Transcript Lindenmayer Systems (L

Evolutionary Design of
Cancer Chemotherapies
Minaya Villasana, and
* Gabriela Ochoa
http://www.ldc.usb.ve/~gabro/
References
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M. Villasana and G. Ochoa (2004) Heuristic design of cancer
chemotherapies. IEEE Transactions on Evolutionary Computation (In
Press)
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M. Villasana y A. Radunskaya (2003) A Delay Differential Equation
Model for Tumor-Growth, Journal of Matheatical Biology, vol 47, pp
270-294
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Houck, C. and Joines, J. and Kay, M. (1995) A Genetic Algorithm for
Function Optimization: A Matlab Implementation. North Carolina State
University
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Nikolaus Hansen and Andreas Ostermeier (2001) Completely
Derandomized Self-Adaptation in Evolution Strategies, Evolutionary
Computation, 9:2, pp 159-195
Content
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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
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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)
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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
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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
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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)
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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
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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
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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
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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)
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They appear naturally when considering the
cell cycle
Previous work supports use of DDE in
modeling cell proliferation
τ
S
Tumour
I
u
M
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
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
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
Basin of attraction
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The system has up to 5
fixed points (depending on
parameter values)
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One fixed point is always
present: S = 0,M = 0, I
=k/d1 (k = bone marrow production of
immune cells, d1 natural cell death)
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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
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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
TI(tf) + TM(tf) + 1/tf ∫TI(tf)+TM(tf)dt
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Min
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Subject to Equations in (1),
Added restriction: I ≥ Imin
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Characteristics of the solutions
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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
1
0
Problem encoding (first approach)
Bounds of the control variables
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According to the medical literature, Taxol
maximun tolerated dose is 5 days of infusion
Problem encoding
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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)
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Algorithms
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Two evolutionary algorithms (freeware implementations
in Matlab)
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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
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GAOT, tested in a series of problems against SA. GA with real valued
encoding superior to both binary GA and SA
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Several genetic operators suited for float encoding (Michalewicz, 92)
(frequency of application as suggested by GAOT)
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Mutation: uniform (4), non-uniform (4), multi-non-uniform(6)
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Recombination: simple (4), arithmetic (4), heuristic (2)
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Selection: Normalized geometric ranking
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Generational GA,
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Population size: 30
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Fixed termination criterium: 100 generations
CMA Evolution Strategies
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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,
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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
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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