Quantile Loss Function Empowered Machine Learning Models for

Predicting Carotid Arterial Blood Flow Characteristics

T. RAJA RANI1, WOSHAN SRIMAL1, ABDULLAH AL SHIBLI2,

NOOH ZAYID SUWAID AL BAKRI3, MOHAMED SIRAJ4, T. S. L. RADHIKA5

1Foundation Programme Department,

Military Technological College,

Muscat,

OMAN

2Applied & Research Department,

Military Technological College,

Muscat,

OMAN

3MTC Clinic,

Military Technological College,

Muscat,

OMAN

4Systems Engineering Department,

Military Technological College,

Muscat,

OMAN

5Department of Mathematics,

BITS Pilani Hyderabad,

Hyderabad,

INDIA

Abstract: - This research presents a novel approach using machine learning models with the quantile loss

function to predict blood flow characteristics, specifically the wall shear stress, in the common carotid artery

and its bifurcated segments, the internal and external carotid arteries. The dataset for training these models was

generated through a numerical model developed for the idealized artery. This model represented blood as an

incompressible Newtonian fluid and the artery as an elastic pipe with varying material properties, simulating

different flow conditions. The findings of this study revealed that the quantile linear regression model is the

most reliable in predicting the target variable, i.e., wall shear stress in the common carotid artery. On the other

hand, the quantile gradient boosting algorithm demonstrated exceptional performance in predicting wall shear

stress in the bifurcated segments. Through this study, the blood velocity and the wall shear stress in the

common carotid artery are identified as the most important features affecting the wall shear stress in the

internal carotid artery, while the blood velocity and the blood pressure affected the same in the external carotid

artery the most. Furthermore, for a given record of the feature dataset, the study revealed the efficacy of the

quantile linear-regression model in capturing a possible prevalence of atherosclerotic conditions in the internal

carotid artery. But then, it was not very successful in identifying the same in the external carotid artery.

However, due to the use of idealized conditions in the study, these findings need comprehensive clinical

verification.

Key-Words: - Machine-learning algorithms, Quantile loss function, Boosting algorithms, Numerical model,

Blood flow, Carotid artery, Wall Shear Stress.

Received: June 12, 2022. Revised: September 5, 2023. Accepted: September 27, 2023. Published: October 10, 2023.

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T. Raja Rani, Woshan Srimal,

Abdullah Al Shibli, Nooh Zayid Suwaid Al Bakri,

Mohamed Siraj, T. S. L. Radhika

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Abbreviations/Acronyms:

AI: Artificial Intelligence

CCA: Common Carotid Artery

D: Blood Density

ECA: External Carotid Artery

ICA: Internal Carotid Artery

InPr: Inlet Pressure (Blood pressure at the CCA

entrance)

MAE: Mean Absolute Error

MAPE: Mean Absolute Percentage Error

ML: Machine Learning

MSE: Mean Squared Error

Out1Vel: Blood velocity at outlet1, i.e., ICA

Out2Vel: Blood velocity at outlet2, i.e., ECA

PARDISO: Parallel Direct Sparse Solver

RFR: Random Forest Regressor

RMSE: Root Mean Square Error

SVR: Support Vector Regressor

Vel_CCA: Blood velocity in CCA

Vel_ECA: Blood velocity in ECA

Vel_ICA: Blood velocity in ICA

VIF: Variance Inflation Factor

Vis: Blood viscosity

WSS: Wall Shear Stress

WSS_CCA: Wall Shear Stress in CCA

WSS_ECA: Wall Shear Stress in ECA

WSS_ICA: Wall Shear Stress in ICA

1 Introduction

The prediction of carotid artery blood flow has

garnered significant attention in the medical

community owing to its relevance in

cerebrovascular diseases and neurological disorders.

Conventional regression models for estimating

blood flow have shown limitations, as they neglect

outliers and heteroscedasticity, leading to

suboptimal predictions. However, recent progress in

machine learning techniques utilizing the quantile

loss function presents promising avenues to

overcome these challenges and improve prediction

accuracy.

A comprehensive review of the existing

literature reflecting the current state of the art was

undertaken and detailed as follows: [1], explored

the potential of artificial intelligence (AI) in

predicting cardiovascular disease (CVD). Their

study summarizes machine learning (ML)

applications in CVD, including direct prediction

based on risk factors and medical imaging and ML-

based hemodynamics for indirect CVD assessment.

Their review discusses research challenges and

envisions future AI technology development in

cardiovascular diseases. The study, [2], proposed a

simulation-based framework to achieve Deep

Learning (DL) based hemodynamic prediction of

healthy and diseased carotid arteries. The

methodology demonstrated accurate DL predictions

by utilizing high-quality point cloud datasets and an

advanced DL network, aligning well with CFD

simulations while significantly reducing

computational costs.

The review of the work by, [3], highlights the

application of AI in cardiovascular imaging,

focusing on coronary atherosclerotic plaque

analysis. It encompasses various areas such as

plaque component analysis, identification of

vulnerable plaque, myocardial function detection,

and risk prediction. The review discusses current

evidence, strengths, limitations, future directions for

AI in cardiac imaging of atherosclerotic plaques,

and insights from other fields. The study, [4],

presented a valuable guide for readers approaching

AI algorithms in carotid atherosclerosis. Their study

revealed that the application of AI using US

(UltraSound), CTA (Computed Tomography

Angiography), and MRI (Magnetic Resonance

Imaging) would offer a new strategy for rapid and

objective diagnosis. However, limitations such as

small cohorts, noise, and difficulties in model

comprehension might hinder widespread clinical

use. They suggested the necessity of multi-center

studies to validate AI's role in symptomatic carotid

plaque detection, especially when considering MRI

techniques.

The study, [5], developed a machine-learning

model to predict the blood flow waveform in the

internal carotid artery (ICA). The model was trained

on patient data, relying on state-of-the-art Doppler

manometry measurements for obtaining accurate

results. The study, [6], used machine learning

models to detect arterial disease, including stenoses

and aneurysms, from peripheral measurements of

pressure and flow rates across the network. The

study, [7], proposed a model to elucidate individual

patients' cerebral circulation using blood flow

simulation incorporating clinical data. Their

approach enabled obtaining the probability of

different outputs, considering patient condition

uncertainty. By combining machine learning with

blood flow simulation, predictions were performed

43,000 times faster on a desktop computer, allowing

real-time surgical risk assessment. Their prediction

results revealed the relationship between collateral

circulation and life-threatening surgical outcomes.

The study, [8], investigated and compared the

performances of different ML techniques for

detecting the presence of a carotid disease by

analyzing the Heart Rate Variability parameters of

opportune electrocardiographic signals selected

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from the available databases. The study, [9],

introduced a 50-layer residual network as a feature

generator for identifying carotid stenosis. The Deep

convolutional neural networks classified

sonographic images into four categories based on

the features related to the ICA blood flow rate. The

study, [10], modeled high-fidelity blood flow in

understanding CVD. The potential for data-driven

patient-specific blood flow modeling in

computational and experimental cardiovascular

research has been highlighted with an emphasis on

the challenges and opportunities in the field. The

study, [11], combined both statistical and machine

learning methods to reduce information redundancy.

By this, they could enhance accuracy in disease

diagnosis. Their study developed Graph theory-

inspired ML models to identify significant features

for prediction models.

Motivated by the capability of ML algorithms to

gain valuable insights from data, this research

endeavors to employ these algorithms on medical

datasets generated from the simulations of the

numerical model for the human carotid artery

developed in this study. The model used an

incompressible Newtonian fluid representation for

blood and considered the artery as an elastic pipe,

allowing for altering its material properties during

simulations facilitated by the COMSOL software.

Machine-learning regression models were trained

following the development of the numerical model

for an idealized human carotid artery and data

generation.

The structure of this article is as follows:

Section 2 introduces the problem identification and

research objectives. In Section 3, a comprehensive

explanation of the adopted methodology is provided.

Sections 4 and 5 present the research results, along

with their interpretation. Finally, Section 6 presents

the conclusive findings of the research, highlighting

its limitations, practical applications, and potential

future research directions in the field.

2 Problem Formulation

While most of the studies in the field of medical

data analysis concentrated on medical image

classification or hemodynamic prediction by

utilizing available clinical datasets on the carotid

artery, this work takes a different approach. It

integrates Computational Fluid Dynamics (CFD)

with data analysis, aiming to analyze datasets

generated from the developed models simulating the

carotid artery under various anatomical and

physiological conditions. The primary focus of this

study is on translational research, aiming to predict

blood flow characteristics accurately while avoiding

the need for invasive or costly methods such as MRI

and Ultrasound. By addressing this, the study aims

to fulfill the requirement for non-invasive

techniques in evaluating vascular changes in the

carotid artery, as highlighted in the work by, [12].

A quantitative and experimental approach is

being proposed, with computer-based simulations

utilized to achieve this goal. The research question

is defined as descriptive and seeks an answer to the

following:

Is it feasible for Machine Learning models trained

on simulated data to accurately detect the presence

of atherosclerosis in the carotid artery?

2.1 Research Objectives

Carotid artery blood flow is paramount in

maintaining cerebral perfusion and brain health.

Accurate carotid artery blood flow prediction is

valuable in understanding cerebrovascular diseases

and guiding clinical interventions. Therefore, the

primary focus of this research is to explore the

efficacy of machine learning models in predicting

carotid artery blood flow and to assess the

effectiveness of the quantile loss function in

improving prediction accuracy and robustness.

1. To develop and validate a numerical model for

the human carotid artery.

 Simulate the model for both healthy and

unhealthy conditions.

 Compute the wall shear stress (WSS) and

compare their values with the clinical

results.

2. To estimate WSS in the carotid artery using ML

algorithms.

 Develop ML models that best fit the data.

 Compute the metrics and quantile loss

function on the train and test datasets to

identify the most reliable ML prediction

model.

3. Since WSS acts as an initial and independent

indicator of atherosclerotic changes, [12], the

goal is to forecast these changes based on the

calculated estimates of WSS.

Exposures and Outcomes:

The independent variables in this study encompass

fluid-related parameters such as blood density and

viscosity, flow-related parameters like the inlet

pressure and outlets' blood velocities, and material

parameters such as the artery's density, Bulk

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Modulus, and Poisson ratio. The outcome to be

measured is the WSS in all three arterial segments.

3 Methodology

This section presents a detailed, step-by-step

procedure from data collection to data analysis

designed to accomplish the proposed objectives.

3.1 Prepare Data Collection Instrument

The carotid artery is a vital supplier of blood to the

face, neck, and brain. Its main vessel, the Common

Carotid Artery (CCA), undergoes bifurcation into

the External Carotid Artery (ECA) and Internal

Carotid Artery (ICA), as depicted in Figure 1. The

ECA primarily supplies blood to the face and neck,

whereas the ICA delivers blood to the brain.

Fig. 1: Anatomy of Carotid Artery

This study has formulated a numerical model to

accurately depict the blood flow dynamics in the

carotid artery. The model incorporates essential

elements of the blood circulatory system, where the

blood is represented as a Newtonian fluid, and the

artery is treated as an elastic circular pipe with

bifurcation. This model was implemented using the

COMSOL Multiphysics software, enabling the

simulation of the blood flow problem within an

idealized artery, as illustrated in Figure 2. Data

collection has relied on clinical data about the

artery's anatomy and the physiological behavior of

blood flow to ensure realism.

Fig. 2: Geometry of the Artery in COMSOL

The current work focused on the stationary

Navier-Stokes equation and implemented it using

the laminar flow interface. Viscosity and density

values were determined based on available human

blood data to ensure accuracy. The elastic behavior

of the artery was characterized using anatomical

data specific to the carotid artery.

The P1-P1 linear finite element Galerkin

method was chosen for the discretization process for

effectively handling velocity and pressure variables.

To create a physics-based mesh, 40637 triangular

elements and 4192 quadrilateral elements were

employed, resulting in 76758 degrees of freedom.

Among these, 51172 degrees of freedom were

utilized to determine the velocity, while the

remaining 25586 degrees of freedom were used for

the pressure variables. For more detailed mesh

parameters, please refer to Table 1.

Table 1. Parameters of the Extremely Fine Mesh

Number of elements

44829

Number of vertex elements

17

Number of edge elements

2466

Average element quality

0.8004

Minimum element quality

0.08677

Mesh area

15.76cm2

Newton's method has been employed to tackle

the resulting non-linear flow problem, which proved

effective in finding a solution. Zero initial and no-

slip boundary conditions were enforced on the

boundary walls, along with no backflow

phenomena. The PARDISO solver has been used to

efficiently solve the system, greatly facilitating the

computation process and leading to reliable results.

Additionally, we activated the stability settings

within the software to guarantee the numerical

stability of the generated Galerkin finite element

model. Furthermore, convergence of solutions for a

given set of model parameters was ensured

throughout the data collection process.

Table 2 presents the input parameters used in

the COMSOL model builder to simulate the human

carotid arterial blood flow, [13], [14], [15], [16],

[17]. Values corresponding to human blood

characteristics are assigned to the model parameters,

density, and viscosity to represent blood as a

Newtonian fluid. Regarding the elastic properties of

the carotid artery, the density, bulk modulus, and

Poisson ratio are assigned values relevant to the

human carotid artery, as indicated in Table 2. The

dimensions of the artery, including length, diameter,

and other relevant measurements, are derived from

information provided in the literature.

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Furthermore, Table 2 and Table 3 provide

essential ranges for the inlet pressure and output

velocities necessary for generating realistic blood

flow patterns in the simulation. These ranges are

crucial in achieving biologically meaningful results

in the model.

Tables 3 (a) and (b) present the data necessary

for conducting simulations. The inlet pressure is set

to 100, 120, and 130 mm Hg. As for the blood

velocities at outlet 1 (ICA) and outlet 2 (ECA), they

are assigned values within the specified range

(normal or healthy), as indicated in the eighth row

(third and fourth columns) of Table 2.

Table 2. Data on Characteristics of Blood and

Carotid Artery, [13], [14], [15], [16], [17]

Fluid (blood)

Properties

Artery Properties

Density

(kg/m3)

1060

Density

960

Viscosity (Pa.

s)

0.004

Bulk

Modulus

(N/m2)

1.2 × 108

Poisson

Ratio

0.45

ARTERY

CCA

ICA

ECA

Diameter

(mm)

6.10 ±

0.8

4.8 ± 0.3

3.0 ± 0.6

Length (cm)

13.6 ±

1.2

8.6 ± 1.4

Velocity

(m/sec)

-

0.187-

0.295

0.121-

0.185

Bifurcation Angle (36 ±11)º

Table 3(a). Dimensions of the Simulated Artery

ARTERY

CCA

ICA

ECA

Diameter (mm)

6.2

4.8

3.0

Length (cm)

14.2

9.0

Table 3(b). Data for running simulations

3.2 Review the Collected Data for Quality and

Completeness

The data types and counts of the numerical model

parameters in segment CCA are summarized in

Table 4. It is evident from the table that there is no

missing data in this segment. Similar findings were

observed in the other two segments, ECA and ICA.

Exposures:

Vis: Blood viscosity (Pa.s)

D: Blood Density (kg/m3)

InPr: Blood pressure at the CCA entrance (mm Hg)

Out1Vel: Blood velocity at exit 1, i.e., ICA (mm

Hg)

Out2Vel: Blood velocity at exit 2, i.e., ECA(mm

Hg)

Outcomes:

Vel_CCA: Blood velocity in CCA(m/s)

Vel_ICA: Blood velocity in ICA(m/s)

Vel_ECA: Blood velocity in ECA(m/s)

WSS_CCA: Wall Shear Stress in CCA (Pa)

WSS_ICA: Wall Shear Stress in ICA (Pa)

WSS_ECA: Wall Shear Stress in ECA (Pa)

Table 4. Information on numerical model

Parameters from Python Code

dtypes: float64(9), int64(3) , memory usage: 8.6 KB

3.3 Data Management

Following the format presented in Table 5, the data

has been systematically collected from the

numerical model. Each set of exposures was

evaluated using the COMSOL software, and the

corresponding results were recorded in their

respective columns within the table.

Table 5. Data collection format

3.4 Data Analysis

The first step is to compute the statistics of

exposures to get some basic information on the

exposures. Table 6 provides a comprehensive

breakdown of this data.

InPr

(mm

Hg)

Out1vel

(ICA)

(m/s)

Out2ve

l (ECA)

(m/s)

D

(kg/𝒎3)

Vis

(Pa. s)

Bifurc

ation

Angle

(degre

es)

100

0.241

0.153

1060

0.0035

30 º

120

0.193

0.122

1075

0.004

130

0.1205

0.077

0.0045

Column

Non-null

Count

Dtype

0

Vis

90non-null

float64

1

D

90non-null

int64

2

InPr

90non-null

int64

3

Out1vel

90non-null

float64

4

Out2vel

90non-null

float64

5

Vel_CCA

90non-null

float64

6

Vel_ICA

90non-null

float64

7

Vel_ECA

90non-null

float64

8

WSS_CCA

90non-null

float64

9

WSS_ICA

90non-null

float64

10

WSS_ECA_

90non-null

float64

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Table 6. Descriptive Statistics of Exposures

Statistic

s

Vis

D

InPr

Out1ve

l

Out2ve

l

count

90

mean

0.0042

1062

112.2

0.1736

0.1102

std

0.0004

5.0

11.4

0.0675

0.0427

min

0.0035

1060

100

0.0723

0.046

25%

0.004

1060

100

0.1205

0.077

50%

0.004

1060

120

0.193

0.122

75%

0.0045

1060

120

0.241

0.153

max

0.0005

1075

130

0.241

0.153

3.5 Data Validation

Following the completion of the previous stage, the

subsequent step involves data validation. During this

phase, the reliability of the developed numerical

model as a data source is thoroughly assessed and

evaluated. This critical process ensures the accuracy

and trustworthiness of the generated data for further

analysis and interpretation. The current research also

examined scenarios where the blood flow in ICA or

ECA experienced a reduction due to conditions like

atherosclerosis. Consequently, states with 20%,

50%, or 70% reductions in blood velocity at the

outlets were analyzed; these findings would shed

light on the impact of various health conditions on

blood flow distribution within the carotid artery

network.

For validating the model, WSS values were

computed for the CCA under healthy conditions and

found in the interval (0.850 Pa to 3.464 Pa), which

is towards the right of the mean, 0.850 +/- 0.195 Pa,

as reported by [18].

The subsequent section introduces an in-depth

analysis of the collected (simulated) data utilizing

advanced ML algorithms. The flow chart in Figure 4

illustrates the step-by-step process employed during

this analysis, offering a clear and systematic

overview of the entire procedure.

4 Results and Discussions for WSS in

CCA

This section presents plots depicting the shear stress

in the carotid artery for specific parameters. These

visualizations offer a comprehensive understanding

of the flow dynamics within the artery.

Subsequently, we delve into data analysis using

machine learning (ML) algorithms. Figure 3 depicts

shear stress contours when InPr is 100, out1vel

=0.241, out2vel is 0.077, D =1060, and Vis=0.0045.

The initial step involved is conducting a univariate

analysis of the features within the dataset. This

analysis encompassed an exploration of their range

and central tendency. Furthermore, descriptive

statistics were thoroughly examined to identify any

potential outliers in the data. This meticulous

scrutiny determined that noteworthy disparities

between the exposures' maximum and 75th

percentile values were absent. This absence of

outliers underscores the robust nature of the dataset,

rendering it well-suited for subsequent analysis.

Following the comprehensive univariate analysis of

the features, a parallel assessment was executed on

the target variables using bar charts. The results of

this evaluation indicated an absence of skewness in

the data distribution, thereby signifying a well-

balanced distribution of the target variables.

Fig. 3: Shear Stress Contour

Moving forward, the next phase involved

conducting a multivariate analysis. This phase

encompasses the exploration of interdependencies

among multiple dependent variables or features

concerning an outcome or target variable. This

approach facilitates a more holistic comprehension

of the intricate relationships embedded within the

dataset. Each feature underwent computation of its

Variance Inflation Factor (VIF) score to gauge the

extent of correlation. This process allowed for

assessing the degree of correlation among the

features. Notably, the analysis revealed that the

features exhibited a modest level of correlation, as

evidenced by VIF scores below 5 for each feature.

Subsequently, the first model, namely the linear

regression model, was developed using Vis, D, InPr,

Out1vel, and Out2vel as features and WSS_CCA as

the target variable.

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Fig. 4: Flow Chart

Various metrics such as MAE (Mean Absolute

Error), RMSE (Root Mean Square Error), R2, and

adj-R2 scores were calculated for both train and test

data sets and presented in Table 7. The assumptions

on the linear regression model have been carefully

examined and deemed satisfactory. As a result, the

linear regression model appears to be well-suited for

the dataset. Nonetheless, in addition to this, non-

linear regression models, ensembled, and boosting

models were also applied and evaluated.

A Python code was developed to evaluate the

metrics of polynomial regressors with various

degrees. To determine the optimal-fit polynomial,

graphs depicting the Mean Squared Error (MSE)

versus the polynomial degree were plotted for both

the training and testing data, as illustrated in Figure

5.

Fig. 5: Graphs depicting MSE vs. degree of the

polynomial

The analysis of these graphs showed that a

third-degree polynomial exhibited the best

performance as a non-linear prediction model for the

given data. In the subsequent steps, the SVR

(Support Vector Regressor), RFR (Random Forest

Regressor), Adaboost, Gradient boosting, and XG

boost algorithms were utilized, and their

corresponding metrics were calculated and

displayed in Table 7.

Observations:

1. The RMSE on the test dataset is the lowest

for the polynomial model, showing that this

model can predict the target value most

accurately.

2. MAE is the average error between the

predicted and actual values and is the

minimum for the polynomial model.

3. A significant reduction in MAE was

observed after hyper-tuning the SVR model

using the RBF (Radial Basis Function)

kernel with the parameters C=100 and

gamma=0.01. This improved performance

4. R2 for the training dataset is significantly

higher for the SVR, decision tree, RFR,

RFR-tuned, and all the boosting models,

showing that all these models have over-

fitted the data.

5. MAPE is 1.1 for the SVR-tuned model,

which means, on average, the prediction is

off by 1%.

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Table 7. Metrics of Regressor Models

Regresso

r Models

Data

Set

RMSE

MAE

R2

Adj-R2

MAPE

Linear

Train

0.2148

0.1829

0.9525

0.9488

-

Test

0.2559

0.2373

0.9405

0.9157

-

Poly

degree 3

Train

0.1295

0.0768

0.9827

0.9814

-

Test

0.1574

0.0983

0.9775

0.9681

-

Support

Vector

Train

0.1961

0.1412

0.9603

0.9573

8.2001

Test

0.2913

0.2194

0.9229

0.8907

7.4285

Support

Vector

Tuned

Train

0.1640

0.1191

0.9723

0.9702

9.0159

Test

0.1764

0.1241

0.9717

0.9599

1.1255

Decision

Tree

Train

0.3523

0.2884

0.8770

0.8677

38.0824

Test

0.4239

0.3322

0.7948

0.7093

-27.4147

Random

Forest

Train

0.1336

0.0938

0.9823

0.9810

17.5722

Test

0.2481

0.1907

0.9297

0.9004

-29.4441

Random

Forest

(Tuned)

Train

0.1383

0.1002

0.9810

0.9796

17.6054

Test

0.2491

0.1945

0.9292

0.8997

-32.1883

Ada

Boost

Train

0.2281

0.1887

0.9484

0.9445

29.3324

Test

0.2935

0.2300

0.9017

0.8607

-33.2171

Gradien

t

Boosting

Train

0.1177

0.0660

0.9863

0.9852

15.7419

Test

0.2186

0.1615

0.9454

0.9227

-24.6019

XG

Boost

Train

0.1104

0.0441

0.9879

0.9870

14.7957

Test

0.2697

0.1939

0.9169

0.8823

-31.9448

While MAE or MAPE plays a crucial role in

identifying acceptable regressor models, it is equally

essential to have a more comprehensive evaluation

performed to ascertain the most dependable

regressor model, [19]. By integrating the quantile

loss function, prediction uncertainty can be

adequately considered, and deeper insights can be

gained into the performance of the models.

Moreover, with this comprehensive approach,

reliable predictions can be made on unfamiliar

datasets.

During the crucial final stage, the main

objective was to quantify the prediction uncertainty

associated with the model. The Quantile loss

function was utilized to accomplish this and provide

valuable insights into the degree of uncertainty

surrounding the point estimation. The outcomes for

the 0.1, 0.5, and 0.9 quantiles are illustrated in

Figure 6, offering a comprehensive view of the

models' performance concerning uncertainty. The

plot analysis revealed that the linear Regression

model exhibited the lowest quantile loss, signifying

its superior capability to capture prediction

uncertainty effectively.

Developing a Python code to implement the

quantile linear regressor model also enabled the

computation of interval estimates for the target

variable. The results of these interval estimates are

presented in Table 8. These results provide insights

into the efficacy of the linear regression model in

capturing the possible prevalence of atherosclerotic

conditions in ICA. Notably, the WSS values

highlighted in Table 8 were not in the range

associated with healthy cases, as detailed in section

3.5. However, the identification of a similar

condition in the ECA proved to be less successful.

5 Results and Discussions for ICA

and ECA

Figure 7 (Appendix) presents the flow chart for data

analysis generated for ICA and ECA. In contrast to

the case of CCA, where only the fluid and blood

vessel properties act as exposures, here we assumed

that the features and target variables in CCA and

ECA are the features of ICA. Similarly, the features

and target variables of CCA and ICA constitute the

feature set for ECA. This assumption is considered

appropriate for the present problem, as any stenotic

conditions in one of the segments impact the blood

flow characteristics in the other carotid artery

segments.

As a consequence of this assumption,

multicollinearity was observed among the feature

sets of both ICA and ECA. Consequently,

regularized regression models, including Ridge,

Lasso, and Elasticnet, were applied to the dataset.

Additionally, the dataset was subjected to a

Decision tree, Random forest, and boosting

algorithms to determine the most suitable predictor

model for assessing the WSS in both bifurcated

segments. This process also aided in identifying the

significant features of the target variables WSS_ICA

and WSS_ECA. It was found that WSS_CCA and

Vel_CCA had the most significant impact on

WSS_ICA, whereas Vel_CCA and blood pressure in

CCA influenced WSS_ECA, see Table 9(a) & Table

9(b).

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Fig. 6: Plot of Quantile Loss function for the ML developed to predict

Table 8. Quantile predictions of WSS_CCA

Exposures

Target Variable (WSS_CCA)

Vis

D

InPr

Out1vel

Out2vel

Actual

0.1

Quantile

0.5

Quantile

0.9

Quantile

0.004

1060

100

0.1202

0.153

0.679323

0.660288

0.684708

0.845187

0.004

1060

100

0.0717

0.153

0.638733

0.490863

0.476632

0.638733

0.0045

1060

100

0.0717

0.153

0.576934

0.571987

0.765744

0.0035

1060

100

0.1931

0.153

0.853178

0.829056

0.902331

1.028715

0.0045

1060

120

0.2414

0.046

1.128829

1.000332

1.128829

1.18801

0.005

1060

120

0.2414

0.153

1.61143

1.28635

1.458091

1.61143

0.0035

1060

100

0.2414

0.077

0.85576

0.943403

1.023779

0.0035

1060

100

0.0717

0.153

0.454918

0.404791

0.381278

0.511722

0.0045

1075

120

0.1202

0.077

0.640726

0.004

1060

130

0.0717

0.046

0.336452

0.004

1060

100

0.1931

0.122

0.874334

0.857199

0.929918

1.06985

0.004

1060

120

0.2414

0.046

1.033475

0.91426

1.033475

1.061

0.005

1060

120

0.0717

0.153

0.693363

0.729826

0.88884

0.0045

1060

120

0.0717

0.153

0.634472

0.607292

0.634472

0.761829

0.004

1060

100

0.1202

0.077

0.518568

0.51827

0.518568

0.634654

0.004

1060

100

0.2414

0.046

1.064914

0.883903

0.97099

1.064914

0.004

1060

120

0.2414

0.122

1.271533

1.056278

1.199615

1.271533

0.0045

1060

100

0.2414

0.046

0.969974

0.969975

1.066344

1.191925

Table 9(a). Important Features for WSS_ICA Identified by Gradient Boosting Model

Feature Importance (ICA)

0

Out1vel

0.834239

1

WSS_CCA

0.064805

2

Vel_CCA

0.058523

3

Vel_ECA

0.020121

4

log WSS_ECA

0.011729

5

Out2vel

0.008123

6

Vis

0.001929

7

InPr

0.000529

8

D

0

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The reliable predictor model for all the target

variables was identified in the subsequent steps.

The quantile loss function was plotted for all the

ML models developed to predict the target

variables. The plot of the quantile loss function for

the models developed to predict WSS in ICA is

shown in Figure 8 (Appendix), while Figure 9

(Appendix) displays the same for WSS in ECA.

Table 9(b). Important Features for WSS_ECA

Identified by Gradient Boosting Model

Feature Importance (ECA)

0

InPr

0.500923

1

Out2vel

0.292016

2

Vel_CCA

0.142237

3

log

WSS_ICA

0.02052

4

Vel_ICA

0.019159

5

WSS_CCA

0.014632

6

Vis

0.008388

7

Out1vel

0.002126

8

D

0

These graphs indicate that the most reliable

predictor model is the Gradient Boosting algorithm.

Consequently, the quantile intervals were computed

using the Gradient Boosting Quantile algorithm for

WSS_ICA and WSS_ECA, and they are presented

in Table 10 (Appendix) and Table 11 (Appendix).

The importance of recognizing these

observations relies on a numerical model of an

idealized carotid artery, formulated with specific

assumptions detailed in this paper's methodology

section, which cannot be overstated. Consequently,

additional investigation and validation are

imperative to corroborate the findings.

5.1 Limitations of this Study

The model proposed in this paper is specifically

designed for an idealized carotid artery. Several

key assumptions have been considered to ensure

numerical tractability and practical feasibility.

These assumptions include treating the blood as a

Newtonian fluid, assuming the arterial segments to

be circular elastic pipes with a constant radius, and

considering the blood flow steady. These

simplifications enabled efficient computational

handling and facilitated reasonably reliable insights

into the blood flow dynamics within the carotid

artery.

6 Conclusions

This study focused on developing a numerical

model to simulate blood flow in the carotid artery

under different medical conditions associated with

stenosis or atherosclerosis. ML models were

tailored to the simulated data and used to predict

blood flow characteristics in the artery. Utilizing

the quantile loss function designed to assess

prediction uncertainty, the Linear regression model

better estimates wall shear stress within the CCA.

On the other hand, when considering the ICA and

ECA, the gradient boosting algorithm emerged as

the most effective model for predicting wall shear

stress.

Through this study, we identified the blood

velocity and the WSS in CCA as the most

important features affecting the wall shear stress in

ICA, while the blood velocity and the blood

pressure affected the WSS in ECA the most.

Results also showed that the quantile linear

regression model could effectively detect

atherosclerotic conditions in ICA for a given set of

exposures. But then, such a condition in ECA was

not successfully identified. However, it is important

to recognize certain limitations in this study, as

specific assumptions were made to ensure

numerical tractability for the problem. Therefore,

the future scope of this work involves extending it

to a patient-specific model and utilizing advanced

data-analytic tools like ANN to enhance its

reliability as a reference source for the medical

community. The model's accuracy and applicability

would significantly improve for personalized

medical assessments and interventions by

incorporating patient-specific data.

Acknowledgment:

The authors thank their respective institutions for

providing the necessary facilities for this work.

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Mohamed Siraj, T. S. L. Radhika

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APPENDIX

Fig. 7: Flow Chart

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Fig. 8: Plot of Quantile Loss function for the ML developed to predict WSS_ICA

Fig. 9: Plot of Quantile Loss function for the ML developed to predict WSS_ECA

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Table 10. Quantile predictions of WSS_ICA

Exposures

Target Variable (WSS_ICA)

Vis

D

InPr

Out1vel

Out2vel

Vel_CCA

Vel_ECA

WSS_CCA

WSS_ECA

Actual

0.1

Quantile

0.5

Quantile

0.9

Quantile

0.0045

1060

100

0.1202

0.077

0.074543

0.088532

0.580231

0.662184

0.226674

0.160194

0.233145

0.280177

0.0045

1075

120

0.1202

0.077

0.075314

0.144331

0.640726

2.1063

0.254805

0.174545

0.228197

0.285173

0.0035

1060

100

0.0717

0.153

0.108956

0.179861

0.454918

2.235889

0.051677

0.06892

0.067079

0.229342

0.0045

1060

100

0.2414

0.077

0.108724

0.219893

1.050035

1.060807

0.600673

0.443052

0.664104

0.644709

0.004

1060

130

0.1931

0.122

0.122154

0.231282

0.949304

5.115034

0.402931

0.344504

0.378675

0.372254

0.0045

1075

120

0.2414

0.122

0.134627

0.267775

1.388943

4.024413

0.536366

0.471021

0.533134

0.553694

0.0045

1060

120

0.2414

0.153

0.15496

0.29696

1.492572

4.715618

0.464327

0.410518

0.465193

0.491097

0.0045

1060

100

0.1202

0.153

0.12139

0.211839

0.756643

2.518693

0.151853

0.130723

0.16931

0.229326

0.0035

1060

100

0.2414

0.046

0.091876

0.193223

0.786211

0.537146

0.557896

0.289502

0.501484

0.508154

0.005

1060

120

0.0717

0.046

0.043588

0.084415

0.391744

1.232727

0.178367

0.144455

0.161412

0.272236

0.0035

1060

100

0.2414

0.153

0.162901

0.294481

1.035599

1.815765

0.348964

0.344433

0.389457

0.479634

0.005

1060

120

0.1931

0.153

0.141493

0.262393

1.328196

4.738597

0.37826

0.323486

0.390723

0.332452

0.0045

1060

120

0.0717

0.046

0.043801

0.084641

0.355445

1.133036

0.159392

0.14206

0.158376

0.291015

0.004

1060

120

0.1931

0.122

0.12296

0.235956

1.029243

3.286256

0.337066

0.355958

0.361726

0.393369

0.005

1060

120

0.1202

0.077

0.074938

0.143758

0.698544

2.289476

0.286675

0.174522

0.230012

0.271204

0.0045

1060

120

0.1202

0.077

0.07527

0.144261

0.639244

2.1008

0.255183

0.16861

0.223877

0.284382

0.004

1060

100

0.2414

0.153

0.155719

0.290189

1.369311

2.47078

0.491992

0.410518

0.433442

0.485729

0.004

1060

100

0.1202

0.153

0.12307

0.2124

0.679323

2.457015

0.131503

0.131523

0.140757

0.229243

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Table 11. Quantile predictions of WSS_ECA

Exposures

Target Variable (WSS_ECA)

Vis

D

InPr

Out1vel

Out2vel

Vel_CCA

Vel_ICA

WSS_CCA

WSS_ICA

Actual

0.1 Quantile

0.5 Quantile

0.9 Quantile

0.0045

1060

100

0.1202

0.077

0.074543

0.149824

0.580231

1.742598

0.081353

0.119975

0.128186

0.29356

0.0045

1075

120

0.1202

0.077

0.075314

0.146314

0.640726

1.946125

0.277067

0.260029

0.294618

0.37747

0.0035

1060

100

0.0717

0.153

0.108956

0.14498

0.454918

0.431412

0.295152

0.247443

0.278725

0.369031

0.0045

1060

100

0.2414

0.077

0.108724

0.251491

1.050035

4.373594

0.134003

0.126406

0.134317

0.318121

0.004

1060

130

0.1931

0.122

0.122154

0.235182

0.949304

2.999843

0.709028

0.379176

0.712675

0.698942

0.0045

1075

120

0.2414

0.122

0.134627

0.267429

1.388943

3.930097

0.550008

0.410095

0.54931

0.548763

0.0045

1060

120

0.2414

0.153

0.15496

0.282304

1.492572

3.429711

0.650533

0.435107

0.645038

0.655274

0.0045

1060

100

0.1202

0.153

0.12139

0.187694

0.756643

1.193764

0.33483

0.252674

0.342368

0.358923

0.0035

1060

100

0.2414

0.045

0.091876

0.232939

0.786211

4.078895

0.065181

0.083116

0.078782

0.294

0.005

1060

120

0.0717

0.045

0.043588

0.090326

0.391744

1.389675

0.157107

0.211608

0.16612

0.410847

0.0035

1060

100

0.2414

0.153

0.162901

0.281886

1.035599

2.618967

0.236766

0.209832

0.235724

0.34013

0.005

1060

120

0.1931

0.153

0.141493

0.24371

1.328196

2.826104

0.653889

0.438416

0.613141

0.566112

0.0045

1060

120

0.0717

0.045

0.043801

0.090234

0.355445

1.249647

0.143685

0.134091

0.142067

0.37729

0.004

1060

120

0.1931

0.122

0.12296

0.227379

1.029243

2.534565

0.443784

0.379176

0.48513

0.496302

0.005

1060

120

0.1202

0.077

0.074938

0.146681

0.698544

2.175203

0.302649

0.260029

0.294055

0.411042

0.0045

1060

120

0.1202

0.077

0.07527

0.14636

0.639244

1.948855

0.276301

0.260029

0.294618

0.37747

0.004

1060

100

0.2414

0.153

0.155719

0.289628

1.369311

3.622354

0.328089

0.267149

0.325061

0.496317

0.004

1060

100

0.1202

0.153

0.12307

0.187177

0.679323

1.042109

0.326153

0.260591

0.311447

0.358339

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2023.20.16

T. Raja Rani, Woshan Srimal,

Abdullah Al Shibli, Nooh Zayid Suwaid Al Bakri,

Mohamed Siraj, T. S. L. Radhika

E-ISSN: 2224-2902

169

Volume 20, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present

research at all stages, from formulating the problem

to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors declare no conflicts of interest.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2023.20.16

T. Raja Rani, Woshan Srimal,

Abdullah Al Shibli, Nooh Zayid Suwaid Al Bakri,

Mohamed Siraj, T. S. L. Radhika

E-ISSN: 2224-2902

170

Volume 20, 2023