Biomathematical Modeling in Clinical Intelligent Diagnosis and Treatment

**HemExPred**

The endeavour, namely the Predictive Modeling of Hematoma Expansion Event Occurrence (HemExPred), aims to discern and forecast the likelihood of a hematoma expansion event transpiring. Drawing upon a comprehensive array of patient-specific factors, encompassing personal and disease history, time of onset, pertinent treatments, and imaging findings, HemExPred ingeniously calculates the probability of a hematoma expansion event (HEE) manifesting within 48 hours. In this research, we employ many feature extraction methodologies, including the techniques of principal component analysis, random forest, and lasso regression. By leveraging the potential of these methodologies, we meticulously extract the ten most pivotal features elucidated by each method, skilfully concatenating and intersecting them to form a robust feature set. Subsequently, these discerned features serve as the foundation for training support vector machine, random forest, and logistic regression models, which in turn bestow the ability to predict the probability of a hematoma expansion event occurring within the specified 48-hour window.

Let us allocate n follow-up visits for each patient, n∈[1,∞] allowing for any positive integer value. To determine the time interval, t, between each patient's follow-up examination and the onset of illness, we shall employ the following mathematical formula:

$$t = t_m + t_i - t_0 \quad i \in [1,n]$$

Here, $t_m$ represents the time interval between the onset and examination at the first visit, as indicated in Table 1. $t_i$ corresponds to the time point of examination at the i follow-up visit for the specific patient, while $t_0$ denotes the time point of examination at the first visit for that particular patient.

We have established specific criteria to define the event's occurrence, focusing on whether the hematoma volume exhibited notable changes within 48 hours of the patient's onset. To meet the criteria, any one of the following two conditions must be satisfied: (1) an absolute increase in volume greater than or equal to 6 ml, or (2) a relative increase in volume greater than or equal to 33%. To determine the occurrence of the event, we employ the following formula:

$$F \left( d_0, d_i \right)_{HEE} = \begin{cases} 1, & (d_i - d_0) > = 6 \text{ or } (d_i - d_0) \div d_0 > = 0.33 \\ 0, & \text{else} \end{cases}$$

Here, $d_0$ represents the volume of the clot at the first examination, while $d_i$ corresponds to the volume of the clot at the examination during the i follow-up visit. By evaluating the outcome of this formula, we can effectively determine whether the event of interest has taken place or not.

The illustrious flowchart of the HemExPred model, as depicted in Figure 2, shall serve as a visual representation of the steps as mentioned above. This diagram shall provide a comprehensive overview, guiding one through our model's intricate sequence of operations. The Random Forest algorithm, highlighted in references Schonlau M, et al. [27] and Wager S, et al. [28], is an effective classification tool in predictive modeling. It uses ensemble learning to create and merge multiple decision trees, providing a clear visual process that details the construction and synthesis of these trees for accurate predictions.

Click to enlarge

EdemaVolReg

This model, also known as EdemaVolReg, designed to analyse changes in edema volume over time, utilizes fitted regression models to construct regression curves that capture the dynamics of edema volume. By exploring the relationship between hematoma volume, volume of edema, and treatment, the researchers seek to unravel the intricate interplay between these factors.

To accomplish this, the researchers embarked upon a series of steps. First, we diligently performed data cleaning, ensuring the integrity and reliability of the dataset. Feature matrices were constructed, serving as the foundation for training linear regression and polynomial regression models. The researchers conducted experiments using linear and polynomial regression models, aiming to identify the most suitable model and optimal parameters. Through rigorous analysis of the experimental results, we discerned the most appropriate model and parameter configurations, ensuring the accuracy and efficacy of their regression modeling endeavors.

Linear regression, an elementary statistical technique, serves as a foundational pillar in the realm of data analysis. It is expertly wielded to explore the intricate interplay between two variables, wherein one is designated as the dependent variable, often symbolized as y, while the other assumes the role of the independent variable, denoted as x.

$$ y = \beta_ {0} + \beta_ {1} x _ {1} + \beta_ {2} x _ {2} + \dots + \beta_ {p} x _ {p} + \dot {o} \tag {3} $$ The pursuit of linear regression lies in the quest for an optimal straight line, one that captures the essence of the relationship between these variables. Its purpose is to minimize the sum of distances, akin to errors, between the observed data points and the sublime trajectory of this line.

Polynomial regression, an extension of linear regression, serves as a formidable tool for encapsulating nonlinear associations present within datasets. Its fundamental premise involves augmenting the model by incorporating higher orders of the original features as novel attributes. Consequently, a meticulously tailored curve is fitted to the data, surpassing the limitations of a mere straight line.

$$ y = \beta_ {0} + \beta_ {1} x _ {1} + \beta_ {2} x _ {2} ^ {2} + \dots + \beta_ {d} x _ {d} ^ {d} \tag {4} $$ It is imperative to embark upon a comprehensive exploration to delve into the intricate nuances of the evolutionary trajectory of edema volume in patients over time and its interindividual variations. In this regard, an initial prerequisite entails characterizing individual patients and unravelling potential divergent evolution paths. This scholarly endeavour undertakes a meticulous selection of pivotal patient features, subsequently employing a constellation of clustering algorithms, including K-means, hierarchical clustering, and Gaussian mixture models, to discern distinct patient subgroups.

With a dedicated focus on each patient subgroup, this study takes a stride forward by employing linear and polynomial regression models. These models fit the edema volume versus time curves for each subset. By harnessing the power of these models, explicit insights into the temporal trends of edema volume within each subgroup can be gleaned. The intricate procedural details are meticulously elucidated in Figure 3 below for comprehensive comprehension.

Click to enlarge

PrognosisPred

The Patient Prognosis Prediction Model (PrognosisPred) is a formidable tool to predict the outcome for patients afflicted with hemorrhagic stroke, drawing upon many influential factors. Following a patient’s initial consultation with a medical practitioner, wherein subsequent follow-up visits may occur, we employ two prediction schemes and construct two distinct models.

Firstly, PrognosisPredI considers solely the information gleaned during the patient’s inaugural visit to the doctor. This model focuses exclusively on the data obtained during this initial encounter to create a predictive framework. On the other hand, PrognosisPredII adopts a more comprehensive approach by amalgamating information derived from both patients’ visits. By incorporating and organizing the data from these two consultations, novel feature inputs are generated and subsequently employed for prediction. This holistic approach enables a more robust and nuanced prognosis evaluation.

These prediction schemes’ intricate mechanisms and intricacies are meticulously elucidated in Figure 4 and Figure 5, elegantly presented below for comprehensive comprehension.

ExtraTreesRegressor Reza M, et al. [29] is an exemplary ensemble learning method rooted in decision trees. It represents a specialized implementation of the ExtraTrees algorithm tailored for addressing regression tasks. The foundational workflow of ExtraTreesRegressor is elegantly delineated below. Before constructing the ExtraTreesRegressor model, it becomes imperative to establish a random subset to ascertain the number of decision trees (n_estimators) and the maximum depth (max_depth) of each tree. Subsequently, for every tree within the ensemble, a random subset is meticulously crafted by employing a process of random sampling with replacement from the original training set.

Within the realm of random feature selection, utilizing a random subset necessitates the random selection of a subset of features for training each decision tree. This stochastic feature selection process mitigates bias and bolsters the model’s diversity. Typically, the number of elements employed for each decision tree is regulated by the hyper parameter known as max_features, which can be defined as a fixed number of features or expressed as a percentage of the total feature set.

Click to enlarge

Click to enlarge

The decision trees are then forged based on the aforementioned random subset and random feature selection techniques, with each tree being trained on its respective random subset. During training, the decision tree partitions the feature space into distinct regions, guided by a designated splitting criterion (e.g., most minor square deviation). This partitioning process aims to minimize the variability of the prediction target, thereby ensuring optimal predictive performance.

$$H(X) = -\sum \log[P(x)]H(x)$$ (5)

Where denotes the entropy $P(x)$: denotes the probability that event $x$ occurs.

Harmonizing the predictions obtained from multiple decision trees is an integral step within the realm of ExtraTreesRegressor. Upon the culmination of training for each decision tree, the ExtraTreesRegressor algorithm artfully amalgamates their individual prediction outcomes for regression quandaries. This amalgamation takes the form of an average, where the predictions from each decision tree harmoniously converge to yield the ultimate prediction outcome. In the context of regression conundrums, the final prediction result materializes through the calculated average of the prediction results from each decision tree. This process can be succinctly encapsulated by the following formula:

$$y_{\text{pred}} = \frac{1}{n} \times \sum_{i=1}^{n} y_i$$ (6)

Where $n$ denotes the number of decision trees and $y_i$ denotes the prediction result of each decision tree.

The fundamental tenet that underlies ExtraTreesRegressor lies in imbuing each tree with a distinctive essence. Achieving this is accomplished through constructing multiple decision trees, while ingeniously introducing an element of randomness during the training process. This deliberate injection of randomness serves a twofold purpose: mitigating variance and elevating the model’s resilience and steadfastness.

**Datasets**

The dataset emanates from the rich troves of data from Question E of the esteemed 20th China Graduate Student Mathematical Modeling Competition, fondly known as the "Huawei Cup." This dataset comprises five distinct tables, each offering a unique facet of information. Table 1 stands as a repository of patients’ comprehensive profiles, encompassing their details, medical history, and treatment regimens. Table 2, on the other hand, encapsulates the vivid realm of patient imaging information, explicitly of hematoma and edema volumes and their corresponding locations. Lastly, Table 3 unveils the intricate nuances of shape and grayscale distribution concerning the patients’ imaging information concerning hematoma and edema, thereby presenting a comprehensive compendium of insights for each image.

**Pre-Processing**

The tabular data, harnessed by invoking the Pandas library in Python, underwent a thorough analysis, revealing a pristine state untainted by missing values, outliers, or multicollinearity. Thus, it remained unadulterated and unaltered in its original form. In order to delve into the inquiry of whether follow-up information exerts an influence on the model’s predictions, two distinct data pre-processing techniques were employed to explore this profound query.

The first kind of data table, aptly christened “PreOne,” entailed constructing a predictive model devoid of the patient’s follow-up information. In this approach, the table was systematically segmented, one entry at a time, based on the ascending order of each image’s running number. Consequently, a tapestry of insights emerged, enabling an investigation into the interplay between the patient’s initial examination and the subsequent follow-up visits.

The second kind of data table, “PreTwo,” sought to establish a cohesive connection between the patient’s initial examination and the subsequent follow-up visits. Recognizing that merely correlating the first examination with the first follow-up visit would not optimally utilize the available data, this approach skilfully orchestrated a chronological correlation of the patient’s two examinations, regardless of whether they were the initial encounters. The ultimate objective was to discern the intricate correlation and interdependence between the two examinations.

Notably, the variable denoting sex underwent a distinctive encoding scheme, adopting a one-hot encoding methodology. Specifically, male and female genders were denoted by the values 0 and 1, respectively. Furthermore, the column of blood pressure was partitioned into two distinct columns, meticulously capturing the highest and lowest blood pressure readings. These columns were aptly designated as “Blood Pressure High” and “Blood Pressure Low,” respectively, enhancing the granularity and comprehensibility of the dataset.

**Correlation Analysis**

Pearson’s correlation coefficient, a statistical measure elucidated [30], serves as a metric to gauge the linear interplay between two numerical variables. Spanning a scale that ranges from -1 to 1, this coefficient manifests as a numerical reflection of the association between the variables under scrutiny.

The Spearman rank correlation method endeavors to evaluate the degree of association between two variables by considering the ordinal ranking of the data [31]. Unlike the unprocessed numerical values, this statistical approach duly acknowledges the significance of the variables’ rank order.

Kendall’s correlation Lin J, et al. [32], a nonparametric statistical technique, serves as a valuable tool for evaluating the correlation between two ordered data sets. The values yielded by Kendall’s correlation span the range of -1 to 1, where 1 signifies a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 denotes the absence of any discernible correlation.

**Feature Extraction**

Principal Component Analysis (PCA) is a statistical methodology that condenses and streams data, there by effectuating its transformation into a novel set of uncorrelated variables known as principal components. These main components epitomize the directions that exhibit the highest variance within the initial dataset.

Random Forest-based feature extraction methodology primarily relies on the calculation of the esteemed Gini Index to determine the top ten features in terms of their importance for subsequent model training and prediction.

The loss function for Lasso regression is defined as follows:

$$L = L_{\text{quare}} + L_{\text{L1}}.$$

Where $L_{\text{quare}}$:

$$L_{\text{quare}} = \frac{1}{2n} \sum_{i=1}^{n} (y - X \times \beta)^2$$

$$L_{\text{L1}} = \alpha \times \sum |\beta|$$

In this formula, $n$ is the number of samples, $y$ is the target variable, $X$ is the identity matrix, $\beta$ is the characteristic coefficients, and $\alpha$ is the regularization parameter.

In the realm of Lasso regression and the random forest-based feature importance ranking method, it has come to light that the selected features do not invariably align within the same genre. Consequently, this paper has employed two distinctive feature extraction techniques. The first approach entails concatenating the elements acquired through both methods, thereby amassing a comprehensive set of features denoted as “feature union.” The second approach involves extracting ten features independently obtained from each method and subsequently identifying their intersection. This intersection of features is aptly referred to as “feature intersection.” By adopting these two feature extraction methodologies, this study aims to explore the potential synergies that arise from their combined utilization.

**Performance Evaluation Metrics**

The metric known as $R^2$ or the coefficient of determination, serves as a powerful indicator of the regression model’s proficiency in elucidating the fluctuations observed within the dependent variable. Spanning a range from 0 to 1, the $R^2$ value assumes its true potential in quantifying the model’s capacity to explicate the variability inherent in the said dependent variable. A higher $R^2$ value signals a superior ability of the model to account for and elucidate the intricate patterns and fluctuations within the dependent variable, thus enhancing our understanding of the underlying phenomena.

$$R^2 = 1 - \frac{\sum (y - \hat{y})^2}{\sum (y - \bar{y})^2}$$

Where $y$ is the true value, $\hat{y}$ is the predicted value, and $\bar{y}$ is the average of the true values.

$R_1^2$ is a correction to $R^2$ that takes into account the number of independent variables used in the model. Adjustment $R_1^2$ reflects the extent to which the independent variables explain the dependent variable, with values ranging from negative infinity to 1. $R_1^2$ larger value means a better fit.

$$R_1^2 = 1 - \left(1 - R^2\right) \frac{n - 1}{n - p - 1}$$

Where $n$ is the number of samples in the test set and $p$ is the number of features.

Root Mean Square Error (RMSE) and Mean Square Error (MSE) are metrics used to measure the prediction error of a regression model, both of which measure the average difference between the predicted and true values of the model. A smaller value of RMSE indicates a higher predictive accuracy of the model. Suppose there are $N$ samples, where the true value of the $i$ sample is $y^{(i)}$ and the predicted value is $\hat{y}^{(i)}$. The mathematical formulas for RMSE and MSE are:

$$\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i$$

$$RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} \left(y^{(i)} - \hat{y}^{(i)}\right)^2}$$

$$MSE = \frac{1}{n} \sum_{i=1}^{n} \left(y_i - \hat{y}_i\right)^2$$

Experiment Settings

In the initial phase of the analysis, the dataset is partitioned into training and testing sets, adhering to an 8:2 ratio. Subsequently, feature extraction techniques, namely Principal Component Analysis (PCA), Random Forest, and Lasso Regression, are employed to identify the most influential ten features using each method. These selected features are then subjected to a concatenation and intersection process.

Three prominent metrics are employed to evaluate the relevance of the extracted main features: Pearson’s correlation coefficient, Spearman’s rank correlation coefficient, and Kendall’s correlation coefficient. Each of these metrics assesses the degree of correlation between the features. Features displaying high correlation are subsequently eliminated from consideration. Once this filtering process is complete, the remaining data is utilized to train a machine-learning model.

Statistical Analysis

Feature extraction is meticulously conducted on the PreOne dataset, wherein the feature variables undergo a meticulous screening process. Subsequently, the intersection and concatenation operations are deftly applied to glean the utmost value from these feature variables. Three paramount features are distilled from the intersection, while the concatenation yields a prodigious set of 18 salient features. A diverse array of machine learning techniques is employed to validate and predict. The comprehensive outcomes are eloquently presented in Table 1, which directly showcases the results of these experiments and the subsequent model validation. A discerning analysis reveals that the ElasticNet model surpasses all others in concatenation-on-based feature extraction. In contrast, the BayesianRidge model emerges as the preeminent choice for intersection-based feature extraction.

Curvilinear trajectories of edema volume throughout the temporal axis were adeptly fitted for each patient. Upon meticulous evaluation, it was determined that despite exhibiting elevated mean squared error (MSE) values, the simple linear model outperformed other models when polynomial regression calculations were applied. Allow me to provide you with the results:

$$ f (x) = 0. 0 0 0 1 x ^ {5} - 0. 0 0 0 1 x ^ {4} + 0. 0 0 0 1 x ^ {3} - 0. 2 1 0 2 x ^ {2} + 1 1 2. 9 5 5 9 x + 2 0 1 6 4. 9 7 2 2 $$ (14) The volumetric progression of edema, observed over a span of time, was meticulously modeled for all patients. Notably, amidst various modeling techniques, it was found that the simple linear model demonstrated superior performance despite exhibiting comparatively high MSE values when subjected to polynomial regression calculations. The results of the clustering are shown in Figure 6 below.

Click to enlarge

The study adopted the procedures outlined in Figures 4 & 5 employing various models such as Support Vector Machine, Random Forest Classification, BayesianRidge, KernelRidge, and AdaBoostRegressor to train the extracted features. The performance of these models was evaluated, as exemplified in Table 2. Notably, the KernelRidge model exhibited commendable performance on the PreOne dataset, albeit with a degree of uncertainty. Conversely, the ExtraTreesRegressor model demonstrated robust performance across a diverse range of scenarios on the PreTwo dataset, showcasing its generalizability.

Both experiments established significant correlations between the initial examination’s MRS scores, subsequent follow-up, and patient treatment. These findings surpassed the results obtained by pairing the examination data pairwise. Interestingly, the initial examination’s outcomes correlated more strongly with hematoma volume than the MRS scores. It is crucial for individuals with related medical conditions to consistently monitor their health and promptly seek medical attention when necessary.

Interpretability

In determining the occurrence of hematoma expansion within a span of 48 hours, we sought to extract the features depicted in Figure 7 herewith, subsequently evaluating their similarity. It is evident that variables such as ‘age,’ ‘original_shape_Maximum2DDiameterColumn’, ‘original_ shape_Maximum2DDiameterRow’, ‘NCCT_original_ firstorder_10Percentile’, and ‘Cholesterol_low’ exhibit a positive correlation with prognostic MRS scores. Conversely, ‘Time between onset and first imaging,’ ‘original_shape_ LeastAxisLength,’ ‘original_shape_MinorAxisLength,’ and ‘Cholesterol_high’ demonstrate a negative correlation with prognostic mRS scores. Notably, ‘Cholesterol_low’ and ‘NCCT_original_firstorder_10Percentile’ emerge as indices of heightened significance, warranting attending physicians’ attention when tending to patients with a primary diagnosis.

Click to enlarge

Following the extraction of features from the ProTwo dataset, to generate the heatmap delineated in Figure 8 below, noteworthy observations come to light. Notably, variables such as ‘HM_ACA_L_Ratio_1’, ‘age,’ ‘original shape Elongation,’ ‘original shape Least-AxisLength,’ and ‘Cholesterol_low’ emerge as the top-ranked features in the Random Forest’s feature importance ranking. Remarkably, these results align harmoniously with the prior Lasso regression-based feature extraction outcomes. Furthermore, the Pearson correlation heatmap reveals certain intriguing associations. Specifically, a positive correlation is evident between ‘original_shape_ LeastAxisLength_2’ and ‘original_shape_MinorAxisLength_2’, while a negative correlation is observed between ‘NCCT_ original_firstorder_Minimum_2’ and ‘NCCT_original_-

firstorder_Minimum_2’. Noteworthy as well, ‘original_shape_ MajorAxisLength_1’ demonstrates a positive correlation with both ‘original_shape_Maximum2DDiameterSlice_1’ and ‘original_shape_Maximum2DDiameterRow_1’. These findings underscore the intricate interplay of variables within the dataset, shedding light on their respective relationships.

In our relentless pursuit to delve deeper into the intricate interplay between patients’ prognostic MRS scores at 90 days post-onset and the various facets of their disease and treatment-related information, alongside the outcomes of multiple imaging examinations, we have undertaken a comprehensive analysis. By scrutinizing the data, we aim to identify the factors that exhibit the strongest association with patients’ prognosis, facilitating more effective clinical diagnosis and treatment.

We have obtained the feature_union feature importance index through meticulous calculations, which serves as evidence of the rigor and accuracy of our model mentioned above. The analysis results are presented in Figure 8 below, providing valuable insights into the relative significance of the various features under consideration.

The profound exploration of the correlation between patients’ prognostic MRS scores at the 90-day mark following disease onset and a myriad of disease-related information, treatment-related details, and the outcomes of diverse imaging examinations has yielded compelling results. Notably, ‘HM_volume_1’, ‘HM_volume_2’, ‘HM_ MCA_R_Ratio_1’, ‘ED_volume_1’, and ‘HM_ACA_R_Ratio’ have emerged as highly correlated with the MRS as mentioned earlier scores in Figure 9. This finding validates the accuracy of influential factors, which exhibit significant correlations with prognostic mRS scores as determined through statistical analyses conducted within this study.

Click to enlarge

Figure 8: Heatmap of ProTwo data for feature extraction and correlation calculation Consequently, when conducting imaging assessments of patients afflicted with hemorrhagic stroke, it is imperative to direct our attention towards crucial indicators such as ‘HM_ volume,’ ‘ED_volume,’ ‘HM_MCA_R_Ratio,’ and ‘HM_ACA_R_ Ratio.’ The significance of ‘HM_volume’ outweighs other characteristics, underscoring its paramount association with a patient’s 90-day MRS score at the onset of the disease. Furthermore, based on the Pearson correlation coefficient, it becomes apparent that a positive correlation exists between ‘HM_volume’ and MRS scores. In simpler terms, larger values of ‘HM_volume’ correspond to higher MRS scores, indicating a more substantial hematoma volume and, subsequently, a poorer prognosis. It is worth noting that the presence of edema surrounding the hematoma contributes to further damage to brain tissue.

While ‘HM_MCA_R_Ratio’ and ‘HM_ACA_R_Ratio’ influence the progression, severity, and prognosis of hemorrhagic stroke, their specific etiology and mechanisms necessitate further investigation. In clinical diagnosis, it is crucial to diligently monitor changes in hematoma volume and prioritize research on therapeutic modalities or medications geared toward controlling it.

Click to enlarge