Scientia et Technica Año XVI, No xx, Mes XX de Año XX. Universidad Tecnológica de Pereira. ISSN 0122-1701 1
Fecha de Recepción: 26/01/2024
Fecha de Aceptación:
Abstract—Optical emission fluorescence spectroscopy allows
for determining biochemical changes in healthy and pathological
biological tissue, either in vivo or in biopsies. The aim of the study
is to analyze the chemical and physical properties of cervical
tissue. A mathematical model is presented to examine and observe
the fluorescence emission of tissue. During the development of
precancerous states, the optical properties of the tissue can be
altered not only by the light dispersion and the fluorescence
increase in the epithelium but for the fluorescence reduction in the
stroma. The Beer-Lambert Law was used to describe light
propagation in the tissues. Four components of cervix tissue were
identified: collagen, elastin, NADH, and flavins. By applying the
developed model, it was possible to characterize each fluorophore
present through Gaussian sub-spectra, providing support for the
medical diagnosis of precancerous lesions in cervical tissue. The
model yielded predictions with a good spectral fit, and the
contribution of each fluorophore showing significant differences
in the signal parameters, particularly for collagen and NADH.
Index Terms—Detection cancer, Fluorophore, Mathematical
Model, Optical Fluorescence Spectroscopy, Precancerous Tissue.
Resumen—La espectroscopía de fluorescencia por emisión
óptica permite determinar cambios bioquímicos en tejidos
biológicos normales y patológicos, ya sea in vivo o en biopsias. El
objetivo de este estudio es analizar las propiedades químicas y
físicas del tejido cervical. Se presenta un modelo matemático para
examinar y observar la emisión de fluorescencia del tejido.
Durante el desarrollo de estados precancerosos, las propiedades
ópticas del tejido pueden alterarse no solo por la dispersión de la
luz y el aumento de fluorescencia en el epitelio, sino también por
la disminución de fluorescencia en el estroma. Para describir la
propagación de la luz en los tejidos se utilizó la Ley de Beer-
Lambert. Se identificaron cuatro componentes en el tejido
cervical: colágeno, elastina, NADH y flavinas. Al aplicar el modelo
desarrollado, fue posible caracterizar cada fluoróforo presente
mediante subespectros gaussianos, brindando soporte al
diagnóstico médico de lesiones precancerosas en el tejido cervical.
El modelo arrojó predicciones con un buen ajuste espectral,
evidenciando diferencias significativas en los parámetros de señal
de cada fluoróforo, especialmente en el caso del colágeno y el
NADH.
Índice de términos— Detección de cáncer, Espectroscopia óptica
de fluorescencia, fluoróforos, modelo matemático, Tejido
precanceroso.
I. INTRODUCTION
ERVIX cancer is the 4
th
most common cancer in women
around the world. Its development is very slow, and it tends
to begin with a lesion called cervical intra-epithelial neoplasia
(CIN), from which several years can go until it becomes cancer.
However, this kind of lesion can be identified in an early stage,
and that way, strong actions can be taken to face the illness on
time [1]. Early detection of CIN represents a fundamental
especially role in the mortality reduction related to cervix
cancer, especially during the last 50 years [2]. Opposite of it,
nowadays, cervix cancer keeps being a relevant threat to
woman’s health [1], [2].
For cervix detection, different diagnosis methods have been
used such as cytology, HVP molecular detection, and
colposcopy [3]. However, the techniques mentioned above are
not efficient enough to detect cervix cancer in its early stages,
and many times, a histopathological analysis of biopsies is
required for the final diagnosis. For example, a sensibility rate
of 32 to 90% and 94% of specificity is associated with cytology
analysis according to [4], [5] due to the limited number of tests
and reading errors. Other diagnosis methods present lower
percentages of sensibility and specificity. To improve the
variables mentioned above, it is necessary to find other
diagnosis methods [5].
An important technique that was used during previous
decades to detect cervix pre-cancerous lesions was fluorescence
spectroscopy. This technique can offer high sensitivity as well
as a specific and accurate diagnosis without extracting the tissue
[6], [7]. Even, when there is huge empirical evidence that
sensitivity suggests that the mentioned technique can be used to
discriminate between normal and dysplastic cervical tissue,
there is no wide information to understand the differences in the
biological tissue of fluoresce spectrum of normal and dysplastic
tissue [8]. The work can be done by developing algorithms to
simulate the spectrum.
Mathematical Model for Analyzing the Fluorescence
Emission of Biological Tissue
Modelo matemático para el análisis de la emisión de fluorescencia del tejido
biológico
B. Segura Giraldo ; M. Londoño Orozco ; S. G. Chacón Chamorro
DOI: https://doi.org/10.22517/23447214.25682
Scientific and technological research paper
C
Scientia et Technica Año XVI, No xx, Mes XX de Año XX. Universidad Tecnológica de Pereira
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The aim of this research is to simulate the normal and
pathological fluorescence spectrum of cervix tissue by using
mathematical models based on the Beer-Lambert law, to study
the fluorescence emission of molecules in the tissue such as
collagen, elastin, NADH, and flavins. Results are compared
with the ones obtained for in-vivo tissue.
II. METHODOLOGY
The following methodology was conducted to conduct the
research.
a. Equipment and materials
The system used for the research was composed of an optical
fiber probe, a spectrograph, and a computer interface. This
equipment was used to record fluorescence spectral data of
cervix in-vivo tissue.
Characteristics of laser are:
Wavelength: 337.1 nm
Pulse: 5 ns
Repetition rate: 33 Hz
Pulsed transmitted energy: 300 µJ
b. Objective Sample
For this research, a pilot test for normal and pathological
cervix tissue of 50 patients between 17 and 60 years old was
conducted. Patients had previous results of cervical cytology.
Informed consent was obtained for each patient and the study
was reviewed and approved by specialists of the IPS-
Universidad de Caldas – Unidad de Cáncer de Cuello Uterino
y Cáncer de Mama.
Fluorescence spectra were taken according to the pre-
established protocol conducted by experts. Figure 1 shows the
cervix and the four points where measurements were done. On
each point, 30 spectra were taken, and then, during signal
processing, an average spectrum was obtained.
Fig. 1. Sections and points of the cervix to take the spectrum.
III. THEORETICAL FRAME
a. Optical Spectroscopy Fluorescence
Spectroscopy comes from electromagnetic interaction with
matter. That produces state transitions for molecules and level
transitions for atoms. Transitions can be electronic in the visible
spectrum (UV), vibrational in the infrared (IR) or rotational in
the radio waves. Spectroscopy relates three processes that are
going to be explained in more detail in the following sections:
radiation absorption, light emission, and dispersion.
Fluorescence is the ability of certain molecules to absorb
energy and emit electromagnetic radiation with different
wavelengths. That is why, optical fluorescence spectroscopy is
a method to analyze the fluorescence of a sample using a beam
of light that can be normally found in the ultraviolet spectral
range. Fluorescence is widely used in analytic measurements,
and photochemical analysis of biological systems, alimentary
products, pharmaceutical products, clinical samples, and
scientific research [9], [10].
b. Physics Principles
Spectrophotometric methods are based on the Beer-Lambert
Law and are used to analyze various media, including
biological tissues. The law defines that the totality of light
emitted by a sample can decrease due to the number of
absorption materials in its trajectory (concentration), the
distance that light must go through (optical path distance), and
the probability that the photon in the particular wave amplitude
can be absorbed by the material (extinction coefficient) [11].
Lambert-Bourger law is derived from Beer-Lambert law and
can be expressed in (1), which relates to the light absorption of
an optically diluted homogenous medium and its thickness,
(1)
Where represents the intensity, the distance,
the
absorption coefficient and is a successive layer of the
absorbent medium. Transmitted light intensity through the
distance is described by (2)
, (2)
being
the initial light intensity. Another important concept
in spectrophotometric theory is the absorption length (inverse
of the absorption coefficient) which can be defined as the
distance required for the beam of light to decrease in
of the
initial light intensity. That allows a redefinition of the
transmitted light intensity shown in (3)
(3)
where is the extinction coefficient.
In a sample, a radiation beam passes through a solution layer
containing a specific absorbent species with defined thickness
and concentration. Beam power is attenuated due to the
Scientia et Technica Año XVI, No xx, Mes XX de Año XX. Universidad Tecnológica de Pereira
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interaction between photons and absorbent particles, generating
absorbance of the solution, which is a fraction of the incident
radiation [12] and can be defined as (4)
(4)
The equation above expresses the quantity of energy emitted
that crosses a body in a certain amount of time. Absorbance
measure is made in spectrophotometers with the aim of
generating an absorption or emission spectrum. The intensity of
electromagnetic radiation distribution in terms of wavelength is
used to determine the characteristics of the sample [13], [14].
c. Fluorescence of biological tissue in a single
layer
In 1852, August Beet determined that the absorption
coefficient has a linear relation with the concentration of a
diluted substance within the medium, represented by its
concentration
and a constant , that is, as presented in (5)
. (5)
From the above, (2) can be rewritten such as (6)
(6)
Where is known as the specific extinction coefficient.
When extending the definition for substances in a sample,
the total absorbance corresponds to the individual sum of the
extinction coefficient times their respective concentrations
times the distance. This relationship is formulated in Equation
(6), which follows the same structure as Equation (7).
(7)
The analysis of the research is focused on the study of emitted
photon energy distribution. That is why, expressing the
fluorescence intensity for each absorbed photon in the medium
The emission characteristics, in terms of the emitted photon
wavelength, are important [15] and can be defined as shown in
(8)
(8)
where
is the quantum yield,
is the fluorescence
emission spectrum function that reflects the probability
distribution of different transition vibratory levels from state
to state
. The emission spectrum is characterized by each
fluorophore in the biological tissue.
In a practice sense, the fluorescence intensity
measured at a certain wavelength
is proportional to
and to the number of photons that are absorbed at a
excitation wavelength
. It is convenient to replace the number
of absorbed photons for the absorption intensity
, which
is defined as the difference between intensities of incident
(
) and transmitted (
) light. In that sense (9),
(9)
From the above, fluorescence intensity can be represented
like in (10)
, (10)
Then, by using (6), the above expression can be rewritten as
(11):
(11)
The factor depends on several parameters, especially in the
visualization optical configuration and the bandwidth of the
monochromator.
Measures of the variations of
in terms of
, for a set
excitation wavelength, reflects the changes in
and then,
provides the fluorescence spectrum.
Equation (11) can be modified to obtain a less complex
expression using the expansion of exponential series
, gettingin like (12)
(12)
The relation above allows to observe the fluorescence intensity
is proportional to the concentration at a low absorbance and
then, the linear variation is lost with the absorbance increase.
Additionally, when fluorescence spectroscopy is used to make
quantitative evaluation of the fluorophore’s concentration, the
proportionality between fluorescence and concentration
intensity, only diluted solution must be taken into account.
IV. RESULTS AND DISCUSSION
The fluorophores from the cervix tissue that contribute to the
emission fluorescence signal under light excitation of 337,1 nm
are collagen, elastin, NADH and flavins [16]. Figure 2 shows
emission and absorption spectrum for each fluorophore in the
cervix tissue. Spectrum were obtained from spectroscopy
measurements and the following signal processing with
gaussian fit. Figure 2a shows absorption spectrum between 200
and 500 nm, and Figure 2b shows emission spectrum between
300 and 600 nm.
a) Absorption spectrum
b) Emission spectrum
Scientia et Technica Año XVI, No xx, Mes XX de Año XX. Universidad Tecnológica de Pereira
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Fig. 1. Spectrum with gaussian fit for each fluorophore (collagen, elastin,
NADH and flavins) in the cervix tissue.
From the previous spectrum, different statistical and
geometrical characteristics were extracted and are shown in
Table I. There,
and
refer to wavelength in the central
absorption and emission spectrum respectively.
and
are
the absorption and emission intensities, respectively.
and
corresponds to the width at mid height of each curve in the
absorption and emission spectrum, respectively.
TABLE I
EXTRACTED CHARACTERISTICS FROM ABSORPTION AND EMISSION SPECTRUM
OF FLUOROPHORES IN THE CERVIX
Fluorophore
Absorption
Emission
Collagen
336.35
1
34.85
393.85
1
60.69
Elastin
352.03
1
46.40
405.48
1
73.08
NADH 1
st
peak
262.61
1
25.8
474.28
1
55.42
NADH 2
nd
peak
345.51
0.45
54.41
Flavine 1
st
peak
260.17
1
28.77
557.44
1
65.51
Flavine 2
nd
peak
387.20
0.45
112.29
Flavine 3
rd
peak
457.57
0.5
100.58
Based on data from Table I, contribution factor for each
fluorophore at excitation wavelength (
) and
emission wavelength (
) are extracted and
denoted as:
and
. Equations (13) and (14) show the
expression of the previous values following the definition of
Beer-Lambert law.
(13)
(14)
where depends on each fluorophore: Collagen: ,
Elastin: , NADH: , Flavins:
Combining (7) and (4), total spectrum absorption with
fluorophores contribution can be determined by (15) and (16),
(15)
(16)
Where
represents the contributions of each fluorophore,
which are unknown variables in the model and that should be
calculated to contribute to the research of normal and
pathological cervical tissue.
Then, total absorbance and transmission in wavelength range
of 200 to 700 nm are calculated considering steps of 0.3nm,
which fits with the spectrometer resolution used for the in-vivo
measures. That way, contribution factors with wavelength
function
are calculated
(17)
Using the contribution factor, it results that total absorbance
and transmittance can be expressed as in (18) and (19),
(18)
(19)
From those contributions, fluorescence emission intensity
(
is calculated in (20) due to each fluorophore in the cervix
tissue
. (20)
Where
,
, and
are intensity, central wavelength, and
width at mid high of the emission peaks of each fluorophore
from cervix tissue, respectively.
Additionally,
shows in (21) is a Raman band dispersion
of water and an adjustment variable inside the system
(21)
,
,
, and
are intensity, central wavelength, width
at mid height and the contribution to dispersion band,
respectively. For obtaining the parameters and their
dependencies on fluorophore, each filtered and averaged
spectrum was fitted using the mathematical model developed in
this research, which simulates the characteristic fluorescence
spectrum through (13), (14) y (20). This model is based on a
sum of Gaussian bands, each representing the contribution of a
specific fluorophore to the processes of absorption, excitation,
emission, and transmission described throughout the theoretical
development.
The fitting process employs a nonlinear least squares method
to extract a set of spectral features from the fluorescence data,
including fluorescence intensities, full width at half maximum
(FWHM), central wavelengths, and relative contributions of
each endogenous fluorophore (such as collagen, elastin,
NADH, and flavins) present in the cervical tissue. This
procedure is applied individually to each measurement point
obtained from the spectral fluorescence data collected on
cervical tissue samples. Figure 3 shows the adjustment. The
spectrum shows similar shapes that the ones on the literature
[17].
Scientia et Technica Año XVI, No xx, Mes XX de Año XX. Universidad Tecnológica de Pereira
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Fig. 3. Original fluorescence spectrum and adjustment with the mathematical
model.
The correlation coefficients R and R² are calculated as
normalized measures of the relationship between the variables
representing the filtered and averaged spectrum and the fitted
spectrum. Based on these values, the spectra corresponding to
analysis points with a correlation index below 0.9 are discarded.
Applying the optical fluorescence spectroscopy technique,
an initial test was done in normal and pathological cervical
tissue of 50 patients between 17 and 60 years old. To obtain the
spectrum from spectral tissue information, results from the
patient’s histopathology were used with the idea to find the
range of each category: normal and pathological. As shown in
Figure 4, there exist a notable separation between the normal
and pathological spectrum, as well as the differences between
maximum and minimum intensities.
a) Normal spectrum
b) Pathological spectrum
Fig. 4. Spectrum with gaussian fit for each fluorophore: collagen, elastin,
NADH, and flavins.
Now, Table II shows the results from simulation process
using the proposed method, where parameters for Equation 20
are obtained from the patient’s previous classification, followed
by the classification in normal and pathological, and the
extraction of the mean and standard deviation to register
,
,
, and
. It is important to notice that collagen decrease in
epithelial cells that preserve the cell structure produce an
increase on NADH levels, altering normal homeostasis of the
epithelium. Biological changes are analyzed and recognized
using fluorescence emission spectroscopy, showing the
difference between normal and pathological cervical tissue.
TABLE II
PARAMETERS OBTAINED WITH THE MATHEMATICAL MODEL, MEAN AND
STANDARD DEVIATION (SD) FOR NORMAL AND PATHOLOGICAL TISSUE.
Parameter
Fluorophore
Normal
Pathological
Mean
SD
Mean
SD
Collagen
0.613
0.074
0.539
0.068
Elastin
0.672
0.110
0.586
0.129
NADH
0.588
0.107
0.542
0.186
Flavins
0.689
0.174
0.597
0.177
Collagen
382.878
3.615
384.530
5.716
Elastin
417.023
5.489
415.874
6.252
NADH
459.613
17.847
456.316
17.403
Flavins
498.730
19.411
500.228
16.989
Collagen
35.146
10.738
31.765
11.383
Elastin
47.541
11.877
43.936
17.876
NADH
47.794
18.571
45.351
17.456
Flavins
62.418
14.706
58.517
16.521
Collagen
3.590
1.146
2.537
1.119
Elastin
3.425
0.975
2.193
0.913
NADH
5.049
1.896
3.933
1.268
Flavins
5.785
2.220
4.483
1.983
V. CONCLUSION
Fluorescence spectral information obtained from the technique
implementation and the mathematical model developed in the
research, suggest that contributions in the cervix tissue such as
collagen, elastin, NADH, and flavins. The mathematical model
implemented allowed to correlate the fluorophores
contributions in each spectrum. In normal tissue, the collagen
contribution is higher than the NADH contribution, while in
pathological tissue, the NADH contribution is higher than the
collagen one. The previous one is a characteristic that makes
possible the identification of normal and pathological tissue.
There is evidence that the combination of fluorescence
spectroscopy with the adjustment model with gaussian
decomposition can be an alternative tool to support medical
diagnosis and recognition of intraepithelial damage in cervix
tissue.
ACKNOWLEDGMENT
The author gratefully acknowledges the financial support of
MinCiencias. This work was also helped by the research group
of Magnetismo y Materiales Avanzados of Universidad
Nacional de Colombia sede Manizales, the group of Cáncer de
Cuello Uterino y Cáncer de Mama at Universidad de Caldas and
the group Automática at Universidad Autónoma de Manizales.
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[13] B. Segura. “Espectroscopía óptica de fluorescencia aplicada al soporte de
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Segura Giraldo, Belarmino is a PhD in
Engineering, Master in physics, and
Specialist in University Teaching.
Professor-researcher at the Universidad
Nacional de Colombia Sede Manizales. He
supports academic and research processes in
the areas of biophysics, physical
instrumentation, biomaterials, digital
signals, and image processing, among
others. He began his work on the fluorescence spectroscopy
technique during his PhD studies with population in Caldas. He
has been directing and researching on several projects based on
the development of low cost and portable equipment for
diagnosis support of breast and cervix cancer. Currently, he
keeps working on the fluorescence spectroscopy technique, as
well as others, with the aim of optimizing the techniques so they
can be used in medical the environment. ORCID
https://orcid.org/0000-0001-9205-8573.
Londoño Orozco, Mariana is a MSc
Physics student at Universidad Nacional
de Colombia Sede Manizales. She is a
Physicst Engineer and Mathematician.
She has worked on biophysics for more
than 3 years. She participated in a Young
Researcher program from Ministerio de
Ciencia, Tecnología e Innovación during
2021-2022 where she contributed to the development of
equipment for diagnosis support techniques for breast and
cervix cancer. She is currently member of Grupo de
Instrumentación Física where she keeps working of the
optimization of the techniques and development of in vivo tests.
https://orcid.org/0000-0002-3260-5333.
Sofía Chacón is a physical and electronic
engineer from the Universidad Nacional de
Colombia, Manizales. She holds a master’s
degree in electronic engineering from the
Universidad de Nariño (Colombia) and is
certified in Artificial Intelligence. She
worked as a Joven Investigadora on a
research project focused on cervical cancer
detection techniques, where she applied fluorescence optical
spectroscopy and mathematical modeling of biological
systems. She also served as a research assistant on project
which focused on energy transactions for multiple agents. Her
main research interests include mathematical modeling of
systems, fluorescence optical spectroscopy, energy
management systems, optimization in power systems, game
theory applications, distributed energy management, and peer-
to-peer energy markets. ORCID https://orcid.org/0000-0002-
5687-6883.