1,2,4 2 2 2,3 2
J. Alan Calderón Ch ; Julio Tafur-Sotelo ; Benjamín Barriga-Gamarra ; John Lozano ; Gonzalo Solano
PROPOSAL FOR AN EVENT RECONSTRUCTION ALGORITHM TO STUDY THE EVOLUTION OF
CANCEROUS CELLS
PROPUESTA DE ALGORITMO DE RECONSTRUCCIÓN DE EVENTOS PARA ESTUDIAR LA
EVOLUCIÓN DE LAS CÉLULAS CANCERÍGENAS
The Biologist
(Lima)
The Biologist (Lima), 202 , vol. (2),2 20 165-173.
ORIGINAL ARTICLE / ARTÍCULO ORIGINAL
1Applied Physics, Institute for Physics, Technical University of Ilmenau, Ilmenau 98693, Germany
2Pontificia Universidad Católica del Perú, Mechatronic Master Program and Energy Laboratory, Lima 32, Peru.
3Northern (Artic) Federal University named after MV Lomonosov, Arkhangelsk, Russia.
4Aplicaciones Avanzadas en Sistema Mecatrónicos JACH S.A.C.*Corresponding and main author:
alan.calderon@pucp.edu.pe
J. Alan Calderón Ch: https://orcid.org/0000-0002-6486-5105
Julio Tafur-Sotelo: https://orcid.org/0000-0003-3415-1969
Benjamín Barriga-Gamarra: https://orcid.org/0000-0002-7781-6177
John Lozano: https://orcid.org/0000-0002-8430-9480
Gonzalo Solano: https://orcid.org/0000-0002-0656-1031
The Biologist (Lima)
ISSN Versión Impresa 1816-0719
ISSN Versión en linea 1994-9073 ISSN Versión CD ROM 1994-9081
165
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ABSTRACT
Keywords: cancerous cells evolution – event reconstruction algorithm – 3D image reconstruction
In this work, a general algorithm is proposed to design the reconstruction of chemical/physical/biological
process events as one of the most complicated today: the evolution of cancer cells. Studying the evolution
of the boundary curves it is possible to make a three-dimensional (3D) integration, in addition the 3D
figure obtained can be explained through a mathematical model to estimate its geometric evolution after
physical/chemical reactions. In this work, there are analyzed images of each stage of the process based on
the evolution of cancer cells. Each image was processed in order to obtain a mathematical equation as a
reference to understand the geometry of the 3D structure based on its 2D image for each stage. On the
other side, with this information and the processing of each stage image, a mathematical equation was
achieved to describe the geometry of the structure between stages by "Optimal Prediction Analysis" which
is so important to gain understanding of the geometry of the structure with the internal process.
DOI: DOI: https://doi.org/10.24039/rtb20222021355
Este artículo es publicado por la revista The Biologist (Lima) de la Facultad de Ciencias Naturales y Matemática, Universidad Nacional Federico
Villarreal, Lima, Perú. Este es un artículo de acceso abierto, distribuido bajo los términos de la licencia Creative Commons Atribución 4.0
Internacional (CC BY 4.0) [https:// creativecommons.org/licenses/by/4.0/deed.es] que permite el uso, distribución y reproducción en cualquier
medio, siempre que la obra original sea debidamente citada de su fuente original.
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Cancerous cells evolution is plenty studied
nowadays (Tume-Farfán et al., 2013, 2014;
Villareño-Do-Dominguez et al., 2020), because of
its quite impact in human health since last century.
In order to study cancer cell evolution, it was
proposed many techniques that needed advanced
mathematic analysis. These techniques were
correlated with human genome and by medical
doctor supervision they can be a good support to
achieve successful diagnostics (Ribeiro & Shah,
2006). Nevertheless, the mathematic models use
methodologies to propose every cause of the
cancerous cell evolution, which can get an error
during the estimation of the diagnostic. For this
reason, in this work, it is proposed the algorithm
that was based in the achieved experience (3
Dimensions) of nanostructures geometric analysis
(Al-Haddad et al., 2015; Calderón et al., 2021). By
the designed algorithm, which depends of a
mathematical model that was based in multiple
input/output variables with support of image
processing, it is achieved estimations of growing
up of different structures and as a consequence to
get diagnostic in possible situations during the
evolution of the researched structure.
It means, the designed algorithm that was based in
the 3D reconstruction by an optimal prediction of
every input variable (not only images) can improve
its own result as geometrical or physical parameter,
that it is looked for a support for medical doctor
interpretations to study cancerous cells evolution.
The Biologist (Lima). Vol. 20, Nº2, jul - dic 2022
RESUMEN
Palabras clave: algoritmo de reconstrucción de eventos – evolución de células cancerígenas –reconstrucción de imágenes 3D
En este trabajo se propone un algoritmo general para diseñar la reconstrucción de eventos de proceso
químico/físico/biológico como uno de los más complicados hoy en día: la "evolución de las células
cancerígenas". Estudiando la evolución de las curvas de frontera es posible hacer una integración en tres
dimensiones (3D), además la cifra 3D obtenida se puede explicar a través de un modelo matemático para
estimar su evolución geométrica después de reacciones físicas/químicas. En este trabajo, hay imágenes
analizadas de cada etapa del proceso basadas en la evolución de las células cancerosas. Cada imagen fue
procesada con el fin de obtener una ecuación matemática como referencia para entender la geometría de la
estructura 3D basada en su imagen 2D para cada etapa. En el otro lado, con esta información y el
procesamiento de cada imagen de etapa, se logró una ecuación matemática para describir la geometría de
la estructura entre etapas mediante "Análisis de Predicción Óptima" que es tan importante para obtener la
comprensión de la geometría de la estructura con el proceso interno.
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Calderón et al.
INTRODUCTION In this work is proposed a general algorithm to
d e s i g n e v e n t r e c o n s t r u c t i o n o f
chemical/physical/biological process, such as one
of the most complicated nowadays “cancerous
cells evolution”.
The main objective in this article is to propose an
algorithm that works with different events of
cancerous cells evolution stage. The sequence of
steps is described to be implemented in different
software platforms, as for example Matlab, which
was used in this work due to the versatility to
operate with mathematical analysis calculations,
process image and sequence of logic in order to
simulate events connections. Furthermore, to
achieve optimal predictions (not only 3D
reconstructions) means to get optimal prediction
support about events to study cancerous cells
evolution. This is the reason, why it is proposed to
monitory every suggestion of medical doctor,
regarding the processed figure, as an input variable,
from which can be possible to interpret changes of
the mathematical model of the dynamical analysis
consequence of the processed figures. (Ljung,
1994; Pearson, 1995; Wang, 2009; Li et al., 2010;
Calderón et al., 2019).
As it was described in articles (Lei et al., 2007;
Sidow et al., 2015), 3D image reconstruction from
2D image reconstruction was achieved by Level
Set Functions (LSF) analysis that means by
MATERIALS AND METHODS
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algorithm to study cancerous cells evolution
The Biologist (Lima). Vol. 20, Nº2, jul - dic 2022
as a set of a period of time, which depends on the
treatment time that is suggested by the medical
doctor.
it means “Video” is represented by equation 4
After to evaluate 3D initial sides view, “h” is a
_o
proposed value. Therefore, with the capability to
make 3D reconstruction for every stage of real 2D
image, it was obtained of the cancerous cells
evolution process, so by “System Identification” it
was possible to get physical and geometrical
parameters of the 3D nanostructure image that was
reconstructed during evolution of the process,
carefully recognizing that real evolution of the
process is elapsed many ranges of time, because of
the quantity of internal process during real cell
evolution, as it is depicted in Figure 1. Therefore, in
this figure is showed the model as a system
“TF_i”, in which an input variables matrix “U”
c a u s e s r e s p o n s e s “Y” t h a t n e e d s
corrections/compensations that was achieved by an
adaptive matrix of weights “W”, which is obtained
for every sequence of package of figures “Y_i ”.
equation 1 is summary of all the system of
equations by equivalent of energy balance, in order
to have a general equation to represent changes on
time, furthermore, the changes on set of curves,
where “g” means the indicator function border and
“α” is the weight of curve area changes (Lei et al.,
2007).
For this reason, in order to propose a 3D
reconstruction algorithm, it was necessary to get
borders by correlating variables to get desired
errors during geometrical reconstructions in every
axis. It means that border changes analysis by LSF
can be fixed by weights of optimal predictions that
were applied to an identified mathematical model
of the geometry dynamic that was achieved and
described by equation 2.
For all “Error” value belong to Error , Error ,...,
_o _1
[
Error then it is possible to find the function “ ”
_m ]
to warrant 3D reconstruction. With sequence value
of “ ” it is simple to create a movement sequence of
figures “Video” that is composed by following
equation 3, in which every instant “t” can be joined
(1)
(2)
(3)
(4)
Figure 1. Proposed scheme to describe event reconstruction algorithm.
is the regression coefficients matrix and “W” is the
adaptive weight coefficient matrix.
By solving equation 6, it is obtained the matrix of
parameters “θ” that is depicted by equation 7
In figure 2 is shown first part of the flowchart for
the algorithm that was designed, in which is
necessary to choose parameters as a dependence of
sampling time, computing time, response time and
estimated growing time as optimal value studied in
this research. The methodology, which was
analyzed in order to design this algorithm, was
proposed by authors (Wang, 2009; Calderón et al.,
2019). The algorithm needs border curves as a
potential reflection according to get 2D to 3D
reconstruction. Through every 3D reconstruction it
was obtained mathematical information by
“System Identification”.
Furthermore, schematic proposal above is
supported through a mathematical base, which are
described in following equations. Therefore, it is
required a model that could work with different
changes through the operations (as derivatives),
the model to represent a nonlinear system as
dependence of many input/output variables. It
means, it was proposed a polynomial model that is
described in equation 5. “P” and “n” are the
derivatives and order respectively, “a” and “b”
are the parametric coefficients, which contain
information of the system. For this scenery the
“system” is defined as the package of figures as
dependence of variables “y” and “u” in time
domain “t” (Pearson, 1995; Calderon et al, 2019).
Therefore, by the costing function equation “J”
analysis, it was achieved an adaptive prediction
over processed image, as it is showed by equation
6, for which “Y_c” is the expected output variables
matrix, “Γ_c” is the output variables matrix, “θ”
168
(5)
(6)
(7)
Figure 2. Proposed flowchart to explain the event reconstruction designed algorithm (Calderón et al., 2019), part 1.
The Biologist (Lima). Vol. 20, Nº2, jul - dic 2022
Calderón et al.
169
Therefore, with these mathematical equations,
because of identification, it was possible to design
subroutines in the main algorithm to achieve the
optimal predictions of images for every stage of the
cancerous cells evolution. Furthermore, by a
simple subroutine, it was possible to organize
“video sequence” and to get a video of all the
process after filtering geometrical information,
which were predicted as not a part of the final 3D
image, in order to achieve the optimal
reconstruction, as it is depicted in figure 3 as a
second part of the proposed algorithm.
Figure 3. Proposed flowchart to explain the event reconstruction designed algorithm (Calderón et al., 2019), part 2.
The expected result of the algorithm execution is
shown in figure 4, in which is represented how
geometrical parameters change, while time elapses
(Calderón et al., 2019). Thus, by analysis of every
input variable as the part of the adaptive costing
error analysis in the main algorithm, it was looked
for a 3D reconstruction that was depicted in figure
4.
Figure 4. Geometrical representation of the “cancerous cells evolution” (Calderón et al., 2019).
algorithm to study cancerous cells evolution
The Biologist (Lima). Vol. 20, Nº2, jul - dic 2022
170
Ethical aspects: This research has not ethical
conflicts in the proposed article, it was cited every
bibliographic reference for every analysis
described, moreover the image from which was
made the three-dimension reconstruction, as a
consequence of the algorithm designed for this
research, this image was proportionated by the
Medical Julio Guevara. Hence, it is warranted that
the ethical aspects are applied for this research.
In order to prove the designed algorithm, it was
made different experiments over aluminum
elaborating electropolishing and anodization
(Wang, 2009; Inan & Marshall, 2011; Calderón et
al., 2019). These processes are based in chemical
RESULTS AND DISCUSSION
procedures, which produce chemical changes over
aluminum foil as it was given by “Anodic
Aluminum Oxide” (AAO) (Poinern et al., 2011),
furthermore, some physical changes in their
geometry. It is necessary to use microscope to
evaluate reactions over the aluminum foil.
Nevertheless, for this research, it was used a
“Leitz” microscope at resolution in micrometers,
owing to watch geometrical deformations in
millimeters scale. After to verify the results that
were achieved by the algorithm, it was proved the
capacity to achieve 3D reconstruction in a real
figure of lung cancer cells (Kraeft et al., 2000).
In this context, figure 5 shows a piece of aluminum
foil (figure 5A) and its 3D reconstruction in figure
5B, for which colors red, yellow and light blue
represent interval scales between 0.9mm, 0.6mm
and 0.3mm respectively.
Figure 5. 3D reconstruction of Aluminum foil.
After electropolishing and anodization processes
(Calderón et al., 2019), according to study the
behaviour of physical/chemical changes over a
structure (aluminum foil for this experiment), it
was achieved AAO that is showed in figure 6A and
its respective 3D reconstruction that is depicted in
figure 6B, for which colors red, yellow and light
blue represent interval scales between 0.9mm,
0.6mm and 0.3mm respectively.
Figure 6. 3D reconstruction of AAO.
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It was necessary to achieve deep chemical changes
in copper sulphate, as it is depicted in figure 7A and
its 3D reconstruction showed in figure 7B, for
which colors red, yellow and light blue represent
interval scales between 0.9mm, 0.6mm and 0.3mm
respectively.
Figure 7. Copper sulphate over AAO foils and its 3D reconstruction.
In spite of the geometrical changes, it was verified
the 3D reconstruction from figures above and it
was increased time from last reaction at 20%.
Therefore, it was damaged the foil (figure 8A) that
was necessary to evaluate geometrical
reconstructions in more complicated situation as it
was achieved in figure 8B, for which colors red,
yellow and light blue represent interval scales
between 0.9mm, 0.6mm and 0.3mm respectively.
Figure 8. Damaged foil after to increase time reaction at 20%.
Finally, according to evaluate the performance of
the algorithm, it was analyzed a lung cancer cell in
figure 9, from which there are tumor cells
surrounding lung cancer cells as depicted by figure
9A1, 9A2, 9A3, and 9A4. Therefore, the algorithm
was evaluated and it was achieved their respective
3D reconstruction, as it is showed by figures 9B,
9C, 9D and 9E, from which was obtained the
possibility to see an estimation of geometrical
parameters as the size that was not possible to see
in 2D. Furthermore, the mathematical model as
dependence not only on figures that were
introduced in the algorithm, it means that every
diagnostic of the medical doctor can be translated
as an input variable to explain the final result of the
model in order to achieve estimation.
Figure 9. Lung cancer cells and its 3D reconstruction.
algorithm to study cancerous cells evolution
The Biologist (Lima). Vol. 20, Nº2, jul - dic 2022
Al-Haddad, A.; Wang, Z.; Xu, R.; Qi, H.;
Vellacheri, R.; Ute Kaiser, U. & Lei, Y.
2015. Dimensional dependence of the
optical absorption band edge of TiO
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Calderón, J.; Tafur, J.; Barriga, B. & Lozano, J.
2019. Event reconstruction algorithm
proposal to study sensors elaboration based
rd
on nanostructures. Proceedings of the 23
World Multi-Conference on Systemics,
Cybernetics and Informatics, 2: 78-83.
Calderón, J.; Tafur, J.; Barriga, B.; Guevara, J.;
Lozano, J.; Lengua, J. & Solano, G. 2021.
Event reconstruction algorithm for
C o r o n a v i r u s ( C O V I D - 1 9 ) 3 D
reconstruction, according to study its
reaction through antiviral analysis
treatment. The Biologist (Lima), 19: 87-96.
Inan, U. & Marshall, R. 2011. Numerical
Electromagnetics, The FDTD method.
Cambridge University Press.
Kraeft, S.; Sutherland, R.; Gravelin, L.; Hu, G.;
Ferland, L.; Richardson, P.; Elias, A. &
Chen, L. 2000. Detection and analysis of
cancer cells in blood and bone marrow using
a rare event imaging system. Clinical
Cancer Research, 6: 434-442.
Lei, Y.; Cai, W. & Wilde, G. 2007. Highly ordered
nanostructures with tunable size, shape and
properties: A new way to surface nano-
patterning using ultra-thin alumina masks.
It was evaluated the algorithm in order to get
geometrical image details that cannot be simple to
see through the microscope. Therefore, a 3D image
reconstruction can be a support for oncology
analysis. Nevertheless, this is only an algorithm
support.
It was achieved numerical data from the estimated
physical variables (medical interpretation of tested
images), for which it can be expected geometrical
parameters of the expected evolution of cancerous
cells and to model mathematical equations as the
diagnostic that needs to be evaluated by medical
doctor.
It is suggested to verify this algorithm with more
different cases of cancer cells photos, in order to
find applications of predictive behavior of the
researched mathematical model, which supports
the designed algorithm. These applications could
be a support for predictive diagnostic of medical
doctors.
It is expressed deep warm gratefulness to
Aleksandra Ulianova de Calderón due to her
support according to understand the importance of
the necessity that the engineering can be quite
important solution to connect the researchers of the
country with its development technology, but
always with respect of the own ancestral
knowledge of all ethnic communities. There is
expressed special thankful to DGI (“Dirección de
Gestión de la Investigación”) researching office
from PUCP (“Pontificia Universidad Católica del
Perú”), because of its financial support in this
research through the financing FONCAI. It is
expressed much gratitude with the Medical Doctor
Julio Guevara because his teachings, time and his
complete support analysis for the development of
this article in the special context of the
compression of the cancer, moreover it is
expressed gratefulness to him owing to the
proportionated images for the 3-dimension
reconstruction proposed in this research. It is
dedicated special gratitude to Hugo Medina,
because of his teachings in Science Physics for
many generations of engineers, he did and makes
that physics laws could be so easy to get
172
ACKNOWLEDGMENT
understanding of nature and current life, such as for
this research. With a very good base of laws of
nature, it was possible to obtain a fundamental to
correlate advanced mathematics with the
formalism that engineering applications always
need. Furthermore, it is declarated thankful to
Alexánder Zutta, Lili Gamarra, Daniel Menacho
and Darío Huanca, owing to their support in
experimental tasks and simulation analysis. It is
expressed special thankful to Rodrigo
Urbizagastegui owing to his compromise with the
main author and coauthors to discuss the
applications of engineering in the necessities of the
country.
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Received January 2, 2021.
Accepted March 27, 2022.
algorithm to study cancerous cells evolution
The Biologist (Lima). Vol. 20, Nº2, jul - dic 2022
