IEEE Computational Intelligence Magazine - November 2021 - 95

only for a single NP. The population size
is P = 20, reproduction selection is implemented
as a tournament of size T = 2,
and replacement selection is the inverse.
Offspring individuals replace the selected
individual only if that individual is less fit,
namely it results in a higher amount of
remaining cancer cell agents. The mutation
procedure is executed on one randomly
selected gene by modifying it with
a random step size of s = [‒5;5]%. The
computational budget is set as 1000 evaluations
with PhysiCell, thus, given the 5
run static sampling approach, the population
evolves for 200 generations.
The results of the evolution with the
EA are depicted in Figs. 11 (example
run) and 12 (average of 5 runs). Fig -
ure 11(a) shows that the average fitness
of the population (remaining cancer
cells agents) converges to c. 880 agents
after 200 generations of the evolution of
the population, from c. 950 agents of the
initial random population. That is an
improvement of c. 7.3%. Quite similar
results are provided by all the tests executed,
as illustrated in Fig. 12(a) showing
the average and 95% confidence
levels of the cumulative results. Note
here that the confidence levels of 95%
are calculated with the mean and standard
deviation of the sample and indicate
the probability (here 95%) with
which the estimated interval contains
the true value of the parameter. For the
best individual in the population the
final fitness seems to converge close to
850 agents, while the one initially randomly
generated has a fitness of c. 900,
an improvement of 5.5%, for one example
run (Fig. 11(b)). The cumulative
results depicted in Fig. 12(b) reveal that
during the rest of the runs the improvement
is slightly smaller.
In order to enable the variation of
genome length-and hence more than
one types of NPs per solution-the
mutation operator described in Section
III is incorporated. The mutation can
therefore alter a randomly chosen gene
allele or add one type of NPs (with randomly
chosen parameters) to the simulated
treatment. Both cases have the same
possibility of occurring, which is 50%. A
maximum of 10 NPs types is allowed
here. As explained before, despite the
amount of types of NPs being tested per
treatment, a total of 50 worker agents are
injected in the simulated treatment. The
initial population used for all tests is the
same as for the previous tests (only gene
allele mutation) and consisted of solutions
with only one type of NPs. Note
that the existence of an additional type
(or multiple types) of NPs potentially
alters the fitness landscape significantly
due to the complex interactions with the
cancer cell agents.
With the ability to add multiple types
of NPs, the optimization process reaches
better results within the 200 generations
as depicted in Figs. 13 (example run) and
14 (average of 10 runs). In Fig. 13(a), the
average fitness of the population is
depicted for an example run, which converges
to c. 450 agents at the end of the
evolution, while the same metric of the
initial random population is c. 950 agents.
Thus, the alternation in the number of
types of NPs injected leads to an
improvement of c. 52.6%. Investigating
the cumulative results of average and 95%
confidence levels of all the runs that are
portrayed in Fig. 14(a), there is evidently
a further improvement of the average fitness
of the population (average of 431
agents). Moreover, there is an improvement
of 52.2% when comparing the final
fitness of the best individual discovered
during one run (approximately 430
agents) with the best individual randomly
generated in the initial generation (c. 900
identical in all runs), as outlined in
Fig. 13(b). In Fig. 14(b) where all the
results of 10 runs are considered, it can be
seen that during the rest of the runs the
improvement is similar and slightly better
(average of 405 agents).
Despite the fact that the maximum
number of NPs types can be 10, the
composition of the best solutions usually
converges to lower complexity. Taking
into consideration the average of all
Average Fitness of Population
1,000
800
850
900
950
020406080 100 120140 160180 200
Generations
(a)
Best Individual Fitness of A Population
1,000
800
850
900
950
020406080 100 120140 160180 200
Generations
(b)
FIGURE 12 Average and confidence levels (95%) results from 5 runs of EA. (a) Evolution of
average fitness of the population and (b) evolution of the best individual in the population.
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 95
Best Fitness of Population
Average Fitness of Population

IEEE Computational Intelligence Magazine - November 2021

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