A Comparative Study of Machine Learning and Algorithmic Approaches to Automatically Identify the Yield Point in Normal and Aneurysmal Human Aortic Tissues

Abstract

The stress–strain curve of biological soft tissues helps characterize their mechanical behavior. The yield point on this curve is when a specimen breaches its elastic range due to irreversible microstructural damage. The yield point is easily found using the offset yield method in traditional engineering materials. However, correctly identifying the yield point in soft tissues can be subjective due to its nonlinear material behavior. The typical method for yield point identification is visual inspection, which is investigator-dependent and does not lend itself to automation of the analysis pipeline. An automated algorithm to identify the yield point objectively assesses soft tissues’ biomechanical properties. This study aimed to analyze data from uniaxial extension testing on biological soft tissue specimens and create a machine learning (ML) model to determine a tissue sample’s yield point. We present a trained machine learning model from 279 uniaxial extension curves from testing aneurysmal/nonaneurysmal and longitudinal/circumferential oriented tissue specimens that multiple experts labeled through an adjudication process. The ML model showed a median error of 5% in its estimated yield stress compared to the expert picks. The study found that an ML model could accurately identify the yield point (as defined) in various aortic tissues. Future studies will be performed to validate this approach by visually inspecting when damage occurs and adjusting the model using the ML-based approach. [DOI: 10.1115/1.4064365]

1. Introduction

Mechanical behavior has been extensively studied in traditional engineering materials and biological soft tissues of normal and diseased tissues during mechanical testing. Uniaxial extension testing to failure has been used extensively to understand various aspects of the force–displacement and the stress–strain relationship in isotropic engineering materials [1]. The linear elastic region of the typical stress–strain curve for an engineering material exhibits reversible strain before reaching the proportional limit, referred to as the yield point. After exceeding the yield point, the material exhibits strain hardening until material failure occurs at the ultimate point. In ductile materials such as stainless steel, the yield point is identified using a 0.2% offset of the linear elastic region and projecting it on the stress–strain curve [2]. However, this definition cannot be appropriately translated when evaluating the stress–strain curves for biological soft tissues owing to the nonlinearity of the behavior, greater noise in data, and lack of congruency as to whether such a point exists for a specified material.

For biological soft tissues, the stress–strain curve demonstrates a nonlinear response attributed to the makeup of the extracellular matrix [3]. The stress–strain curve of biological soft tissues shows two distinct regions representing the interaction of elastin and collagen. These predominate microstructural proteins provide mechanical resistance in both low- and high-strain areas (Fig. 1) [4–6].

Fig. 1.

Typical nonlinear stress–strain behavior of biological soft tissues. Initial toe region that transitions to an elastic and eventual plastic region until the ultimate point is reached. Stress-strain curve of a biological soft tissue with various regions that exhibit different loading scenarios on collagen fibers. The yield point is generally chosen as the transition from the linear elastic region to the plastic region until failure.

Aneurysmal aortic tissue behaves differently than normal aortic tissue as much of the elastin content has degraded (therefore stiffer, with reduced extensibility) throughout the course of disease progression [7,8]. The typical normal aortic tissue response from the extension is nonlinear with two distinguishable moduli for the toe and heel region [9]. Additional experimental studies highlight damage to vascular tissues in layers before the high-stiffness region [10] and the overall plastic response through viscoelastic experiments in the porcine aorta [11]. The yield point is defined as the transition from the elastic to a plastic region that occurs [12,13] and could provide critical information about the behavior of biological soft tissues if correctly identified. Currently, an expert must manually identify the yield point through a qualitative process where a reduction in slope denotes when the tissue has yielded. However, this is a subjective process without clear quantitative guidelines, which serves as a motivation for the current study.

Machine learning algorithms continually improve and have expanded into biomedical research for their high-accuracy predictive capabilities [14,15]. Supervised ML is commonly criticized for its “black box” nature; however, ML models are beneficial when appropriately trained, as they provide an accurate prediction based on a predefined list of input variables. Uniaxial extension data present practical challenges by the continued discretized loading of the soft tissue specimen to failure [16,17].

This study presents an automated algorithmic approach to calculate the yield point and present the necessary preprocessing steps to train an ML classification model to identify the yield point from uniaxial extension testing data using tree-based pipeline optimization, a tool for automachine learning [18]. The resulting ML model can be deployed to identify the yield point in new uniaxial extension tests (that were not used during training) of various biological soft tissues. Additionally, this study introduces several previously derived rule-based algorithms to determine the yield point without needing ML for comparison. Data preprocessing mimicked a closed-loop feedback system often found in closed-loop feedback control systems - proportional, integral, and derivative (PID) controller [19].

2. Materials and Methods

2.1. Specimen Procurement, Preparation, and Mechanical Testing.

The tissue samples and mechanical testing protocol used in this study were previously reported by Raghavan et al. [7] and Ninomiya et al. [20] of aneurysmal and nonaneurysmal aortic tissue. The study focused on properties of both longitudinal and circumferential orientations in various regions of the aorta. Mechanical extension tests were performed using an Instron Spec2200 tabletop tester (Instron Corporation, Norwood, MA). Clamped specimens were preconditioned by loading and unloading the specimen between 5% extension at 20% of the specimen’s length/min. Finally, the uniaxial extension test was performed at a rate of 20% strain/min (0.33%/second), ~3.5 mm/min, or ~0.583 mm/second) until failure while recording data at 1 Hz. The Cauchy stress–strain and tension-strain were calculated using Eq. (1) and Eq. (2), respectively [21]

$$\text{Stress}=\left(\frac{F}{w_o{t}_o}\right)\left(1+\epsilon \right)$$ (1)

$$\text{Tension}=\left(\frac{F}{w_o}\right)\left(\sqrt{1+\epsilon }\right)$$ (2)

where $F$ is the force in Newton, $w_o$ is the initial width to the initial thickness, and ε is the strain defined as the linear displacement divided by the initial gauge length, $l_o$. The derivation for the tension equation is found in Appendix A.

2.2. Rule-Based Yield Point Identification Algorithm.

We evaluated three rule-based algorithms to identify the two yield point candidates: a traditional rule reported for Hookean materials (Christensen method [12]) and two rule-based methods that we developed ($Y_P$ and $Y_E$.) are shown in Fig. 2.

Fig. 2.

(a) Example of yield point identification proposed by Christensen by interpolating a window of points that exhibit a 5% reduction in stress. (b) Graphical representation of $Y_p$ where the ratio of stress/strain is used to identify the highest slope region from the origin. The maximum slope is where $Y_p$ is identified. (c) YE is identified by processing the data through a 3% strain window to identify when a 3% reduction occurs.

2.2.1. Christensen Yield Point.

Christensen [12] modified the offset method to identify the stress–strain curve in Hookean materials (Fig. 2(a)) prior to the 0.2% offset yield method typically used in engineering materials. The point at which the linearly extrapolated stress value exceeded the actual observed stress value by 5% was denoted as the criterion for establishing a yield point.

2.2.2. Peak Yield Point, $Y_P$.

In this method, the yield point was identified as the point where the stress/strain ratio, or the current stress over the current strain value, reaches its peak, $Y_P$(Fig. 2(b))

$$Y_P={\max }_{i=1}^n\left(\frac{{\sigma }_i}{{\epsilon }_i}\right)$$ (3)

where $i$ denotes the discrete points in a single uniaxial extension test, with $n$ being the constrained such that ${\sigma }_{\left(i=n\right)}\le {\sigma }_F$ (failure strength).

2.2.3. Elastic Yield Point, $Y_E$.

In this second method, the yield point was identified as the first instance where the stress/strain ratio drops by 3% or more while averaged also within a 3% window of strain (Eqs. (4a)–(4c)). The rationale behind $Y_E$ is that it provides a localized strain window or “neighborhood” of when stress values reduce, indicating that the material has yielded (Fig. 2(c))

$$k_i=\frac{{\sigma }_i}{{\epsilon }_i}$$ (4a)

$$k_{i-w}\text{is}\text{the}\text{moving}\text{average}\text{of}k\left(\epsilon \pm 0.03\right)$$ (4b)

$$\text{if}\left(\frac{k_i^{ma}-k_{i-1}^{ma}}{k_i^{ma}}\right)\ge 3\%\text{and}\left(\frac{k_i^{ma}-k_{i+1}^{ma}}{k_i^{ma}}\right)\ge 3\%,\text{then}{\epsilon }_i=Y_E$$ (4c)

where $k_i$ is the ratio of stress to strain, $i$ is denoted by the maximum number of discrete points in a uniaxial extension test set, $w$ is the 3% window of points, and a 3% drop in stress is the location of $Y_E$ (defined by the strain ${\epsilon }_i$, where the reduction of stress occurs). Finally, $k_i^{ma}$ is the moving average of the strain window. In extension curves where this criterion does not yield a pick (occurs rarely <1%), the first rule-based pick, $Y_P$, was chosen by default.

2.3. Data Processing.

Three different biomechanics experts identified the yield point manually from 100 stress–strain curves from mechanical testing and reconvened to adjudicate a finalized ground truth label. The rest of the stress–strain curves were labeled by a single expert using the criteria from the consensus to compile a database of 279 experimental datasets for training and 43 for testing from holdout validation. The dataset was split randomly to include aneurysmal, nonaneurysmal, longitudinal, and circumferential-oriented samples.

The dataset included a title indicating test sample identification, stress–strain values converted from force–displacement data from uniaxial extension testing, and the human expert-selected yield point picks identified through visual discretion. Noise from extension testing was reduced through a filtering process using a 3rd order Savitzky–Golay filter. The filter was applied to each specimen following the calculations of stress and strain. The ML model was trained using 22 unique features (Table 1) that included the rule-based algorithms, the Christensen yield point, the elastic yield point $\left(Y_E\right)$, and the peak yield point $\left(Y_P\right)$.

Table 1.

Preprocessed variables used for training the machine learning classification model that includes these algorithm-based yield points

Variable	Equation
First derivative	$\frac{d{\sigma }_n}{d{\epsilon }_n}$
Second derivative	$\frac{d^2{\sigma }_n}{d{\epsilon }_n^2}$
Forward and previous first derivative	$\frac{d{\sigma }_{n\pm 1}}{d{\epsilon }_{n\pm 1}}$
Forward and previous second derivative	$\frac{d^2{\sigma }_{n\pm 1}}{d{\epsilon }_{n\pm 1}^2}$
Stress to failure stress ratio	$\frac{\sigma }{{\sigma }_F}$
Instantaneous strain energy	${\int }_{\epsilon =0}^n{\sigma }_{n}d\epsilon$
Strain energy to total strain energy ratio	$\frac{S.E}{\text{Total}S.E.}$
Forward and previous strain energy change	${\int }_{\epsilon =0}^{n+1}{\sigma }_{n+1}d\epsilon$
Stress to strain ratio	$\frac{{\sigma }_n}{{\epsilon }_n}$
Strain to failure strain ratio	$\frac{\epsilon }{{\epsilon }_F}$
Elastic yield point	$Y_E$
Yield point	$Y_P=\max \left(\frac{{\sigma }_n}{{\epsilon }_n}\right)$

Additional variables were extracted based on an analog to a PID controller. Additional static variables were calculated from the stress–strain and tension-strain curves. Variable window sizes were used during the training phase with a 1, 2, 3, 4, and 8% window, with the center being the ground-truth yield point (Fig. 3).

Fig. 3.

(a) Stress-strain curve where the yield point is labeled and (b) is an example of an 8% window being used surrounding the adjudicated ground-truth yield point pick. This window was perturbed to be 1,2,3,4, and 8%.

2.4. Statistical Analysis of the Training and Testing Datasets.

Several statistical approaches were used to characterize the training and testing dataset to identify the yield point. A normalized histogram was tabulated for ground truth yield points $\left(Y_{GT}\right)$ training/testing, $Y_E$ training/testing, $Y_P$ training/testing, and for the $Y_{\text{ML}}$ predictions during the training/testing phase of the study. The median, mean, and standard deviation was also tabulated for the previously mentioned quantities (Table 2) and additional information regarding the curves using skewness and Kurtosis. A paired Student’s t-test and Pearson coefficients were calculated and plotted for paired data regarding training and testing.

Table 2.

Tabulated results of the training and testing dataset for $Y_{\text{GT}}$ (ground truth), $Y_E$(elastic), $Y_P$(peak), and $Y_{\text{ML}}$ (machine learning), with regard to their respective mean and standard deviation (strain), median strain, kurtosis, and skewness of each frequency distribution

		$Y_{\text{GT}}$	$Y_E$	$Y_P$	$Y_{\text{ML}}$
Training	Mean ± std (strain)	0.420 ± 0.249	0.453 ± 0.273	0.498 ± 0.269	0.446 ± 0.278
	Median (strain)	0.353	0.385	0.441	0.387
	Kurtosis	5.329	5.643	5.630	5.230
	Skewness	1.494	1.537	1.481	1.318
	p-value	N/A	0.39	0.0081	0.33
Testing	Mean ± std (strain)	0.457 ± 0.169	0.489 ± 0.179	0.603 ± 0.219	0.492 ± 0.164
	Median (strain)	0.46	0.49	0.54	0.48
	Kurtosis	3.58	2.95	2.78	3.17
	Skewness	0.73	0.48	0.57	0.65
	p-value	N/A	0.13	0.0039	0.24

P-values reported compare each metric tro the ground truth.

2.5. Machine Learning Testing and Training.

The processed data were input into MATLAB’s Classification Learner toolbox and using separate python libraries scikit-learn [22] and XGBoost Classifier [23]. The machine learning model was trained using multiple algorithms within the toolbox, attempting all models. The training dataset of 279 samples was performed using a k-fold cross-validation fixed at 10-fold per trained model to test and minimize potential overfitting. The testing dataset consisted of 43 samples selected randomly with the criteria that the expert reviewers were completely blind to the dataset. It is important to note that the 43 samples were taken from the same research institution and processed in the same manner. Model output gave a small region of interest along the strain axis. A mean clustering method was implemented to iteratively sweep across strain values using a 3% window to determine a singular point as the model’s final pick. Uniaxial extension testing was performed on 43 different human soft tissue samples and experts again determined yield points. The 2% window and 20 unique parameters were calculated as input features for evaluating the model (Table 1). The model’s test set results were compared to all three rule-based algorithms – the Christensen yield point $\left(Y_C\right)$, the peak yield point $\left(Y_P\right)$, and the elastic yield point $\left(Y_E\right)$. Notably, the rule-based picks $Y_E$ and $Y_P$ were included as features for model training.

The functional use of the trained models allows for the evaluation of “new” data to predict or identify the yield point. During the training phase, each trained model is exported and reserved for use during the hold-out validation testing. The MATLAB based classification models were exported as *.mat files, whereas the python-based models were exported using pickle or *.json file. The testing datasets from 43 human samples were input into the trained models to identify which stress/strain value indicates the yield point. The data were compiled for each model and the accuracies tabulated (when compared to the human picked yield points).

3. Results

The overall distribution of the training and testing set for $Y_{\text{GT}}$, $Y_E$, and $Y_P$ are shown in Fig. 4. Generally, the training dataset strain range at the yield point was higher than the range of the testing dataset. Figure 4(d) shows $Y_{\text{GT}}\text{vs}{Y}_{\text{GT}}$, the distribution for the testing set fell within the training set. Table 2 lists the various metrics regarding the ground truth, $Y_E$, $Y_P$, and $Y_{\text{ML}}$ measuring the overall spread of the data for the training and testing datasets. The training dataset was skewed right, whereas the testing dataset was less skewed in either direction. It was found that there was no statistical significance for both the training and testing datasets when comparing $Y_E$ and $Y_{\text{ML}}$ to $Y_{\text{GT}}$. However, by comparison, $Y_P$ to $Y_{\text{GT}}$ was statistically significant for both training and testing.

Fig. 4.

(a) Normalized histogram of the $Y_{\text{GT}}$ adjudicated yield point for the training and testing set, (b) normalized histogram for ${\text{Y}}_{\text{E}}$ training and testing set, (c) normalized histogram for $Y_P$ training and testing set, and (d) normalized histogram of the $Y_{\text{GT}}$ adjudicated yield point and ML predicted yield points. The distribution for the testing dataset nominally fell within the training dataset range as well as the ML predicted strains and ground truth testing sets.

Observing the machine learning model’s output for a single specimen revealed a proportional relationship between the window size and the number of predictions the model gave (Fig. 3). Stress-strain curves from a single specimen’s extension testing using five different window sizes surrounding the human yield point pick label marked with a red asterisk. In this case, each window size, including the smallest 1% window, correctly identified the true yield point picked among the model’s predictions. MATLAB’s Classification Learner application was evaluated using k-fold cross-validation during training.

This study enabled a 10-fold cross-validation scheme, which provided the following validation accuracies for each of the tested window sizes. In the training dataset, trend lines were linear fit (intercept at the origin) and plotted to compare the relationship between the adjudicated yield strain (subsequently yield point) with $Y_p$ and $Y_E$. For the training dataset, $Y_P$ and $Y_E$ overestimated the yield point by 14% and 8%, respectively. Figure 5 shows the Pearson regression plots with the coefficient and p-value reported for all comparisons, including the highest accuracy, and performing model, the ensemble bagged tree model. The testing dataset had the lowest coefficient $\left(r=0.656\right)$ when comparing the $Y_{\text{GT}}$ testing data with $Y_P$ strain but was found to have a positive correlation. Although the range of the yield strains differed in the training and testing datasets, the ML model accurately classified the yield point (compared to $Y_{\text{GT}}$ yield strains).

Fig. 5.

Pearson coefficient and p-values for training and testing. $Y_{\text{GT}}$ are found on the x-axis and the algorithmic or predicted strains on the y-axis: (a) $Y_E$ versus $Y_{\text{GT}}$ for the training dataset, (b) $Y_P$ versus $Y_{\text{GT}}$ for the training dataset, (c) ${\text{Y}}_{\text{ML}}$ versus $Y_{\text{GT}}$ for the training dataset, (d) $Y_E$ versus $Y_{\text{GT}}$ for the testing dataset, (e) $Y_E$ versus $Y_{\text{GT}}$ for the testing dataset, (f) $Y_E$ versus $Y_{\text{GT}}$ for the training data, and (c) $Y_E$ versus $Y_{\text{GT}}$ for the testing dataset.

The model was trained using 279 unique patient datasets from uniaxial extension testing. The trained mode had an internal cross-validation score of 89.9% using ensemble bagged trees of the aneurysmal/nonaneursymal and longitudinal/circumferentially oriented specimens. The ensemble bagged trees hyperparameters included perturbing the random seed, number of k-folds, a maximum number of splits to be 25% of the dataset and using 50 learners. A separate dataset of 43 cases reserved for validation (and not used in training) was input into the trained classifier yielding a test accuracy of 92.2%. The evaluation was performed by calculating the percent error in strain from the ground truth pick. The machine learning model determined yield point picks with an average percent difference error in strain of 1.84 ± 0.934%. The new rule-based selections had average percent errors in strain of 20.6% for $Y_E$ and 5.06% for $Y_P$, respectively.

In comparison, Christensen’s method resulted in an average percent error of 33.4% and, therefore, was not included in figures to avoid clutter. Notably, several successful machine learning picks demonstrated a near-perfect prediction. Comparison with each rule-based method indicates the exact pick as $Y_P$ and a significantly closer pick than $Y_E$. The final model was evaluated based on training and test set hit rates and strain error from the human pick. The testing dataset was plotted with linear fit trendlines (intercept at the origin) and found that $Y_P$, $Y_E$, and ML predicted $Y_{ML}$ overestimated the strain by 26.5%, 6.6%, and 5.5%, respectively (Fig. 6(a)). Finally, Fig. 6(b) represents the typical yield point picks using the rule-based and ML models that show varying levels of agreement against the adjudicated ground truth.

Fig. 6.

(a) Rule-based $Y_P$ and $Y_E$ comparison to the adjudicated ground-truth yield point picks for the training dataset demonstrated an overestimation of the yield point after linear fitting (14% and 8% for $Y_P$ and $Y_E$, respectively) and (b) Rule-based $Y_P$, $Y_E$, and $Y_{\text{ML}}$ comparison to the adjudicated ground-truth yield point picks for the testing dataset demonstrated an overestimation of the yield point after linear fitting (26.5%, 6.6%, and 5.5% for $Y_P$, $Y_E$, and $Y_{\text{ML}}$

The machine learning model demonstrated a training hit rate of 97.1% or 271 out of 279 total cases predicted. A testing dataset of 43 cases (and not used in training) yielded a test hit rate of 100%, with some cases having multiple predictions. The strain error from the ground truth human yield point pick was again calculated, this time using the testing set. In the case of multiple predictions per experimental test, the mean strain value was chosen. The percent difference was 7.20 ± 9.59%, 39.4 ± 48.5%, 10.8 ± 21.4% for $Y_E$ versus $Y_H$, $Y_p$ versus $Y_H$, and $Y_{ML}$ versus $Y_H$, respectively. Although the percent error mean and standard deviation were the lowest for $Y_e$, the overall linear fit (Fig. 6(b)) for $Y_{ML}$ overestimated the adjudicated yield strain the least (when compared with the rule-based selections). Figure 7 shows several cases of how the $Y_{ML}$ picks relate to $Y_H$, $Y_E$, and $Y_P$. The overall spread of yield point picks from the algorithmic and ML model with the adjudicated yield point indicate the overall accuracy and strength of each approach. When $Y_E$ and $Y_P$ were colocated, this indicates that the algorithmic approach using the window could not detect a 3% stress reduction prior to reaching $Y_P$ (which is close to the peak stress). For the experimental data in Fig. 7(b), both $Y_E$ and $Y_{ML}$ did not indicate the human-picked yield point as the specimen had fractured and continued to extend. Finally, $Y_{\text{ML}}$ indicated picks that were within reason of the human-picked yield points (Figs. 7(c) and 7(d)).

Fig. 7.

(a) An example of $Y_{\text{ML}}$ matching $Y_H$ whereas $Y_P$ and $Y_E$ choose the ultimate point, (b) bimodal response where $Y_{\text{ML}}$ and $Y_E$ identify a point prior to the human and $Y_P$ picks, (c) $Y_{\text{ML}}$ slightly underestimating the yield point and $Y_P$ and $Y_E$ choosing the ultimate point, and (d) $Y_{\text{ML}}$ picking a reasonable yield point

4. Discussion

This study outlined the implementation and evaluation of an algorithmic and machine learning-based methodology for yield point identification from extension testing of biological soft tissues. The comparison was achieved through two objectives: first, the development and training of a machine learning-based algorithm to identify the yield point and assess biomechanical properties of elastic tissue behavior, and second, the comparison of the machine learning algorithm and the rule-based methods, $Y_E$ and $Y_P$, against $Y_{\text{GT}}$ human yield point picks in a test population. The machine learning algorithm performed slightly better than the rule-based methods but relied heavily on inputs from the rule-based yield point picks. The rule-based and ML algorithms demonstrated a lower average percent error than Christensen’s method supporting reasonable improvement relative to pre-existing techniques [12]. Furthermore, this continues to base the notion that yield point identification requires alternative soft tissue extension analysis methods compared to traditional engineering materials. Although the machine learning approach is attractive for automation, it may not be entirely necessary until further validation can be introduced to confirm when yielding has occurred.

It is important to mention other groups that have employed machine learning techniques from mechanical tests of biological soft tissues. Luo et al. introduced an ML model to understand the relationship between strength and varying elastic properties in ascending thoracic aneurysms [24]. However, the yield point was not considered for defining elastic properties, differing from the current approach to identifying the yield point. Sugita et al. suggest that the yield point occurs in physiological in vivo stress ranges [25], in which vascular tissues yield, grow, and expand to become aneurysmal to eventual rupture.

In a recent study, Chung et al. published an ML model capable of predicting rupture in abdominal aortic aneurysm that relied on computational finite element models informed by mechanical testing (26). It may be possible that further testing and validation of the ML model to predict yield point could adjust material model parameters to improve computational modeling techniques.

The overall approach using different types of aortic tissue (aneurysmal/nonaneurysmal) and longitudinal/circumferentially oriented strips revealed that the ML model was able to accurately identify the yield point (strain), a potential avenue to explore other types of biological soft tissues under loading scenarios. Irrespective of whether damage may have occurred before the adjudicated $Y_{\text{GT}}$ and predicted $Y_{\text{ML}}$, the ML model was able to identify the yield point accurately. Furthermore, the ML model was able to identify the yield point in different tissue disease states and orientations, indicating that the model was agnostic to the specifications of the specimens.

A point of contention remains regarding the viability of a human, visually inspected yield point pick. The study of the yield point in biological soft tissues remains sparse. The gold standard of human visual identification leaves much to be desired in reproducibility and consistency. It may be worth considering that, in some cases, the yield point simply has too much variability in soft tissues and is not a feasible property for biomechanical analysis. However, the demonstration of this ML model to reproducibly identify the yield point may potentially remove the need for an adjudication panel of experts. As the adjudication process may be limited due to subjectivity, future studies that extract image-based yield point criteria can replace the human-expert picked points while reducing processing time through automated yield point identification.

The study presents the development and training of an ML model to identify the yield point from uniaxial extension testing data with several limitations. There was a total of 322 uniaxial extension testing datasets performed on varying regions of the aorta that also included aneurysmal tissue. The testing dataset did not have the same distribution as the training dataset, meaning that more samples are needed to alleviate these differences reported in Table 2. It was also observed that the percent error was large for the algorithmic approach using $Y_P$, a limitation of the algorithmic approach that may be prohibitive for use in identifying the yield point. However, $Y_P$ was informative for aiding the identification using the ML model. Future studies will investigate the effects of using the same tissue types compared to mixing different normal and diseased tissues to understand whether this trained model can be expanded for other use-cases for biological soft tissues. Additional studies must investigate the effect of stress on the extracellular matrix and when irreversible damage can visually confirm when both elastin and collagen in normal tissues yield.

The yield point has been primarily used to identify the effectiveness of irreversible damage while testing normal and diseased biological soft tissues. It is unclear when damage occurs in arteries due to the constant physiologic loading of tissues during hypertensive systole. Understanding how arterial damage occurs through means of fatigue or material yielding may provide key insights to the progression of certain diseases (e.g., aneurysms). Experimentation to extract viscoelastic parameters may enable additional information to be incorporated into an ML model [11]. Although the method to identify the yield point has been used extensively by biomechanics experts, a significant gap in knowledge exists regarding understanding how damage initiates physiologically. Further interrogation of the yield point and viscoelastic parameters may elucidate their respective functions to the irreversible damage occurring in the extracellular matrix and its constituents, an important reference point for future studies.

5. Conclusions

Several algorithms and an ML approach were developed to objectively identify the yield point. The overall methods predicted the yield point in a wide range of aortic tissue and in normal and aneurysmal tissue specimens. The ML classification model performed well, although it relied heavily on the derived yield point picks that were not statistically different than the $Y_E$ algorithms. The trained model provides an objective method to identify the yield point typically used for constitutive modeling by biomechanics experts. However, additional scrutiny is needed to physically relate the yield point to irreversible tissue damage. Future studies will investigate the applicability of the ML model while studying the relationship between the yield point and microstructural damage of tissues in depth.

Appendix

Appendix A: Tension Derivation

The generalized tension equation is found in the following equation:

$$\text{Tension}=\frac{F}{t_f}$$ (A1)

where $F$ is force in Newtons (N) and $t_f$ is the final thickness of the specimen. The strain equation is found in Eqs. (A2a) and (A2b), where the final form is related to the ratio of the final length with the gauge length

$$\epsilon =\frac{l_f-l_o}{l_o}$$ (A2a)

$$\epsilon +1=\frac{l_f}{l_o}$$ (A2b)

where $\epsilon$ is strain, $l_f$ is the length during extension, and $l_o$ is the initial gauge length of the specimen in the following equations:

$$V_o=V_f$$ (A3a)

$$1=\frac{l_f{w}_f{t}_f}{l_o{w}_o{t}_o}$$ (A3b)

where $V_o$ is the initial volume of the tissue specimen and $V_f$ is the stretched volume. The initial length, width, and thickness of the tissue specimen ($l_o$, $w_o$, and $t_o$) and the final length, width, and thickness are $l_f$, $w_f$, and $t_f$, respectively. Assuming material isotropy, the width and thickness ratios would be consistent in following equations:

$$\frac{w_o}{w_f}-1=\frac{t_o}{t_f}-1$$ (A4a)

$$\frac{w_o}{w_f}=\frac{t_o}{t_f}$$ (A4b)

Equations 2(b) and 4(b) are substituted into Eq. 3(b), that results in following equations:

$$1=\frac{\left(\epsilon +1\right)\left(w_f{t}_f\right)}{w_o{t}_o}$$ (A5a)

$$1=\frac{\left(\epsilon +1\right)t_f^2}{t_o^2}$$ (A5b)

$$\frac{1}{t_f}=\frac{\sqrt{\epsilon +1}}{t_o}$$ (A5c)

Finally, substitute Eq. (A5c) into Eq. (A1), nets Eq. (A6), the tension equation with respect to the initial thickness.

$$\text{Tension}=\frac{F\sqrt{\epsilon +1}}{t_o}$$ (A6)

Data Availability Statement

Data provided by a third party listed in Acknowledgements.