Mutation: boundary (4), 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 criterion: 100 generations

CMA Evolution Strategies

Mutation strength σ, adaptation, distinctive component of ESs. σ (a single number in basic ES) is replaced by an N x N matrix

(Covariance matrix). Several covariance matrix adaptation methods have been proposed

CMA: Cumulative Mutation Strength Adaptation, attempts to de-randomize the process of mutation adjusting. deterministic rather than based on variation and selection accumulates 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

As the perturbation operators, we selected the mutation operators provided by GAOT (boundary, uniform, non-uniform, muli-non-uniform)

Annealing schedule: Tmax = 10, temperature decreasing factor = 0.85

Each temperature was tried 30 times, and the total number of (different) temperatures tested was 100

1.4 12.9 5 12.3 5 11.9 5 11.4 5 11.2 5 30 5 30 5 30 5 30

Extended Bounds

Most runs reached the maximum of 5 days for application times on all cycles. Moreover, from the sixth cycle onwards, resting times also saturated to the maximum of 30 days

With the aim of exploring a wider range of schedules, we ran experiments extending the upper bounds of administration and resting times Administration times: [0.2, 15]

Resting times: [0, 30]

1.1 26.6 7.9 33.5 8.9 42.5 10.1 45.5 10.7 50 11.3 50 11.6 50 11.8 49.9 11.9 50

Summary and Conclusions

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

The three algorithms tested produced efficient drug schedules, suitable for practice

ES has the best speed of convergence and quality of solutions

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