© The Authors, 2021, Published by the Universidad del Zulia*Corresponding author: tamaramolero@unad.edu.do
Tamara Molero Paredes
1,3*
Said Kas-Danouche
2,3
Rev. Fac. Agron. (LUZ). 2022, 39(1): e223913
ISSN 2477-9407
DOI: https://doi.org/10.47280/RevFacAgron(LUZ).v39.n1.13
Food Technology
Associate editor: Dra. Gretty Ettienne
1
Universidad del Zulia. Facultad de Humanidades y
Educación. Departamento de Biología. Venezuela.
2
Universidad de Oriente. Escuela de Ciencias. Departamento
de Matemática. Laboratorio de Matemática Aplicada para la
Industria. Venezuela.
3
Universidad Adventista Dominicana. Facultad de
Humanidades. Escuela de Ciencias. República Dominicana.
Recibido: 10-06-2021
Aceptado: 11-11-2021
Published: 03-02-2022
Keywords:
Aloin
Concentration
Estimate
Mathematical modeling
Estimation of aloin concentration in Aloe vera L. (Aloe barbadensis Mill.) by mathematical
modeling
Estimación de la concentración de aloína en Aloe vera L. (Aloe barbadensis Mill.) mediante
modelización matemática
Estimativa da concentração de aloína em Aloe vera L. (Aloe barbadensis Mill.) por modelagem
matemática
Abstract
Aloin is one of the secondary metabolites that gives plants of the genus
Aloe spp. their healing properties. The concentration of aloin is related to
the fresh mass and, its industrial purication involves laboratory processes
that add extra costs to its commercialization. The objective of this research
was to mathematically modelize the estimation of the aloin concentration in
A. vera L. from the fresh mass. The theory of discrete perfect least squares
approximations was used, considering linear and exponential approximation
functions. For the tabulation of the data, the option of class mark and the
average of the values were used. The analyses of the approximations indicate
that the exponential curves approximate the data better (with R
2
= 75% and
82% for the two options, respectively) than the straight lines (with R
2
= 65%
and 70% for the two options, respectively). The use of these approximations
is recommended to estimate the concentration of aloin in A. vera plants
based on their fresh mass, facilitating the measurement of this secondary
metabolite, and minimizing costs in the industrialization process.
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Rev. Fac. Agron. (LUZ). 2022, 39(1): e223913. January - March. ISSN 2477-9407.
2-6 |
Resumen
La aloína es uno de los metabolitos secundarios que le otorga
a las plantas del género Aloe spp., sus propiedades curativas. La
concentración de aloína está relacionada con la masa fresca y su
puricación industrial requiere procesos de laboratorio que agrega
costos extras a la comercialización. El objetivo de este trabajo fue
modelizar matemáticamente la estimación de la concentración
de aloína en A. vera L., a partir de la masa fresca. Se usó la teoría
de aproximaciones de mínimos cuadrados perfectos discretos,
considerando la función de aproximación lineal y exponencial. Para
la tabulación de los datos se usaron las opciones de marca de clase
y el promedio de los valores. Los análisis de las aproximaciones
indican que las curvas exponenciales aproximan mejor los datos
(con un R
2
= 75% y 82% para las dos opciones, respectivamente)
que las líneas rectas (con un R
2
= 65% y 70% para las dos opciones,
respectivamente). El uso de estas aproximaciones se recomienda para
la estimación de la concentración de aloína en A. vera en función a su
masa fresca, facilitando la medición de este metabolito secundario y
minimizando gastos en el proceso de industrialización.
Palabras clave: aloína; concentración; estimación; modelización
matemática.
Resumo
Aloin é um dos metabólitos secundários que dá às plantas do
gênero Aloe spp., suas propriedades curativas. A concentração de aloin
está relacionada à massa fresca e sua puricação industrial requer
processos laboratoriais que agregam custos extras à comercialização.
O objetivo deste trabalho foi modelar matematicamente a estimativa
da concentração de aloína em A. vera L. a partir da massa fresca. Foi
utilizada a teoria das aproximações de mínimos quadrados perfeitos
discretos, considerando a função de aproximação linear e exponencial.
Para a tabulação dos dados, foram utilizadas as opções de marcação
da classe e a média dos valores. As análises das aproximações indicam
que as curvas exponenciais aproximam melhor os dados (com R
2
=
75% e 82% para as duas opções, respectivamente) do que as retas
(com R
2
= 65% e 70% para as duas opções, respectivamente) . O
uso dessas aproximações é recomendado para estimar a concentração
de aloína em A. vera com base em sua massa fresca, facilitando a
mensuração desse metabólito secundário e minimizando custos no
processo de industrialização.
Palabras-chave: aloin; concentração; estimativa; modelagem
matemática.
Introduction
Aloe vera L (= Aloe barbadensis M.= sábila or aloe) is cultivated
all over the world for agricultural, medicinal, cosmetic and decorative
purposes, being its place of origin the northern limits of the Arabian
Peninsula, located in southern Africa (Klopper & Smith, 2013;
Pandey & Singh, 2016). Aloes have been named since ancient times
because in the Bible it is said that they were employed as perfumes
and in the funerary arts. Numerous studies show that the genus Aloe
was subjected to strong environmental processes diversifying by
speciation, giving rise to the extraordinary diversity of species known
today (Klopper & Smith, 2013; Deng-Feng et al., 2019).
The therapeutic and medicinal properties of A. vera as a
pain reliever, analgesic, bactericide, moisturizer, emollient, anti-
inammatory, antitumor, among others, is well documented, thanks
to the metabolites that are synthesized by secondary routes (Mahor
& Ali, 2016). One of them is aloin, considered the active compound
of these plants, which is found in their exudates with a defensive
function and is the taxonomic marker of the genus Aloe spp, it also
has a laxative action in humans, so the consumption of supplements
based on this compound can have toxic effects if ingested in large
quantities (Minjares & Femenia, 2019).
This compound is found especially in the acíbar or juice of the
leaves and when extracted from this yellow liquid and after a drying
process, it appears as a yellow powder of bitter taste, product of its
chemical structure, characterized by being a glycoside formed from
an anthraquinone (genin) linked with a sugar. Aloin can be presented
in two different isomeric forms, aloin-A (barbaloin) and aloin-B
(isobarbaloin), in such a way that some of them can produce one of the
two types or present a mixture of them. Likewise, their concentration
may vary according to the species, age of the plants, region and time
of collection (Kaparakou et al., 2020).
Depending on the Aloe spp. product to be offered on the market
(medicines, cosmetics, beverages, food, etc.), it is necessary to
determine the aloin concentration and ensure that it complies with
international requirements. In this sense, the European Union
established that, in the case of foods and medicines, the maximum
limit of aloin must be 0.1 mg.kg
-1
(EEC 88/388) (Ofcial Journal of the
European Communities, 1988). The same regulation was established
by the International Aloe Science Council for the United States. In the
cosmetic industry, there is no direct consumption of any part of the
plant, therefore, the use of both acíbar and gel, either concentrated or
powdered, is accepted (Pedroza et al., 2009). The aloin concentration
is more accurately measured by chromatographic methods; however,
these methods of analysis are laborious and expensive to apply at
the industrial level, in which large volumes of plant material are
handled. Therefore, it is convenient to establish an alternative that
allows estimating the aloin concentration in A. vera L., considering
other variables that are easy to handle for both the producer and the
industry, for example, fresh and dry mass of the plant, which allows
knowing the aloin content without the need of resorting to laborious
laboratory protocols.
Mathematical modeling allows to test the consequence of changes
in a system due to it describes phenomena using the language of
mathematics. When using a mathematical model, the rst step is to
identify the signicant variables in the system, which will be included
in the model. The second step is related to the determination of the
equations that govern the system that allow predictions to be made.
Although mathematics offers the facility of proving general results,
these critically depend on the formulation of the equations used.
Small changes in the structure of the equations can involve enormous
changes in the mathematical methods.
The use of simulation and prediction models has become popular
in agriculture and has become a tool both for research, as well as
for producers, technical consultants and industrialists, who can now
mathematically dene the best crop management practice in certain
situations (FAUBA, 2018). In addition, modeling has been used to
improve the effects of industrialization processes of agricultural
products. In this regard, Martinez et al. (2017) optimized the protocol
for removing aloin from A. vera gel by employing three input variables:
temperature, agitation time, and activated carbon concentration. The
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Molero and Kas-Danouche. Rev. Fac. Agron. (LUZ). 2022, 39(1): e223913
3-6 |
results showed that the latter two variables exerted a signicant action
on the removal of aloin from A. vera gel.
Due to the mathematical modeling is an important tool for the
calculation of the yields of crops of interest, the objective of the
present research was to apply the mathematical modeling to estimate
the aloin concentration in A. vera L. starting from the fresh mass
values of the plant.
Materials and methods
Plant material
Ten plants of A. vera were randomly collected in seven
commercial farms located in the western region of Venezuela, three
in the central zone of Falcon state (farm 1: 11°03′13″N69°45′07″W,
farm 2: 11°29′16″N 69°21′08″W and farm 3: 11°14′26″N
70°06′08″W), another located in northern Falcon state (farm 4:
11°52′22″N 69°59′18″W) and three in northern Zulia state (farm 5:
10°51′34″N 71°45′44″W, farm 6: 10°54′56″N 71°49′38″W and farm
7: 10°40′14″N 71°31′36″W). The collection was carried out between
the months of January to March, which corresponds to the dry season
in the country.
According to Ewel and Madriz (1976), the climates of these sites
correspond to a thorny-tropical forest and very dry tropical forest,
with altitudes of 0-200 mamsl, with temperatures above 25 °C and
rainfall between 250 and 800 mm per year, with high solar radiation.
Plants in good health and with at least 10 leaves and approximately
35 cm in height were selected. Subsequently, they were taken to the
nursery of the Facultad de Humanidades de la Universidad Del Zulia,
and they were sown in plastic containers with equal proportions of
sand and organic fertilizer, to achieve vegetative reproduction.
Approximately 10 months after sowing, three sprouts from each
mother plant, for a total of 210 plants evaluated. Later, they were
transplanted into new plastic containers and were irrigated once a
week.
Determination of fresh mass and aloin concentration
When the sprouts were six (6) months old, fresh mass (PF) and
aloin concentration (CA) were determined by high performance liquid
chromatography (HPLC). For this purpose, each one was sanitized
and subsequently weighed on a commercial balance. Then, they were
homogenized in a blender and this content was stored in hermetically
sealed bags in a -18°C freezer. This material was freeze-dried in a
LABCONCO LYLH-LOCK 6 equipment for 72 hours until it reached
complete dehydration. The powder was sent to the chromatography
laboratory of the Facultad de Agronomia de la Universidad Del Zulia,
(Venezuela) to measure CA according to the protocol previously
established by Molero et al. (2016). The CA value corresponds to the
amount of grams per 100 g of fresh plant mass.
Data tabulation
It was planned to use the theory of approximations, to nd the
function that best t the collected data. However, it was observed
that there were data with the same abscissa, which clearly indicates
that they did not represent a function. For the development of the
approximation theory, it is assumed that it starts with a function
(Epperson, 2021; Chapra & Canales, 2015; Burden & Faires, 2011)
which is desired to be approximate or, in its defect, a set of data
(which represent a function) to which is required to approximate by
means of an approximation function. In view of this, it was proceeded
to tabulate the data in classes. The procedure used to tabulate the data
was the following: per each plant analyzed of Aloe vera, PF value was
corresponded to the CA. To describe the classes, the Herbert Sturgesʼ
Rule was used (González, 2017; Sturges, 1926).
k =1+3.322 log(n),
where
k is the number of intervals to be formed and n is the total number
of data collected or, in other words, the total number of samples
taken. To dene the classes it is necessary to know the range that each
class should have. The range is dened, using the number k from the
Herbert Sturgesʼ Rule (González, 2017; Sturges, 1926), as follows:
Range=(Max {Fresh mass}-Min{Fresh mass})/k,
where
Max {Fresh mass} is the maximum value and Min{Fresh mass} is
the minimum value of the set of all the values and data collected from
the PF. The Sturgesʼ rule is a criterion that helps to formally determine
the number of classes needed to represent a data set (González, 2017;
Sturges, 1926).
Once the classes were dened, it was proceeded to determine the
best representative of each class, discriminating between the class
mark and the average of all the fresh masses measured that fall into
each class. We proceeded, then, to work with the two possibilities
as options for the tabulation of the PF, denoted by
, to then make
comparisons and see which of the two options produces a better
approximation.
These two options were used for the approximation functions
proposed in the following paragraphs.
Approximation theory
The theory of discrete least squares approximations was used to
nd the functions that best t the tabulated data. This theory was used
because it allows nding the function that “best” ts a given data set
(Epperson, 2021; Chapra & Canales, 2015; Burden & Faires, 2011).
The two options, described in the previous section, were considered
for both a linear approximation function and an exponential
approximation function.
For the case of the linear function, the objective is to nd the line
that best approximates the tabulated data, with which a function of the
form should be found (Burden & Faires, 2011):
y=a.x+b,
where
=
(
=1
)(
=1
)
=1
2
=1
2
=1
, 1
represents the slope of the linear function, and,
=
2
=1
=1
(
)
(
=1
)
=1
2
=1
2
=1
, 1
represents the intercept of this function with the y-axis. Also, it
is considered that and correspond to the fresh mass and the
aloin concentration, respectively, for each class, and is the
number of classes corresponding to in the previous section.
In the second case, it was intended to nd the exponential curve
that best approximates the tabulated data, with which a function of the
form should be found (Burden & Faires, 2011):
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Rev. Fac. Agron. (LUZ). 2022, 39(1): e223913. January - March. ISSN 2477-9407.
4-6 |
y = be
ax
,
where
=
ln(
)
=1
=1
=1
2
=1
2
=1
,
1
represents the coefcient of the argument of the exponential
function, and
ln
(
)
=
(
2
=1
)(
ln(
)
=1
)
(
ln(
))
(
=1
)
=1
2
=1
2
=1
,
1
where b represents the degree of the slope of the exponential
curve, considering that and correspond to the fresh mass and
the aloin concentration, respectively, for each class, and is
the number of classes that correspond to in the previous section.
To calculate the linear and exponential approximations of the
data, the Excel spreadsheet for Windows environment was used
(Charte 2016) and they were corroborated with InfoStat statistical
package (Di Rienzo et al., 2011). The comparisons between the
four approximations were carried out using the coefcient of
determination (R
2
).
Results and discussion
As it was mentioned in materials and methods, the number
of the data collected was n = 210 for each characteristic studied,
therefore, from the Herbert Sturges Rule it is deduced that k ≈ 9.
From the collected numbers, the maximum and minimum PF values
were 523 g and 100 g respectively, and the average of all the PF data
was 252.2 g. In this sense, the range is approximately equal to 47,
which denes PF.
Once the classes were dened, the CA data were classied into
each class. The maximum and minimum CA values were 0.03 g
and 6.771g.100
-1
g of PF, respectively, with an CA average value
of 1.242g.
Table 1 shows the two options proposed in the methodology for
PF; the rst option, calculating the class mark in each class, and
the second option, calculating the fresh mass average of each class
using all the collected values corresponding to each particular class.
The aloin concentration average for each class is also included.
Table 1. Distribution of aloin concentration with respect to
fresh mass (PF) classes in Aloe vera L.
PF Classes x1-PF marks x2-Prom. PF Class y-Prom. CA.
1 100-147 123.5 117.3 0.93
2 147-194 170.5 174.9 0.93
3 194-241 217.5 215.2 1.34
4 241-288 264.5 259.6
1.54
5 288-335 311.5 310.0 1.60
6 335-382 358.5 357.3 1.82
7 382-429 405.5 400.4 2.93
8 429-476 452.5 446.8 2.75
9 476-523 499.5 520.0 6.77
Total Sum: 2803.5 2801.6 -
The results of the linear and exponential approximations for the
two options are presented below.
Results for the approximations by linear function
For this case it was obtained:
y=0.011443x-1.2754,
for the rst option (class mark) and,
y = 0.011509x-1.2935,
for the second option (averages of the CA values in each class).
Figure 1 shows the linear approximations of the aloin concentration.
The validation of these results with the InfoStat software showed
that: a = 0.01 and b = -1.28, which are clearly rounded values of
those calculated for the rst linear approximation (a = 0.011443 and
b = -1.2754).
Figure 1. Linear approximations of aloin concentration
The red solid line represents the rst of the linear approximation
equations found, and the black dotted line corresponds to the second
one. It can be seen that practically both lines overlap.
Results for the approximations by exponential function
For this case of approximation by exponential function, the
following results were obtained.
y = 0.44886 e
0.00458x
,
for the rst option (class marks) and,
y = 0.45896 e
0.00452x
,
for the second option (averages of the CA values in each class).
Figure 2 shows the two corresponding curves to the calculations
performed for the approximations of aloin concentration by
exponential functions.
Figure 2. Exponential approximations of aloin concentration.
The red solid curve represents the rst of the exponential
approximation equations found and corresponds to the rst option
(class marks). The black dotted curve corresponds to the second
option (CA averages). It can be seen that practically both exponential
curves are overlapping.
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Molero and Kas-Danouche. Rev. Fac. Agron. (LUZ). 2022, 39(1): e223913
5-6 |
The concentration of aloin depends on several factors and for
the industrialization of commercial aloe vera-based products, it
is necessary to quantify the CA data and other plant components
in order to meet the demands and specications required by
international standards.
When observing the graphs of the mathematical models
obtained, both linear (gure 1) and exponential (gure 2), there is
no noticeable difference between the two straight lines or between
the two exponential curves, which correspond to the two tabulation
options. What is observed is that both straight lines overlap, and
both exponential curves overlap. In view of this, the coefcient
of determination, (R
2
), was calculated for each of the models to
analyze their behavior with respect to the data. The coefcient of
determination is a statistic that determines the quality of the model
to replicate the results, and the proportion of variation in the results
that can be explained by the model (Steel & Torrie, 1960). It is
used in the context of a statistical model whose main purpose is to
predict future results or to test a hypothesis. But rst, we proceeded
to validate the models using ANOVA; that is, to test the three
assumptions (Martinez et al., 2021). The rst assumption requires
that the residuals approximate a normal distribution. In gure 3
(a and b), it is observed that they do not have exactly a normal
distribution. However, since in this research the size of the groups is
equal to each other, ANOVA is robust with respect to non-normality
(Glass & Hopkins, 1996; Blanca et al., 2017).
a) b)
Figure 3. Histograms of aloin concentration residuals.
The second case requires that the aloin concentration residuals
do not show a tendency (completely dispersed) with respect to the
fresh mass (gure 4).
Fresh mass
Figure 4. Dispersion of the residual concentration of aloin
regarding to the fresh mass.
For this purpose, the Durbin-Watson (DW) statistic was
calculated for each model. For linear model 1, DW=1.493; for
linear model 2, DW=1.490; for exponential model 1, DW=1.726;
and for exponential model 2, DW=1.793.
The third assumption requires that homoscedasticity be
respected. Similar to normality, when the size of the groups is
equal, ANOVA is robust to heteroscedasticity (Statistics Solutions,
2013). The p value for the linear approximation model 1 (i.e., the
model worked with the data classied with the class marks), is
0.551 and for the linear model 2 (i.e., the model worked with the
data classied with the averages of the CA values in each class)
is 0.563. For the exponential approximation model 1 (the data
classied with the class marks), the p-value found is 0.929 while
for the exponential model 2 (with the averages of the CA values in
each class) it is 0.857. The models can be considered robust because
they predict with acceptable accuracy the aloin concentration with
the following coefcients of determination, R
2
; for the linear
approximation models, with the data classied with the class marks
(linear model 1), R
2
= 0.6540 (≈ 65%) and with the data classied
with the averages of the CA values in each class (linear model 2),
R
2
= 0.6990 (≈ 70%). For the exponential approximation models,
with the data classied with the class marks (exponential model 1),
R
2
= 0.8055 (≈ 81%) and with the data classied with the averages
of the CA values in each class (exponential model 2), R
2
= 0.8712
(≈ 87%).
Based on the previous analysis, the use of the exponential
approximation mathematical model 2 is recommended to estimate
or infer aloin concentration as a function of fresh mass only. This
model is given by the equation of exponential approximation found:
y = 0.44886 e
0.00458x
,
where x represents the fresh mass (PF) of the plant and y
represents the aloin concentration.
In this research, the mathematical model did not discriminate
the plants by their origin, but they were considered as belonging to
a region that conglomerates all the populations into one.
The use of mathematical modeling to analyze the drying of A.
vera leaves has been widely employed (Moradi et al., 2019; Sabat
et al., 2018), as well as for the study of mass transfer during the
gel rehydration phenomenon (Vega-Galvéz et al., 2009) and in the
determination of optimal conditions for the removal of aloin from
the gel (Martínez et al., 2017; Jawade & Chavan, 2013), but no
research is recorded on the application of these models to estimate
the concentration of aloin, or any other secondary metabolite in
Aloe spp. using only the fresh mass of the plant, so comparative
studies of these results with those obtained in other works cannot be
made. However, the robustness of the model was veried.
Conclusions
In this research, based on the mathematical modeling to estimate
the concentration of aloin in A. vera L., from the values of the fresh
mass, it is concluded that the exponential curves found, approximate
the data better than the calculated straight lines. Therefore, this
model is recommended for the estimation of the aloin concentration
in naturally propagated plants from their fresh mass and it can be
used to carry out future investigations about the concentration of
metabolites and other chemical compounds in plants of the Aloe
genus by means of least squares approximations considering
nonlinear polynomial functions. In the same way, this model is
useful for the agricultural producer or interested person since it
will be able to estimate the value of CA starting from an easy-to-
measure parameter, such as the fresh mass of the plant, without
Residuos
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Rev. Fac. Agron. (LUZ). 2022, 39(1): e223913. January - March. ISSN 2477-9407.
6-6 |
having to resort to laborious and expensive laboratory protocols and
techniques, minimizing expenses and production stages.
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