On building machine learning models for medical dataset with correlated features

2.1 Research question

This study embodies a translational approach, aiming to forecast the hemodynamic traits within the human femoral and popliteal arteries. Our investigation is a quantitative and empirical analysis employing computer-based simulations. The primary research inquiry can be articulated as follows:

Can ML models trained on simulated data adeptly predict the blood flow characteristics within the femoral and popliteal arteries, considering the specified flow conditions?

2.2 Research objectives

The objectives of this research are as follows:

1. To develop and validate a numerical model for the human (idealized) femoral and popliteal arteries.

   1. Simulate the model under diverse flow conditions.

   2. Compute appropriate physical quantities and validate the results.

2. To estimate WSS and other physical quantities using ML algorithms.

   1. Develop ML models that best fit the data.

   2. Compute the metrics on the train and test data and identify the most accurate ML prediction model.

   3. Compute the quantile loss function to identify the most reliable predictor model.

3. To use the ML model to predict the physical quantities, WSS, and average blood velocity under the given flow conditions.

2.3 Feature set and target variables

Fluid (blood)-related parameters, such as density and viscosity, flow-related parameters, such as the inlet blood velocity, and material parameters, such as the density, Poisson ratio, and Young’s modulus for the artery, constitute feature set. The target variables are the WSS, average velocity, and average pressure.

3.1 Step 1: Prepare data collection instrument

The femoral artery is one of the large arteries in the human body situated in the thigh and is the leading blood supplier to it and the legs. It includes CFA and SFA. This study developed a numerical model for the blood flow in the femoral and popliteal arteries and applied the proposed concept of varying the vessel properties and the blood characteristics in the arterial network to derive a realistic model.

The key features of the blood circulatory system, namely, the blood, have been modeled as a Newtonian fluid in the first segment (CFA) and Newtonian or non-Newtonian in the second and third component pipes. The arteries are elastic circular pipes with variable material and geometric properties. The model has been simulated with the help of clinical data on the anatomy of the arterial network and the physiological behavior of blood flow through it. COMSOL Multiphysics software is used to develop a 2D model for this blood flow problem in an idealized arterial network consisting of the CFA, SFA, and PA, as shown in Figure 1. We considered the time-independent momentum and continuity equations for incompressible fluids and implemented it using the laminar flow interface under fully developed flow condition. The values of fluid-related parameters, the viscosity and density, are based on the available data on the human blood. Parameters describing the elastic behavior of the network are taken from the anatomical data on the human artery. The P1-P1 (linear) finite element method was applied to discretize both velocity and pressure fields. A grid independence test was conducted, and the findings are outlined in Table 1. From these findings, we concluded that both a normal and an extremely fine mesh are well suited for the study. A normal mesh composed of triangular (7,240) and quads (1,678) elements corresponding to 17,191 degrees of freedom has been used to determine the velocity and pressure variables. The mesh parameters are presented in Table 2.

Figure 1

Geometry of the artery in COMSOL.

The resulting nonlinear flow problem is solved using Newton’s method. We used no-slip conditions on the boundary walls with no backflow, and the PARDISO solver was used to solve the system.

3.2 Step 2: Data collection (by performing simulations)

Table 3 shows the data on the various input parameters to the COMSOL model builder for simulating the blood flow in the human arterial network [2,5,6,8,10,12–14,16,18,19,23,24, 27,29,36].

As blood is modeled as a Newtonian fluid in the first component, the model parameters, density and viscosity, assume the values of human blood. The parameters related to the non-Newtonian model, the yield stress, the model parameter in the Casson fluid model, and flow consistency and flow index in the power-law fluid model are assigned values reported in the literature. The artery is described as an elastic circular pipe, and the parameters describing its elastic nature, density, Young’s modulus, and Poisson ratio are given the values related to the human arteries, as shown in Table 3. The arterial dimensions, length, and diameter are as per the data reported in the literature. Furthermore, the inlet velocities to make the blood flow happen are taken in the range specified in Tables 3 and 4. For the simulations, the inlet velocity has taken 0.08, 0.09, and 0.1 m/s.

Before generating the medical dataset from the developed numerical model, we identify the appropriate fluid models for the blood in SFA and PA.

3.3 Step 3: Data management

Data from the numerical model have been collected as per the format shown in Table 6. For each set of features, the target variables are evaluated using the COMSOL software and noted in the designated columns in the Table 6.

Features

D1: blood density in the CFA ( kg/m 3 )

V1: blood viscosity in the CFA (Pa s)

D2: blood density in the SFA ( kg/m 3 )

V2: blood viscosity in the SFA (Pa.s)

Y: blood yield stress in the PA ( N ∕ m 2 )

M: blood model parameter PA (s)

INVEL: inlet blood velocity at the CFA entrance (m/s)

Target variables

WSS_CFA: WSS in CFA (Pa)

WSS_SFA: WSS in SFA (Pa)

WSS_PA: WSS in PA (Pa)

VEL_CFA: average blood velocity in CFA (m/s)

VEL_SFA: average blood velocity in SFA (m/s)

VEL_PA: average blood velocity in PA (m/s)

Pre_CFA: blood pressure in CFA (Pa)

Pre_SFA: blood pressure in SFA (Pa)

Pre_PA: blood pressure in PA (Pa)

3.4 Step 4: Data analysis

The next step is to compute the statistics of features to obtain some basic information on the features. Table 7 details the same.

4.1 Flow variables in the first and second component pipes

We now proceed to conduct an in-depth analysis of the simulated dataset employing ML algorithms. The sequential methodology is visually depicted in Figure 3.

Figure 3

Flow chart.

This section furnishes an intricate portrayal of the data analysis procedure: an initial univariate analysis was conducted to delve into the characteristics of the dataset, encompassing an assessment of the range and central tendencies. Furthermore, the descriptive statistics for the features were examined to identify any potential outliers. Notably, it was discerned that the maximum and 75th percentile values of the features exhibited negligible differences, implying the absence of outliers within the dataset. Following this, a univariate analysis was executed on the target variables, employing bar charts, which revealed a lack of skewness in the data. Subsequently, our focus shifted toward the subsequent stage of the analysis, a multivariate examination. This phase explored multiple dependent variables or features influencing a singular outcome or target variable.

The degree of correlation among features was assessed by computing the variance inflation factor (VIF) score. It became evident that these features are not correlated, as each feature’s VIF score is not significantly different from unity, which is shown in Table 8. Subsequently, the initial model was constructed: linear regression model incorporating D1, V1, and INVEL are the features and WSS_CFA as the target variable; similarly, D1, V1, D2, V2, and INVEL are the features for the target variable WSS_SFA; and D1, V1, D2, V2, and VEL_CFA are the features for VEL_SFA. The evaluation metrics were computed for the training and testing datasets, including mean absolute error (MAE), root mean square error (RMSE), R ², and adjusted R ² scores. Interestingly, it was observed that the RMSE and MAE values nearly mirrored each other for both the training and testing datasets, as depicted in Table 9. This convergence suggested a strong alignment of the model with the dataset.

Additionally, we delved into nonlinear models, exploring polynomial regression, support vector regressor (SVR), fine-tuned SVR, Random Forest regressor, fine-tuned Random Forest regressor, Adaboost, Gradient boosting, and XGBoost. This thorough exploration aimed to gain deeper insights into the data.

Examining the MAE values detailed in Table 10 highlights that the XGBoost regressor algorithms exhibit remarkably reduced MAE values for WSS_CFA, WSS_SFA, and VEL_SFA, when applied to our dataset. Although the MAE metric is valuable for discerning viable regression models, it is imperative to incorporate the quantile loss function for a more comprehensive assessment, facilitating the identification of the most dependable regression models. This inclusive approach enables the incorporation of prediction uncertainty, fostering a more robust comprehension of the models’ efficacy. Consequently, the final phase of our analysis centers on quantifying the prediction uncertainty inherent in these models. Leveraging the quantile loss function, we explored the extent of uncertainty encompassing point predictions. The outcomes for quantiles 0.1, 0.5, and 0.9 are visually represented in Figures 4, 5, 6, affording a holistic perspective on the models’ performance for uncertainty. Scrutinizing these graphs uncovers that the SVR-tuned model stands out with minimal quantile loss for WSS_CFA, while XGBoost and polynomial (degree = 2) regressor defines the minimum quantile loss for WSS_SFA and VEL_SFA, respectively, underscoring their capacity to predict these target variables for unknown feature datasets accurately. A Python codebase was developed to operationalize these models enabling the derivation of interval estimations for the target variables. The outcomes of these interval estimations are laid out in Tables 11, 12, 13.

Figure 4

Plot of quantile loss per quantile for WSS_CFA.

Figure 5

Plot of quantile loss per quantile for WSS_SFA.

Figure 6

Plot of quantile loss per quantile for VEL_SFA.

4.2 Flow variables in the third component

This section dives into the data analysis related to the third component pipe, namely, the PA. It is important to remember that the features and target variables from the first and second components collectively compose the feature set of the third component. D1, V1, D2, V2,Y, M, VEL_CFA, and VEL_SFA are the features for the target variables WSS_PA and VEL_PA. To conduct our analysis, we followed the systematic procedure outlined in Figure 7. This entailed performing both univariate and multivariate analyses on the set of variables. To gauge the extent of correlation within the feature set, we utilized the VIF score, the results of which are outlined in Table 14.

Figure 7

Flow chart.

The data within the table reveal correlations within the feature set, and hence, regularized regression models were built. The metrics computed are presented in Table 15. We see that both ridge regression and tuned ridge regression are best suited for predicting WSS_PA and VEL_PA.

However, we applied various algorithms, including SVR, SVR-tuned, Random Forest, Adaboost, Gradient boosting, and XGBoost. The corresponding metrics for these algorithms are given in Table 16.

To identify the most dependable predictor model, we graphed the quantile loss for the target variables WSS_PA and VEL_PA across quantiles. These graphical representations are depicted in Figures 8 and 9.

Figure 8

Plot of quantile loss per quantile for WSS_PA.

Figure 9

Plot of quantile loss per quantile for VEL_PA.

These quantile plots indicate that Gradient boosting regression yields reliable predictions for WSS_PA and VEL_PA. Detailed information regarding the quantile intervals is given in Tables 17 and 18.