Computational Optimisation of Cancer Chemotherapies Using Genetic Algorithms

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Computational Optimisation of Cancer
Chemotherapies Using Genetic Algorithms


Andrei Petrovski
1

and John McCall
2

1

School of Computer and Mathematical Sciences, The Robert Gordon University,
ABERDEEN, AB25 1HG, United Kingdom. e
-
mail:
ap@scms.rgu.ac.uk
.

2

address as above. e
-
mail:
jm@scms.rgu.ac.uk
.



1.

Introduction


Chemotherapy is a common mode of treatment which attempts to alter the natural
course of cancer. The design of

chemotherapeutic treatments requires selecting
different combinations of drugs and dosages to be administered over a treatment
period. The designer has to balance the benefits of the treatment against the, often
serious, toxic side effects. At present t
reatments are developed and evaluated
through empirical clinical trials. This process prohibits the full exploration of all
possible treatment options and has led to a large number of patients being treated
in sub
-
optimal ways. The intention of the prese
nt paper is to introduce an
intelligent approach to optimising chemotherapy regimens.


2.

Problem formulation


An optimisation problem can often be formulated in the following canonical form:

Find an optimal value of the objective functional

subject to

the state equation

and the set of constraints

where

is the state vector described by the function
;

is
the vector of c
ontrol variables. The optimal value of

is achieved by
varying the control variables

within the boundaries specified by
(1) and (2). All functions here are to be regarded as discrete, evaluated at time
inter
vals
.


In cancer treatment the state of the tumour at time

is conventionally described
by the number of cancerous cells

comprising the tumour. Therefore the
state vector in this case
is one
-
dimensional (
), and the state equation (1)
becomes:


where

is a real
-
valued function which models the growth of an untreated
tumour;

is the concentration l
evel of anti
-
cancer drugs in the bloodplasma;

is a quality representing the efficacy of the anti
-
cancer drugs used. The
drug concentration

in (3) is the only variable directly controllable by the
oncologist.

Therefore the problem of optimising treatments can be regarded as the
problem of selecting the most effective drug administration regimen
.


The optimal treatment regimen is sought in the form of a discrete dosage program
with

doses given at times
. Each dose is a cocktail of

drugs
characterised by their concentration levels

in the blood
plasma. These levels can be varied within the boun
daries specified by the
constraints (2) which are detailed in (McCall and Petrovski, 1999). The conflicting
nature of these constraints and the intention to develop a model
-
independent
approach to chemotherapy optimisation makes the utilisation of mathemat
ical
optimisation techniques very difficult. In contrast, methods of computational
optimisation (Genetic Algorithms in particular) are most suited to this task.



3.

Genetic Algorithms as chemotherapy optimisers


Genetic Algorithms (GAs) have long been appl
ied in engineering to find high
quality solutions to optimal control problems and more recently applications are
being found in biology and medicine (Baeck, Fogel and Michalewicz, 1997). GAs
are well suited to the concurrent manipulation of models of varyi
ng resolution and
structure due to their ability to search non
-
linear solution spaces with no
requirement for gradient information or
a priori

knowledge relating to model
characteristics. Since the present work is concerned with finding optimal solutions
for a number of different mathematical models of tumour growth, for some of
which the precise values of model parameters are unknown, the abovementioned
properties of Genetic Algorithms are highly advantageous.


The optimal solution to the problem of chemo
therapy optimisation is sought by
GAs in the solution space

of control vectors representing multi
-
drug treatments
. The corresponding representation space

is a Cartesian
product




of allele sets
. Each allele set can take either the form of a 4
-
bit string (4
-
bit
encoding scheme)

,


or a simpler form:
, referred to as the sign encoding sc
heme. The
4
-
bit encoding allows each concentration level

to take an integer value in the
range of 0 to 15 concentration units. (This level of precision is justified by the
difficulty of controlling the drug concentration
in vivo
.)

The sign scheme is used
to represent either no change or an alteration of

to the drug levels

and
it allows searching for an optimal modification of a given treatment regimen.

The quality of the elements of


is quantified by the fitness function
. By matching the fitness function with a given treatment objective it
becomes possible to start an evolutionary search for the elements of

that
yield
the best values of
. The fitness function takes into account all aspects of
chemotherapy optimisation, viz. the treatment objective, the model of tumour
growth and the degree of constraint violation. Given an appropriate fitne
ss
function, the evolutionary search of Genetic Algorithms develops into a versatile
technique applicable to a wide range of problems related to chemotherapeutic
treatment.


4.

Results and Discussion


The versatility and effectiveness of the GA chemotherapy o
ptimisation is
illustrated in this paper by considering four commonly used models of tumour
growth (Marusic
et al
, 1994):



exponential model:
;



von Bertalanffy model:
;



Verhulst model:
;



G
ompertz model:
.


For each of these models two treatment scenarios have been examined, viz. when
the treatment objectives are either to eradicate the tumour (curative treatment) or
to prolong the patient survival time (palliative tre
atment). The objectives of
curative and palliative treatments conflict with each other (Martin and Teo, 1994).
For this reason curative and palliative treatments are optimised separately by GAs.
The results of this optimisation are assessed by comparing t
he characteristics of
the treatment regimens found by Genetic Algorithms with those of a commonly
used schedule CAF. The latter schedule is a cocktail of three anti
-
cancer agents
(Cyclophosphamide
-
Doxorubicin
-
5
-
Fluorouracil); the listing of CAF is given i
n
Dearnaley
et al

(1995).

The first characteristic of a treatment regimen is its curative effect, which can be
expressed as follows:


Secondly, the palliative effect of a given treatment

can be formulated as:


where

is such that

The four tumour growth models and the two treatment scenarios addressed in this
paper define eight different solution spaces to search through. The search is
implem
ented by Genetic Algorithms with a randomly generated initial population
consisting of 50 chromosomes. The roulette
-
wheel selection procedure with a
linear fitness normalisation scheme is applied, and two
-
point crossover and bit
-
wise mutation are used for
chromosome modification. The single best
chromosome of each generation is copied into the succeeding generation.



The application of Genetic Algorithms to chemotherapy optimisation has shown
that irrespective of the tumour growth model underlying cancer
treatment, Genetic
Algorithms can find a treatment regimen which noticeably outperforms one of the
most commonly used chemotherapy schedules


CAF. The comparative results of
CAF and the regimens found by GAs are given in Table 1.


Table 1. Comparison o
f the GA optimal regimens with CAF

Model

Treatment regimen



Exponential

GA regimen

CAF


6.1002

10.3616

35

31

Von Bertalanffy

GA regimen

CAF


7.9120


9.7691

40

40

Verhulst

GA regimen

CAF


7.3572

10.1039

37

32

Gompertz

GA regimen

CAF


4.6843

11.2897

38

24

Therefore, the results of our work show that the underlying model of tumour
growth does not affect the ability of Genetic Algorithms to achieve the curative
and palliative objectives of chemotherapeuti
c treatment. Moreover, the developed
method of treatment optimisation can be applied to other drug therapies where a
dose
-
response model and well
-
formulated constraints already exist.


5.

References


[1]

T. Baeck, D. Fogel and Z. Michalewicz (eds.), 1997. Handboo
k of
Evolutionary Computation. Oxford University Press.

[2]

D. Dearnaley, I. Judson and T. Root, 1995. Handbook of adult cancer
chemotherapy schedules. The Medicine Group (Education) Ltd.

[3]

R. Martin and K. Teo, 1994. Optimal control of drug administration in
cancer chemotherapy. World Scientific.

[4]

M. Marusic
et al.
, 1994. Analysis of growth of multicellular tumour
spheroids by mathematical models.
Cell Proliferation
,
27
, pp. 73
-
94.

[5]

J. McCall and A. Petrovski, 1999. A Decision Support System for Cancer
Chemothe
rapy using Genetic Algorithms. Proceedings of the International
Conference on Computational Intelligence for Modelling, Control and
Automation. IOS Press: ISBN 90
-
5199
-
474
-
5, pp. 65
-
70.

[6]

T. WHELDON, 1988. Mathematical models in cancer research. Adam
Hilger
: Bristol and Philadelphia, 1988.