IEEE Computational Intelligence Magazine - November 2021 - 86

a purely mutation-based approach. Here
the term fitness refers to the value of the
objective function in the given optimization
problem. The insights gained are
then applied within the bio-engineering
domain. In particular, we simulate a
nanoparticle (NP) based cancer treatment,
where multiple different types of
NPs can be included. Given the high
computational cost of running the cancer
simulator, the conclusions drawn from
the abstract model are used as guidelines
to economize computational resources.
There are various procedures in
nature that can produce novel DNA
sequences and hence vary genome
lengths, such as horizontal gene transfers,
recombination events, whole genome
duplications, retrotransposons, and others.
A novel sequence can have no immediate
function and thus can be subject to
genetic drift or can be under positive or
negative selective pressure, due to beneficial
or detrimental effects of mutation
[17], gene dosage effects [18], the subsequent
specialization of a duplicated function
[19], etc. In this paper, following [1],
a simple procedure to vary genome
length is implemented, where the length
of an existing genome is increased with a
chosen number of random genes. By
doing so, the fitness of the altered individual
will be instantly modified as random
contributions are included. To study
these phenomena, a well-known abstract
NK model [20] is utilized. The NK
model allows dynamic alterations in the
size of fitness landscapes of specified ruggedness.
Our results suggest that landscape
ruggedness, the length of the new
sequence added, and the presence of
gene deletion can all affect the evolution
of genome length.
We used these findings to inform
investigations of optimizing the design of
a simulated, NP-based, targeted drug
delivery system for cancer treatment. This
simulation is based on PhysiCell [21], an
open source physics-based cell simulator,
which extends BioFVM [22], a large-scale
transport solver. It should be noted that,
given the high complexity of the simulator
and its long execution times, the evolutionary
optimization should ideally be
kept under a computational budget.
PhysiCell source code was altered to simulate
the injection of multiple types of
NPs with different behaviors within the
same treatment. Since the appropriate NP
properties for a given type of tumor are
unknown, optimization can be applied as
a search through a space of variable size.
Therefore, to optimize both the number
of types of NPs and their different features,
we use a variable-length evolutionary
algorithm.
The optimization problem of designing
robust anti-cancer treatments can be
expressed as:
minimize () where
X [,...,
,],
X
subject to
(, )( ,)
=
xx
xx j
x
(, )( ,)
jj
11 21
45
f X
,
= 110
0 # (,j)1 # 1,
j 110
01 10
0101 10
0101 10
0101 10
#
#
#
#
xj
xj
xj
xj
(, )
(, )
(, )
(, )
2 j
3
4
5
j
j
j
#
#
#
#
1,
,[ ,...,]
,[ ,...,]
,[ ,...,]
=
=
=
=
=
where f is the objective function or fitness
function that is defined as the size
of the tumor after the application of a
treatment (i.e., the number of simulated
cancer cells at the end of a PhysiCell
run) and needs to be minimized.
X represents the design variable vector
(i.e., the genome) which is composed
of sets of five variables for each type of
NPs in the solution. Note that variable
j in the above equation is the amount
of different types of NPs used simultaneously
in a treatment and can take
values from 1 to 10, so the genome has
variable length. Moreover, the bounds
are presented for the following variables:
the attached worker migration
bias
^ 1, j^
x
x
x
tion bias ^ 2, j^
sion ^ 3, j^
^ 4, j^
x
x
hh , unattached worker migrahh
,worker relative adhehh
, worker relative repulsion
time (min) ^ 5, j^
hh , worker motility persistence
hh For further details
.
of the simulation refer to Section IV
and for the parameters of the implemented
standard steady-state EA refer
to Section V.
Our findings indicate that more
complex treatments (i.e., with more
than one NP type) are more effective
86 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2021
[,...,]
[,...,]
[,...,]
than those with a single NP type. Significantly,
the evolved treatment complexity
does not reach the maximum
available complexity. Therefore, bloat
phenomena, although present to a certain
extent, are not prevailing in this
application. Following findings in the
NK model, the average treatment complexity
varies slightly with the increase
in solution length size. The smaller the
number of NP types added per mutation
step, the shorter the length of the
typical solution found. Since there is no
significant difference in the final fitnesses
of the scenarios explored, for practical
reasons (i.e., ease of manufacturing, the
problem of interactions between drugs
and lower toxicity), the solutions of
smaller complexity can be considered as
preferable in vivo.
The paper is arranged as follows: the
next two sections present the NK model
and the behavior of a simple NK mutation-based
approach to variable-length
optimization. The following two sections
introduce the bio-engineering
problem and the results of the optimization
methodology applied to it, respectively.
In the final section conclusions
are drawn.
II The NK Model
The NK model was
introduced to
investigate the characteristics of rugged
fitness landscapes [20]. The basic model
contains two parameters: N, the length
of the binary genome; and K, the number
of genes in a genome -the position
of which are typically chosen at random-
that influence the fitness contribution
of each gene. Increasing K
increases the ruggedness of the landscape,
leading to an increase in the number
of local optima and a decrease in
their typical height [23]. Since the
model assumes a high complexity of the
gene interdependence effect, the only
appropriate tactic is to assign random
values to their contribution to fitness. As
depicted in the example (Fig. 1), a table
is defined with 2 K 1+^h
fitness values randomly
chosen between 0 and 1, where
there is one fitness for each combination
of traits and the fitness contribution of
each gene is found in the table. The total
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