Design and Analysis a Low-Cost Cementitious Waterproofing Mixture Based on the Solution of a Mathematical Model

Journal of Environmental Treatment Techniques

2021, Volume 9, Issue 1, Pages: 268-274

J. Environ. Treat. Tech.

ISSN: 2309-1185

Journal web link: http://www.jett.dormaj.com

https://doi.org/10.47277/JETT/9(1)274

Design and Analysis a Low-Cost Cementitious

Waterproofing Mixture Based on the Solution of a

Mathematical Model

Fatima Alsaleh , Feras Al Adday , Mohamed Bassam Hamami

, 3

Department of Transportation Engineering, Faculty of Civil Engineering, University of Aleppo.

Middle East University, Amman, Jordan

Mathematics Department, Faculty of Science, University of Aleppo

Received: 16/09/2020

Accepted: 13/11/2020

Published: 20/03/2021

Abstract

Most of the investigation results of engineering studies depend on the principle of trial and error in decision-making, which requires a lot

of time and effort and does not guarantee to reach the optimum solution. Whereas, mathematical models provide mathematically proven

optimum solutions to problems. This research aims to exploit the huge quantities of fine recycled aggregates (FRA) from the destroyed

buildings and infrastructure of the city of Aleppo to design a cement-based waterproofing concrete mixes, by developing a mathematical

model. The optimum proportions of the materials included in the composition of cement-based waterproofing concrete mixes have been

founded by this model which is derived and solved by a simulated annealing method. Experimental results showed the efficiency and accuracy

of the proposed model in determining the optimal quantities of the mixture content with a minimum cost according to the required engineering

conditions.

Keywords: Fine Recycled Aggregates, Cement-Based Waterproofing Mixture, Linear and Nonlinear Constrains, Simulated Annealing

Method

above the natural ground level or above the ceiling level [4]. The

Introduction

waterproofing is divided according to the method of its

implementation into two main types, namely positive insulation,

in which the source of moisture is isolated so that it prevents its

access to the structural element, thus preserving its properties. As

for negative insulation, which is done if the leaked water has

entered the element to reduce moisture damage to the cladding

and inner medium of the structural element, this type is not

resorted to except in very special cases [5]. The waterproofing

materials are divided into two main parts: the films are either

bituminous or polyvinyl chloride (PVC) films or polyethylene

Concrete buildings are usually exposed to different

environmental and climatic conditions that may negatively affect

their efficiency. Moisture resulting from the exposure of buildings

to water is one of the most important problems that affect not only

structural damage but also affect the health of the residents of

these buildings [1]. The moisture leaking into the building causes

damage to the internal and external cladding elements, it also has

another destructive role on reinforced concrete elements upon

constant exposure to moisture, as it leads to the migration of salts

from the concrete, affecting the properties of the concrete. Not

limited to this only, but may extend to the reinforcement steel and

lead to rust and weaken its sections [2]. Therefore, waterproofing

is the first step to solving this problem to preserve the structural

elements and increase the service life of the facility.

Waterproofing can be defined as the process by which surfaces of

building elements and joints are treated to prevent water from

leaking into or through them in the event of hydrostatic pressure

(

PE) films of different types of HDPE, LDPE, LLDPE, and

VLDPE. These materials vary greatly according to their chemical

composition, these are divided into waterproofing materials with

a water-based, polymeric-based, bituminous-based, cement-basis,

and other. These materials are called liquid waterproofing

coatings. Or in the form of a mixture of two components, one in

the form of a cement-base and the other in the form of a

polymeric-basis [6, 7]. Cement-based waterproofing mixes are

the most common type due to their ease of use and relatively

cheapness as well as its homogeneity with the concrete elements

covered by [8, 9]. Liquid waterproofing coatings it can be formed

from only one compound, which is a solid (powder) mixed with

water to produce a coating, or two compounds, one of them is

solid and the other is a liquid, which produces the waterproofing

[

3]. While waterproof materials are generally known as materials

that are often in the form of one or more layers or membranes that

are implemented on surfaces or ceilings to prevent water from

seeping into the structural elements with or without hydrostatic

pressure, as for moisture insulation materials they are materials in

the form of layers or strips placed within walls to reduce water

leakage without hydrostatic pressure, and to be executed directly

Corresponding author: Feras Al Adday, Faculty of Engineering, Middle East University, Amman, Jordan, falkhalil@meu.edu.jo

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Journal of Environmental Treatment Techniques

2021, Volume 9, Issue 1, Pages: 268-274

coatings if mixed, it can be polymeric, when mixed with the solid

compound, it forms a high elasticity, weatherproof, and adhesion

film. The solid compound in cement coatings usually consists of

a mixture of cement with fillers powder or natural sand [10, 11].

The waterproofing materials constitute the basic element in

designing insulating concrete mixes [12]. Many factors play a big

role in the process of designing insulating cement mixes, such as

strength, permeability, and durability in addition to economic

feasibility, especially in places that are constantly exposed to

moisture and water submersion that require continuous treatment

specifications according to the Indian code. Also, Gupta, 2013

[25, 26] used artificial neural networks trained in the technique of

backward error propagation to predict the compressive strength

values of 55 concrete mixtures with different ratios of cement,

coarse gravel, sand, water and the fineness modulus at 7,14, and

28 days of curing. Nano-Silica was also added to other concrete

samples according to the same previous method for predicting

compressive strength. Another study on pre-cast concrete mixes

was carried out by Bilgil, A, 2012 [27] using artificial neural

networks to determine the behavioral properties of concrete,

which included Slump value, yield stress, and viscosity data for a

number of concrete mixtures used.

Modified and developed forms of the mathematical models

presented by Papadakis, 2000 [28]. Mathematical models have

been modified to determine the values of the ratios of addition of

some substances such as: Silica-fume and Fly ash with low and

high calcium so that the modified model describes the spread of

carbonation and penetration of chlorides to concrete. The

development of mathematical modeling methods for the

formation of mixtures concrete have been achieved to include

Multiple target mathematical relationship, where the Toklu, 2005

[13, 14, 15]. The basis of the process of designing concrete

mixtures is the accurate determination of the proportions of the

materials involved in it, it is possible to change the specifications

of the physical and chemical mixture by changing the quality and

proportions of the formed materials, especially if the concrete

mixture contains the FRA, due to the difference in the

specifications of those materials from the natural aggregate, such

as the water absorption [16]. The war in Syria has produced, in

recent years, tremendous damage to buildings and infrastructure,

which has resulted in very large quantities of solid waste, which

can be used in many engineering applications [17, 18]. Which

made the disposal and recycling of these solid waste an urgent

economic and environmental necessity. By using them in various

applications to ensure the utilization of the occupied spaces and

the non-depletion of natural resources in the reconstruction phase.

Various studies have proven that the coarse recycled aggregate

can be used in various civil engineering projects, such as asphalt

and concrete mixtures and others [19-22], while the use of FRA

was limited to some applications.

[29] developed

a multi-objective mathematical model to

determine the mixing ratios of aggregate in concrete mixtures at

the lowest cost. The proposed model was solved using heuristic

algorithms after the analytical mathematical methods failed to

reach an optimal solution. As for the insulating cement-based

mixes, whether it is paint or mortar, there is a lack of research that

tried to find mathematical models to reach the best ratios of

materials included in the composition of the concrete mixture,

including: What Zhang and Zheng, 2012 have done [30], where

they prepared and insulating capillary crystal mixes with a

cement-basis, the basic structure of which is made of Portland

cement and fine aggregates, while Portland cement was used with

a substance composed of quartz powder in addition to five other

materials as a coating for the mixture, and then they applied

software to analyze the test data to obtain an optimal relationship

that gives the proportions of the input materials In the

composition of the concrete mixture. Each of the research groups,

Zhang, et.al [31] contributed to developing insulating capillary

crystal mixes with a cement basis and determining the quantities

of active substances by means of orthogonal tests. The

proportions of cement to sand were also determined with the

active materials to obtain the composition of the insulating

material according to the optimal proportions. The results showed

that all the properties of the developed insulation material meet

the standard specifications of bonding stress and leakage

resistance. A new addition in Bohus, et.al's search [32] is the use

of cement with the Fly ash according to experimentally

determined percentage. Similarly, work has been done by

Pushkarova, K., et.al, 2015 [33] to find a crystal mixture to protect

concrete structures and increase their resistance to water and

freezing. Optimum of the proportions of materials included in the

mixture were also experimentally done.

Figure 1: Destruction and rubble in the war-affected areas of the city of

Aleppo [17]

With regard to previous studies, mathematical and statistical

modeling has been used in the design concrete mixtures by many

researchers. Vengadeshwari and Reddy, 2013 [23] applied

heuristic algorithms to design concrete mixtures to obtain the

optimal solution away from the experimental principle adopted in

traditional engineering studies, where the best design has been

developed that determines the proportions of materials used in

concrete mixtures to obtain the mixture with the lowest cost and

weight in addition to reliability. The other uses of artificial

intelligence was made by Kale, Kute, 2014 [24] to design concrete

mixtures based on modeling using artificial neural networks to

avoid the implementation of a large number of experimental

mixtures to choose the best combination of materials involved in

the design of the concrete mixture and to achieve the required

As for the insulating cement mix with an elastic polymeric-

basis studied by Lecha and Joanna Juliab, 2011 [34], Where

Optimum of this mixture were carried out using relationships that

connection the composition to the engineering specifications of

the mixture (flexibility, insulation and water vapor permeability).

Optimum of a mathematical model were done to obtain the

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Journal of Environmental Treatment Techniques

2021, Volume 9, Issue 1, Pages: 268-274

optimal proportions of the mixture through statistical experiments

by evaluating the different combinations of mixtures. The

resulting mixtures were tested depending on the desired output of

the generated statistical function. The results obtained showed the

efficiency of the statistical model in designing the insulating

covering mixtures of cement basis.

Based on the previous studies that have been presented, it is

evident that the cement-based waterproof concrete mixes were

not designed according to a specific mathematical model solution

that fulfills the required engineering conditions at the lowest

possible economic cost. In this paper, an innovative mathematical

model has been proposed to determine the optimal proportions of

the materials included in composition of waterproof concrete

mixes. In addition to making use of the FRA resulting from the

waste of destroyed buildings and infrastructure in the city of

Aleppo, A mathematical model is derived and solved by

simulated annealing method.

2.2.1 Limitations of the Mathematical Model

a. Permeability condition: Water isolation can be achieved in

more than one method, but all methods can be summarized by

reducing the permeability of the material by reducing the

percentage of material voids to a minimum or closing the voids

so that a specific substance enters the pores and closes them

mechanically or chemically as in the case of crystal cement

materials [30]. In this research, work has been done to achieve a

minimum permeability by reducing the percentage of voids in the

samples in order to achieve a maximum water-proof ratio. The

sum of the absolute unit volumes of the materials forming the

cement-base waterproof mixture is equal to one that are designed

with taking into account the homogeneity of the units as shown in

equation (1).

V = V

+ V

= 1 Eq. (1)

are the volume of cement, polymeric

where: V

, V

Materials, tools and research methods

bond, water, and sand and filler, and voids respectively. Thus, the

volume of voids in this mixture can be obtained as follows:

.1 Research materials

The following materials were used in the research according

to the appropriate standard specifications:

Cement: Cement type -32.5 was used and it was ensured that it

meets the Syrian standard specifications 1673/1996 and 75/1998.

Water: Drinking water has been used that meets the Syrian

standard 2007/45.

Sand and fillers: In this research, FRA were used, resulting

= 1 - V Eq. (2)

+ V

•

By substituting the weights instead of volumes, with making

•

some reforms to the equation (2) and adopting the proportion of

voids instead of their volume, equation (2) becomes as follows:

•

from destroyed buildings in the neighborhoods of the Salah al-

Din area, passing through the sieve opening 0.425 mm, which

includes fine sand whose dimensions range between 0.425 mm

and 0.075 µ, in addition to the powder whose diameter is less than

훾_푤푊_푐

푊푝

퐺_푝

푊_푤푊푓

퐺_푤퐺_푓

−

(

)

푉푣

푉

푔

퐺_푐

퐴% =

Eq. (3)

푉

5 µ according to the ASTM E11-16. The specific gravity and

water absorption percentage of the three samples of FRA were

measured at average of 2.6 and 6.98%, respectively. It was

observed that the absorption of FRA of water is higher than

similar samples from NA, which is consistent with the reference

studies.

where: Wc , Wp , Ww , Wf are the weights of cement,

polymeric bond, water, and sand and filler, respectively; γ

is unite weight of water; Gc , Gp , Gw , Gf are the specific

gravity of cement, polymeric bond, water, and sand and

filler, respectively. By making this ratio minimum, and

considering that the unit volumes equal to 1, n is the number of

components in the mixture design, the general form of equation

•

It was also confirmed that the samples are free of impurities

(

clay and exotic materials soluble). As for the sand equivalent, the

average value for the three samples was 69.3% (the acceptable

value for the concrete is 75%, which increases with the increase

in the strength, as it is 80% for the highly strength concrete), since

the required strength of waterproofing mixes are less than

concrete in general, and it decreases whenever the surfaces are not

loaded, the sand equivalent value of the FRA used in the mixture

is considered acceptable.

(

3) becomes as follows:

ꢀ_ꢁ

ꢂ

ꢃ_푖

ꢄ_푖

푛

ꢅꢆꢇ

퐴% = 1 −

∑

→ 푀ꢈ

Eq. (4)

In general, the voids percentage of concrete mixtures according

to the American method (ASTM) of design ranges between 0.5 -

3%, depending on the maximum size of the aggregates and the

workability [35]. These numbers are very large in the case a

waterproofing material. Therefore, the value of the voids

percentage was adopted at the minimum value A = 10 -22 in the

final model, which is the lowest value that gives a solution to the

mathematical model within the solution space.

b. Ratio of sand and filler (FRA) to cement: Several initial

experiments were conducted, starting with the proportions of sand

to cement (Wf / Wc ) approved for regular mortar according to

the ASTM C778-74 and the ISO system, which are 2.75 / 1 and

3/1, and then these ratio reduced according to the results of the

•

Bond: Acrylic dissolved in water of 1.05 specific weight was

used.

.2 Mathematical model for designing dielectric mixtures

The mathematical design model is based on the mathematical

relationships governing the material that give the basic properties

to be achieved. These relationships are classified into two groups.

The first is the relationship that represents the properties or values

necessary for the design to achieve its objective, and the second

is the relationships that can be considered constraints imposed by

the practical and implementation conditions.

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2021, Volume 9, Issue 1, Pages: 268-274

preliminary study of the cement insulation materials to suit the

available materials, in order to achieve a good consistency that

ensures ease of implementation. It has been adopted field of

change Wf / Wc ϵ [0.5-2.5].

푀ꢈꢉ ꢊ(ꢋ) = 0.055푊 + 1.5 푊 + 0.001푊 + 0.01 푊

푐

푝

푤

푓

푊_푐

푊_푝

푊_푤푊_푓

ꢍ ≤ 1 × 10^ꢎ^ꢏ^ꢏ

2.6

−

ꢌ

.981 3.15 1.05

ꢃꢑ

.5 ≤ ꢐ_ꢃ_ꢒꢓ ≤ 2.5

c. The ratio of water to cement: The ratio of water to cement

ꢃꢁ

(

Ww / Wc) in different concrete mixes greatly and effectively

0.3 ≤ ꢐ_ꢃ_ꢒꢓ ≤ 1.2

Eq (6)

affects the compressive strength of concrete mixtures, as the

compressive strength increases with increasing this ratio until

reaches a maximum value that matches the best ratio and then

returns to decrease [34][35][36]. An analytical study was

conducted for research and studies interested in cement insulation

materials, taking into account the specifications of the different

materials included in the designs, in addition to monitoring the

increase in the water percentage at a rate of more than 7.5%, since

FRA used have a high absorption rate, and thus the field Ww / Wc

ϵ [0.3-1.2] was adopted as a limitation for the studied model.

d. Ratio of acrylic material to cement: a preliminary analytical

study completed for different polymeric cement insulating

materials. After performing a number of preliminary experiments

with different mixing ratios, the change field was determined

within the range Wp / Wc ϵ [0.25-0.7].

푊_푝

0.25 ≤ ꢌ ꢍ ≤ 0.7

푊_푐

.4푊 + 0.03 푊 ≤ 0.5푊 + 012 푊

푐 푓 푐 푓

푊 , 푊 , 푊 , 푊 ≥ 0

푐

푓

푝

푤

Thus, it can represent the final form of the mathematical

model for the required design as it is in the general equation

(6). Heuristic algorithms, such as the Optimization Particle

Swarm have been applied to solve this model.

2.3 Optimization Particle Swarm

Optimization Particle Swarm was devised by Eberhart and

Kennedy in 1995 [38, 39].

e. The relationship of the correlation of the percentages of

cement and filler with water was studied so that the permeability

condition is achieved physically, and the restriction was derived

mathematically to achieve the condition within the required field

2.3.1 How the binary optimization particle swarm works:

This algorithm randomly generates a swarm of particles, where

each particle represents an acceptable solution of the possible

solutions to the problem. Therefore, determining which the best

particle is requires the algorithm to search in an iterative manner

using two equations. The first is called the particle velocity

equation (speed equation) [40].

[

37].

f. All variables in the mathematical model fulfill the non-

negativity condition.

.2.2 The target function

The target function in the mathematical model was adopted as

푡ꢔꢇ

= W 푉^푡+ ꢕ 푟 ꢖꢗ푏푒푠ꢘ − 푋 ꢙ + ꢕ_ꢏ푟_ꢏ_푗ꢖ푔푏푒푠ꢘ −

푡

푉

푋

ꢅ푗

ꢇ ꢇ푗

ꢅ

ꢅ푗

ꢅ

an economic cost function (a minimal cost function), which is a

linear programming that represents the price of the mixture in

Syrian pounds according to the price of each component offered

in the local market until the date of preparing the study, price and

specific gravity can be shown in the table1. Thus the target

function in the general formula is given as:

푡

ꢅ푗

ꢙ

Eq. (7)

where: i: Denotes the particle number; j: the number of

elements inside the particle; 푉 : The velocity of the particle i in

푡

ꢅ푗

푡ꢔꢇ

the previous instant t; 푉_ꢅ_푗: The velocity of the particle i in the

푡

next instant t +1; ꢗ푏푒푠ꢘ :The most appropriate value reached

by the i-particle until the iteration t; 푔푏푒푠ꢘ : The most

^푃푇표푡푎푙 ₌_∑푛 푃 푊 → 푀ꢈꢉ

ꢅ

Eq. (5)

푗ꢆꢇ

푗

푡

ꢅ

appropriate value within the swarm has been reached up to

whereas, Pj is the price of the component and its weight is Wj, n

is the number of elements of the mixture. Therefore, the target

function must strive to its minimum value in order to achieve the

economic benefit required in the case of using of FRA.

푡

ꢏ

repetition

푔푏푒푠ꢘ = 푚ꢚ푥 (ꢗ푏푒푠ꢘ , ꢗ푏푒푠ꢘ ,

ꢅ

ꢇ

…

ꢗ푏푒푠ꢘ^푡). r1j and r2j: Random values to ensure diversity of

ꢛꢛ

푡

investigation and fall within the range [0,1]; 푋 : The position

ꢅ푗

of the particle i in the previous instant t; W: A variable

representing a percentage of the particle velocity at the previous

Table1: The specific gravity and price in Syrian pounds for each

component of the mixture

Fine

moment; c

speed of reaching the best solution. In applications that use this

algorithm, the variables (c and c and W) are calibrated

experimentally [41, 42]. Value of W is confined within the

range [0.3- 0.9], as for the two variables c and c , they have a

field [0.4-2]. Consequently, a set of experiments were

conducted to reach the appropriate values. Equation (7)

determines the probability of a change in the values of the

particle's forming elements. The second equation is the equation

for the new state of the particle, as expressed by equation 8 [43].

1 2

and c : Acceleration variables that control the

Component

Cement

Recycled

Acrylic Water

Aggregate

Specific

gravity

.15

2.6

1.05

1.5

Price: SP/gr

0.055

0.01

0.001

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Journal of Environmental Treatment Techniques

2021, Volume 9, Issue 1, Pages: 268-274

푢 < sig [푣 (ꢘ + 1)]

As for the variable W, after performing the experimental

calibration, the best value is 0.4.

ꢅ푗

푋 (t + 1) = {

Eq. (8)

ꢅ푗

푢_ꢅ_푗≥ sig [푣 (ꢘ + 1)]

ꢅ푗

푢 : A random value within the interval [0, 1] generated

ꢅ푗

according to an equal probability function at the beginning of

each iteration.

ꢇ

푠ꢈ푔 ꢐ푣 (t)ꢓ =

Eq. (9)

ꢅ푗

ꢝꢞ (ꢠ)

푖ꢟ

ꢇꢔꢜ

Sigma function 푠ꢈ푔 (v) aims to narrow the numeric values

into confined space [0, 1] in order to improve the performance

of the algorithm [44]. Figure 2 shows a systematic diagram for

working the binary optimization particle swarm.

1 2

Figure 3: Calibration of C , C value

After that, it is entered into an iterative loop to improve the

solutions generated, where in each iteration the following is done:

- Generating a vector (uij) from random values confined to the

interval [0, 1] according to an equal probability function, where

uij) will be used later in the process of changing the state of the

solution.

- Calculate the velocity (v) for each element of the problem

elements of the particle, then calculate the value of Sig (v).

- Conducting a comparison process between the values of (uij)

(

and Sig (v), where the comparison process results in a change in

the position of the problem elements of the particle.

4- Calculation of the best fitting value (pbest) for the particle and

the best fitting value (gbest) at the swarm level (this is equivalent

to knowing the best solution in this iteration) [46].

Then the second iteration is taken, where the same previous

steps are repeated until the best solution is accomplished. As a

result, the weight of each substance in unite volume of the best

mixture shown in the Table 2.

Table 2: shows the weight of each substance in unite volume

Component

Weigh (in

designed cm ): gr

Cement

Filler

Acrylic

Water

0.4239

1.0580 0.1056

0.3388

By substituting the values shown in the previous table, which

represent the components of the cement-based waterproof

concrete mixes, it is noticed that it achieves the mathematical

model, which represents the lowest value of the target function of

Figure 2: Illustrates flow chart of methodology [45]

(

0.1918) SP for each cm .

Results and discussion

Initially, a random generation of particles (acceptable

4 Conclusion

Most of the investigation results of engineering studies

depend on the principle of trial and error in decision-making,

which requires a lot of time and effort and does not guarantee

reaching the optimum solution. Whereas, mathematical models

provide mathematically proven optimum solutions to problems

that can be formulated according to a mathematical model. The

solutions) is generated, and then the velocity equation is provided

with the necessary random variables (r^t1j ,r 2j ), the variables (C

, W), and the initial velocity calculated Figure 3 shows the

calibration curve for the C and C variables, which were obtained

experimentally, as it appears that the best value obtained is 1.9.

272

Journal of Environmental Treatment Techniques

2021, Volume 9, Issue 1, Pages: 268-274

proposed model was solved by using the binary optimization

particle swarm, the laboratory experimental results of the

permeability showed the efficiency of the model in determining

the exact proportions of the materials included in composition of

cement–base waterproofing concrete mixes according to the

imposed engineering conditions at the low-cost.

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Ethical issue

Authors are aware of, and comply with, best practice in

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(

avoidance of guest authorship), dual submission, and

020, 8(8), pp. 4037-4043. doi.org/10.30534/ijeter/2020/02882020.

manipulation of figures, competing interests and compliance with

policies on research ethics. Authors adhere to publication

requirements that submitted work is original and has not been

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Competing interests

The authors declare that there is no conflict of interest that

would prejudice the impartiality of this scientific work.

Feras. Al Adday, 2020. Selecting the Best Method for Adding

Recycled Aggregate. International Journal of Emerging Trends in

Engineering Research vol. 8, no. 6, pp. 2253–2258, 2020.

doi.org/10.30534/ijeter/2020/08862020.

Fatima Alsaleh and Feras Al adday, 2020. Manufacture of

Lightweight Thermal Insulation Concrete Using Recycled

Aggregates and Syrian Pozzolan Int. J. Adv. Trends Comput. Sci.

Authors’ contribution

All authors of this study have a complete contribution for data

collection, data analyses and manuscript writing.

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